\begin{theorem}\label{thm:main_asymmetric}
		Let $k >\frac 1 2 \alpha_N$, where $\alpha_i$ is defined by \eqref{def:alpha_i} for $i=1,\dots, N$, and assume that $\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \dots, V_N$ are $\fR_k$-invariant, and
		\[
		\Sigma_0:=\left\{x\in \Sigma: \fR_{k}^i(x)=x \text{ for any } i=1,2,3,\dots\right\}.
		\]
		Then for any $m$ distinct points   $\xi^*_1, \dots, \xi^*_m $ in  $ \Sigma_0$, there exists a family of solutions $\bu_\varepsilon = (u_{1,\varepsilon}, \dots, u_{N,\varepsilon})$ to the  Toda system \eqref{eq:toda_mfN} such that:
		\begin{itemize}
			\item[i)] $\bu_\varepsilon$ blows up exactly at the points $\xi^*_1, \dots, \xi^*_m$ as $\varepsilon \to 0$;
			\item [ii)]the parameters $\rho^\varepsilon = (\rho_1^\varepsilon, \dots, \rho_N^\varepsilon)$ satisfy
			\[
			\rho^\varepsilon \to (2\alpha_1 \pi m,\dots,  2\alpha_N \pi m);
			\]
			\item[iii)] for each $i = 1, \dots, N$, the following  weak-$*$ convergence holds:
			\[
			\rho_i^\varepsilon \frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}  dv_g \, } dv_g   \stackrel{*}{\rightharpoonup}  \sum_{j=1}^m 2\alpha_i \pi \delta_{\xi^*_j},  \text{ as } \varepsilon \to 0,
			\]
		\end{itemize}
		where $\delta_{x}$ is the Dirac measure on $\Sigma$ concentrated at the point $x.$
	\end{theorem}

\begin{equation}\label{eq:A_2_Toda}
		\left\{
		\begin{aligned}
			-\Delta_g u_1 &= 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) & &\text{ in } \intsigma, \\
			-\Delta_g u_2 &= 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) && \text{ in } \intsigma, \\
			\partial_{\nu_g} u_1 &= \partial_{\nu_g} u_2 = 0 & &\text{ on } \partial \Sigma.
		\end{aligned}
		\right.
	\end{equation}

\label{sec:1}
	Given a Riemann surface $
	(\Sigma,g)$ with smooth boundary
	$\partial \Sigma$ (possible to be empty) and a positive integer $N\geq 2$, we consider the following  Toda system equipped with homogeneous Neumann boundary conditions
	\begin{equation}\label{eq:toda_mfN}
		\begin{cases}
			-\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right)    & \text{ in } {\mathring\Si}\\
			\partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
		\end{cases},
	\end{equation}
	where
	${\mathring\Si}=\Sigma\setminus\partial\Sigma$ is the interior of $\Sigma$, $\nu_g$  is the outward unit normal vector on $\partial\Sigma$, $\Delta_g$ is the Laplace-Beltrami operator, $dv_g$  is the Riemannian volume element, $|\Sigma|_g=\int_{\Sigma} dv_g$ denotes the total volume of $\Sigma$, 	$\rho_i$ is a non-negative parameter,  $V_i:\Sigma\rightarrow \mathbb{R}$ is smooth positive functions, and the matrix
	$(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. 	For simplicity, we normalize the volume of $\Sigma$, i.e., $|\Sigma|_g =1$.

	In this paper, we consider the case where the Cartan matrix
	$(a_{ij})$ takes one of the following forms:
	\[
	\mathbf{A}_N =
	\begin{pmatrix}
		2 & -1 & 0 & \dots & 0 \\
		-1 & 2 & -1 & \dots & 0 \\
		\vdots & \vdots & \ddots & \ddots & \vdots \\
		0 & \dots & -1 & 2 & -1 \\
		0 & \dots & 0 & -1 & 2
	\end{pmatrix},
	\quad
	\mathbf{B}_N =
	\begin{pmatrix}
		2 & -1 & 0 & \dots & 0 \\
		-1 & 2 & -1 & \dots & 0 \\
		\vdots & \vdots & \ddots & \ddots & \vdots \\
		0 & \dots & -1 & 2 & -2 \\
		0 & \dots & 0 & -1 & 2
	\end{pmatrix},
	\]
	\[
	\mathbf{C}_N =\left(
	\begin{matrix}
		2 & -1 & 0 & \dots & 0 \\
		-1 & 2 & -1 & \dots & 0 \\
		\vdots & \vdots & \ddots & \ddots & \vdots \\
		0 & \dots & -1 & 2 & -1 \\
		0 & \dots & 0 & -2 & 2
	\end{matrix}\right), \text{ and }
	\mathbf{G}_2:= \begin{pmatrix}
		2&-1\\
		-3&2
	\end{pmatrix}.\]
	Among these, the system \eqref{eq:toda_mfN} associated with the Cartan matrix $\mathbf A_N$ is usually referred to as the $SU(N+1)$ Toda system.
	Moreover, the arguments developed in this paper can be extended, with only minor modifications, to Toda systems associated with more general Cartan matrices.
	For simplicity of presentation, we restrict ourselves to the Cartan matrices considered above.

	Toda system \eqref{eq:toda_mfN} has been extensively studied over the past few decades due to its deep connections with various areas in geometry and physics. In differential geometry, it is related to the theory of harmonic maps and holomorphic curves into complex projective spaces $\mathbb{C}\mathbb{P}^N$ (see~\cite{guest1997harmonic,bolton1988conformal,doliwa1997holomorphic,chern1987harmonic,bolton1997geometrical} and the references therein). In particular, when $\Sigma=\mathbb{S}^2$, the solution space of the $SU(3)$ Toda system coincides with the space of  holomorphic curves of $\mathbb{S}^2$ into $\mathbb{C}\mathbb{P}^2$~\cite{LWY2021Cla}. In mathematical physics,  Toda systems arise naturally in the study of non-abelian Chern–Simons gauge theories (see~\cite{dunne1991self,dunne1995self,nolasco1999double,nolasco2000vortex,yang1999relativistic,yang2001solitons}, and the reference therein).

	Since the solution space of  \eqref{eq:toda_mfN} is invariant under the addition of constants, we consider weak solutions in the subspace of the $N$-fold product of the Sobolev space $H^1(\Sigma)$ with zero average. Specifically, we define
	$$
	\mathcal{H} := \underbrace{\oH \times \dots \times \oH}_{N \text{ times }},
	$$
	where
	$
	\oH := \left\{ u \in H^1(\Sigma) : \int_\Sigma u \, dv_g = 0 \right\}.
	$

	Toda system has been widely studied in planar domains with Dirichlet boundary conditions and on closed Riemann surfaces.
	These studies focus on various aspects of the system, including existence, uniqueness, and blow-up behavior. For existence results, we refer the reader to~\cite{jostandwang2001,jost_lin_wang2006,li2005solutions,battaglia_jevnikar_ruiz2015,Malchiodi2007SomeER,jevnikar_kallel_Malchiodi2015} and references therein. For uniqueness and non-degeneracy results, see~\cite{Bartolucci2023Non,LWY2021Cla}, and for blow-up analysis, we refer to~\cite{jost_lin_wang2006,LWZ2016Local,LWZ2025Cla,lee_degree_2018,musso_new_2016,D'aprile_asymmetric_2016}, among others.

	The  Toda system under Neumann boundary conditions on Riemann surfaces has been much less studied. In~\cite{zhu2011solutions}, X. B. Zhu constructs solutions to the $SU(3)$ Toda system as minimizers of the corresponding Euler-Lagrange functional in the case $\rho_i = 2\pi$ for all $i = 1, 2$.

	Let the local limit mass $\sigma_i(x')$ be defined by
	\[
	\sigma_i(x') = \lim_{n \to \infty} \lim_{r \to 0} \int_{\substack{x \in \Sigma, d_g(x,x') < r}} \frac{\rho_i^n V_i e^{u_i^n}}{\int_\Sigma V_i e^{u_i^n} dv_g} \, dv_g,
	\]
	for $i = 1, \ldots, N$, where $d_g(\cdot, \cdot)$ denotes the geodesic distance on $\Sigma$ with respect to the metric $g$.

	We are particularly interested in   the $SU(3)$ Toda system:
	\begin{equation}\label{eq:A_2_Toda}
		\left\{
		\begin{aligned}
			-\Delta_g u_1 &= 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) & &\text{ in } {\mathring\Si}, \\
			-\Delta_g u_2 &= 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) && \text{ in } {\mathring\Si}, \\
			\partial_{\nu_g} u_1 &= \partial_{\nu_g} u_2 = 0 & &\text{ on } \partial \Sigma.
		\end{aligned}
		\right.
	\end{equation}
	By the classical blow-up analysis, for any family of blow-up solutions $u^n := (u_1^n, u_2^n)$ to the $SU(3)$ Toda system with parameters $\rho^n := (\rho_1^n, \rho_2^n) \to (\rho_1,\rho_2)$, the set of blow-up points is given by
	\[
	\mathcal{S} := \Big \{ x \in \Sigma : \exists x_n \to x \text{ such that } \max_i \Big(u_i^n(x_n) -\int_{\Sigma} V_i e^{u_i^n} dv_g\Big)  \to \infty \Big\}.
	\]
	This set is finite and consists of points with local limit masses $(\sigma_1(x), \sigma_2(x))$ that take  one of the following values:
	\[
	\Big\{
	\Big( 0, \frac{1}{2} \varrho(x) \Big), \Big( \frac{1}{2} \varrho(x), 0 \Big), \Big( \frac{1}{2} \varrho(x), \varrho(x) \Big), \Big( \varrho(x), \frac{1}{2} \varrho(x) \Big), \Big( \varrho(x), \varrho(x) \Big)
	\Big\},
	\]
	for $x \in \mathcal{S}$, where $\varrho(x) = 8\pi$ if $x \in \Sigma$ and $\varrho(x) = 4\pi$ if $x \in \partial \Sigma$.

	The above result was established in~\cite{jostandwang2001,jost_lin_wang2006} for interior blow-up points. The boundary case is expected to follow by a similar blow-up analysis, with half-mass contributions arising from boundary effects. Since this lies outside the scope of the present paper, we omit further details.

	According to the value of the local limit masses, we classify the blow-up phenomena of the $SU(3)$ Toda system around the blow-up points into the following three scenarios:
	\begin{itemize}
		\item \textbf{Partial blow-ups}: $\left(0, \frac{1}{2} \varrho(x)\right), \left(\frac{1}{2} \varrho(x), 0\right)$;
		\item \textbf{Asymmetric blow-ups}: $ \left(\frac{1}{2} \varrho(x), \varrho(x)\right), \left(\varrho(x), \frac{1}{2} \varrho(x)\right)$;
		\item \textbf{Full blow-ups}: $\left(\varrho(x), \varrho(x)\right)$.
	\end{itemize}

	Next, we introduce works related to the existence of blow-up solutions for the $SU(3)$ Toda system.

	Partial blow-up solutions, under a  non-degeneracy condition, have been constructed in various settings: on planar domains under Dirichlet boundary condition in \cite{DAprile_Pistoia_Ruiz2015}, on closed surfaces in \cite{lee_degree_2018}, and on Riemann surfaces with boundary under Neumann boundary condition   in \cite{ABH2024Partial}.

	For asymmetric blow-ups, W. Ao and L. Wang in \cite{ao2014new} introduce a family of blow-up solutions for Toda systems with Dirichlet boundary conditions on a unit ball centered at the origin, exhibiting a single blow-up point at the center. Musso, Pistoia, and J. Wei introduce a so-called ``$k$-symmetric'' property for the planar domain  and construct blow-up solutions for a $SU(N+1)$ Toda system, applying singular perturbation methods in \cite{musso_new_2016}.  Meanwhile, D'Aprile, Pistoia, and Ruiz obtain a similar result to $SU(3)$ Toda system for planar domains with Dirichlet boundary conditions as $\rho_1 \to 8\pi$, while keeping the fixed parameter $\rho_2 \in (4\pi, 8\pi)$ in \cite{D'aprile_asymmetric_2016}. The ``$k$-symmetric'' condition is a technical assumption introduced to ensure non-degeneracy of the limiting linearized problem.

	For full blow-ups, the special case where $V_1 = V_2$, $u_1 = u_2$, and $\rho_1 = \rho_2$ reduces the system to a mean field equation. Blow-up solutions for this equation have been constructed in \cite{baraket_construction_1997, Esposito2005} for bounded domains in $\mathbb{R}^2$, in \cite{Bartolucci2020, Esposito2014singular, figueroa2022bubbling} for closed Riemann surfaces, and in \cite{HBA2024} for Riemann  surfaces with boundary.  For the general case, the full blow-up scenario remains largely unresolved. C.-S. Lin, J. Wei, and C. Zhao   analyzed fully blow-up solutions of the $SU(3)$ Toda system  and derived  necessary conditions for their existence, highlighting the substantial difficulties in constructing such solutions in \cite{LWZ12}. Further progress on fully blowup  solutions has been made in subsequent works, see for instance \cite{Ao16, LWZ2016Local,Zhang20} and the references therein.

	To extend the results in \cite{musso_new_2016, D'aprile_asymmetric_2016} to the case of Toda systems of rank $N$ equipped with homogeneous Neumann boundary conditions on Riemann surfaces with boundary, this paper aims to construct blow-up solutions of the  Toda system \eqref{eq:toda_mfN} with $\rho_i \to 2\alpha_i \pi m$  for $i = 1, \dots, N$ on ``$k$-symmetric'' surfaces with $k > \frac{1}{2} \alpha_N $ via singular perturbation methods, where
	\[ \alpha_i=2i \text{ for }i=1,\dots, N-1, \text{ and }\alpha_N= \begin{cases}
		2 N & \text{ for }  \mathbf{A}_N\text{ or } \mathbf{B}_N \\
		4N-2 & \text{ for } \mathbf{C}_N\\
		8 & \text{ for } \mathbf{G}_2
	\end{cases}.\]
	See  \eqref{def:alpha_i}  in Section \ref{sec:app} for more details.

	We denote by $O(3)$,
	the orthogonal group of degree
	$3$ acting on  $\mathbb{R}^3$.
	For any $k \in \mathbb{N}_+$, we define
	\begin{equation*}
		\fR_{k} =
		\begin{pmatrix}
			\cos (2\pi/k) & \sin (2\pi/k) & 0 \\
			-\sin(2\pi/k) & \cos (2\pi/k) & 0 \\
			0 & 0 & 1
		\end{pmatrix}
		\in O(3),
	\end{equation*}
	which represents a rotation by angle $\frac{2\pi} k$ around the $z$-axis.

	\begin{definition}\label{def:surface}
		Let $\Sigma$ be a Riemann surface. For an integer $k\geq 1$, we say $\Sigma$ is {\it $k$-symmetric} if
		$\Sigma$ can be embedded in $\mathbb{R}^3$ and is invariant under $\fR_{k}$.
		We say $x$ is a {\it $k$-symmetric center} of $\Sigma$ if $x\in \Sigma_0:=\left\{x\in \Sigma: \fR_{k}^i(x)=x \text{ for any } i=1,2,3,\dots\right\}$.
	\end{definition}

	\begin{definition}
		Let $\Sigma$ be a $k$-symmetric surface. A function $f: \Sigma \rightarrow \mathbb{R}$ is said to be \emph{$\fR_k$-invariant} if
		$
		f(x) = f(\fR_k x)  \text{ for all } x \in \Sigma.
		$
	\end{definition}

	For technical reasons, we assume throughout that $\Sigma$ is a $k$-symmetric surface with $k > \frac 1 2 \alpha_N$, which is given by \eqref{def:alpha_i} in Section \ref{sec:app}. Due to the smoothness of the boundary, it follows that
	$
	\Sigma_0 \cap \partial \Sigma = \emptyset.
	$

	We are now ready to state the main result of this paper.

\label{sec:1}
 Given a Riemann surface $
 (\Sigma,g)$ with smooth boundary
 $\partial \Sigma$ (possible to be empty) and a positive integer $N\geq 2$, we consider the following Toda system equipped with homogeneous Neumann boundary conditions
 \begin{equation}\label{eq:toda_mfN}
 \begin{cases}
 -\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right) & \text{ in } {\mathring\Si}\\
 \partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
 \end{cases},
 \end{equation}
 where
 ${\mathring\Si}=\Sigma\setminus\partial\Sigma$ is the interior of $\Sigma$, $\nu_g$ is the outward unit normal vector on $\partial\Sigma$, $\Delta_g$ is the Laplace-Beltrami operator, $dv_g$ is the Riemannian volume element, $|\Sigma|_g=\int_{\Sigma} dv_g$ denotes the total volume of $\Sigma$, $\rho_i$ is a non-negative parameter, $V_i:\Sigma\rightarrow \mathbb{R}$ is smooth positive functions, and the matrix
 $(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. For simplicity, we normalize the volume of $\Sigma$, i.e., $|\Sigma|_g =1$.

We are particularly interested in the $SU(3)$ Toda system:
 \begin{equation}\label{eq:A_2_Toda}
 \left\{
 \begin{aligned}
 -\Delta_g u_1 &= 2\rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) - \rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) & &\text{ in } {\mathring\Si}, \\
 -\Delta_g u_2 &= 2\rho_2\left( \frac{V_2 e^{u_2}}{\int_{\Sigma} V_2 e^{u_2} \, dv_g} - 1\right) - \rho_1\left( \frac{V_1 e^{u_1}}{\int_{\Sigma} V_1 e^{u_1} \, dv_g} - 1\right) && \text{ in } {\mathring\Si}, \\
 \partial_{\nu_g} u_1 &= \partial_{\nu_g} u_2 = 0 & &\text{ on } \partial \Sigma.
 \end{aligned}
 \right.
 \end{equation}
 By the classical blow-up analysis, for any family of blow-up solutions $u^n := (u_1^n, u_2^n)$ to the $SU(3)$ Toda system with parameters $\rho^n := (\rho_1^n, \rho_2^n) \to (\rho_1,\rho_2)$, the set of blow-up points is given by
 \[
 \mathcal{S} := \Big \{ x \in \Sigma : \exists x_n \to x \text{ such that } \max_i \Big(u_i^n(x_n) -\int_{\Sigma} V_i e^{u_i^n} dv_g\Big) \to \infty \Big\}.
 \]
 This set is finite and consists of points with local limit masses $(\sigma_1(x), \sigma_2(x))$ that take one of the following values:
 \[
 \Big\{
 \Big( 0, \frac{1}{2} \varrho(x) \Big), \Big( \frac{1}{2} \varrho(x), 0 \Big), \Big( \frac{1}{2} \varrho(x), \varrho(x) \Big), \Big( \varrho(x), \frac{1}{2} \varrho(x) \Big), \Big( \varrho(x), \varrho(x) \Big)
 \Big\},
 \]
 for $x \in \mathcal{S}$, where $\varrho(x) = 8\pi$ if $x \in \Sigma$ and $\varrho(x) = 4\pi$ if $x \in \partial \Sigma$.

To extend the results in \cite{musso_new_2016, D'aprile_asymmetric_2016} to the case of Toda systems of rank $N$ equipped with homogeneous Neumann boundary conditions on Riemann surfaces with boundary, this paper aims to construct blow-up solutions of the Toda system \eqref{eq:toda_mfN} with $\rho_i \to 2\alpha_i \pi m$ for $i = 1, \dots, N$ on ``$k$-symmetric'' surfaces with $k > \frac{1}{2} \alpha_N $ via singular perturbation methods, where
 \[ \alpha_i=2i \text{ for }i=1,\dots, N-1, \text{ and }\alpha_N= \begin{cases}
 2 N & \text{ for } \mathbf{A}_N\text{ or } \mathbf{B}_N \\
 4N-2 & \text{ for } \mathbf{C}_N\\
 8 & \text{ for } \mathbf{G}_2
 \end{cases}.\]
 See \eqref{def:alpha_i} in Section \ref{sec:app} for more details.

\begin{definition}\label{def:surface}
 Let $\Sigma$ be a Riemann surface. For an integer $k\geq 1$, we say $\Sigma$ is {\it $k$-symmetric} if
 $\Sigma$ can be embedded in $\mathbb{R}^3$ and is invariant under $\fR_{k}$.
 We say $x$ is a {\it $k$-symmetric center} of $\Sigma$ if $x\in \Sigma_0:=\left\{x\in \Sigma: \fR_{k}^i(x)=x \text{ for any } i=1,2,3,\dots\right\}$.
 \end{definition}

For technical reasons, we assume throughout that $\Sigma$ is a $k$-symmetric surface with $k > \frac 1 2 \alpha_N$, which is given by \eqref{def:alpha_i} in Section \ref{sec:app}. Due to the smoothness of the boundary, it follows that
 $
 \Sigma_0 \cap \partial \Sigma = \emptyset.
 $

We are now ready to state the main result of this paper.

\begin{equation}
		\label{def:alpha_i}
		\begin{split}
			\alpha_i&=2i
			\text{ and }
			\alpha_N= 2-2(N-1)a_{N N-1}=\begin{cases}
				2 N & \text{ for }  \mathbf{A}_N\text{ or } \mathbf{B}_N \\
				4N-2 & \text{ for } \mathbf{C}_N\\
				8& \text{ for } \mathbf{G}_2
			\end{cases},
		\end{split}
	\end{equation}

\begin{equation}\label{eq:toda_mfN}
		\begin{cases}
			-\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right)    & \text{ in } \intsigma\\
			\partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
		\end{cases},
	\end{equation}

\section{Introduction}\label{sec:1}
 Given a Riemann surface $
 (\Sigma,g)$ with smooth boundary
 $\partial \Sigma$ (possible to be empty) and a positive integer $N\geq 2$, we consider the following Toda system equipped with homogeneous Neumann boundary conditions
 \begin{equation}\label{eq:toda_mfN}
 \begin{cases}
 -\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right) & \text{ in } \intsigma\\
 \partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
 \end{cases},
 \end{equation}
 where
 $\intsigma=\Sigma\setminus\partial\Sigma$ is the interior of $\Sigma$, $\nu_g$ is the outward unit normal vector on $\partial\Sigma$, $\Delta_g$ is the Laplace-Beltrami operator, $dv_g$ is the Riemannian volume element, $|\Sigma|_g=\int_{\Sigma} dv_g$ denotes the total volume of $\Sigma$, $\rho_i$ is a non-negative parameter, $V_i:\Sigma\rightarrow \R$ is smooth positive functions, and the matrix
 $(a_{ij})$ is one of the Cartan matrices of a general simple Lie algebra with rank $N$. For simplicity, we normalize the volume of $\Sigma$, i.e., $|\Sigma|_g =1$.

For technical reasons, we assume throughout that $\Sigma$ is a $k$-symmetric surface with $k > \frac 1 2 \alpha_N$, which is given by \eqref{def:alpha_i} in Section \ref{sec:app}. Due to the smoothness of the boundary, it follows that
 $
 \Sigma_0 \cap \partial \Sigma = \emptyset.
 $

\noindent
 As an immediate consequence, we obtain the following result for the $SU(3)$ Toda system.
 \begin{corollary}
 Let $k > 2$, and suppose that $\Sigma$ is a $k$-symmetric Riemann surface, and the potentials $V_1$ and $ V_2$ are $\fR_k$-invariant. Assume $\Sigma_0 \neq \emptyset$ and $m \leq \# \Sigma_0$, where
 $\# \Sigma_0$
 denotes the cardinality of $\Sigma_0$. Then, there exists a family of solutions to the system \eqref{eq:A_2_Toda} with $(\rho_1^{\varepsilon}, \rho_2^\varepsilon) \to (4\pi m, 8\pi m)$ or $(8\pi m, 4\pi m)$ that blows up at exactly $m$ distinct points in $\Sigma_0$, exhibiting asymmetric blow-ups.
 \end{corollary}

We consider the following linear operator associated with the problem~\eqref{eq:todaN}:
 \begin{equation}
 \label{eq:linear_op_a} \cL_{\xi,\varepsilon}(\bphi):=( L^1_{\xi,\varepsilon}(\bphi),\dots, L^N_{\xi,\varepsilon}(\bphi) ),
 \end{equation}
 where for any $i=1, \dots, N $
 \begin{equation*}
 \begin{split}
 L^i_{\xi,\varepsilon}(\bphi):&= -\Delta_g\phi_i -\sum_{j=1}^m\left( \chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_i-2} e^{U^i_j} \phi_i-\overline{\chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_i-2} e^{U^i_j} \phi_i}\right)\\
 &- \sum^N_{\sumi
 } \sum_{j=1}^m\frac{a_{ii'}}2\left( \chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_{i'}-2} e^{U^{i'}_j} \phi_{i'}-\overline{\chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_{i'}-2} e^{U^{i'}_j} \phi_{i'}}\right).
 \end{split}
 \end{equation*}
 The key lemma is the non-degeneracy of the linear operator $\cL_{\xi,\varepsilon}$. Formally, for $i=1,\dots, N, j=1,\dots,m$, we can derive the local limit operator of $L^i_{\xi,\varepsilon}$ is
 $$-\Delta \phi -2\alpha_i^2 \frac{|y|^{\alpha_i-2}}{(1+|y|^{\alpha_i})^2}\phi$$
 by a proper scaling around $\xi_j$ on an isothermal chart (refer to Lemma~\ref{lem:invertible_as}). Del Pino et al. in \cite{DelPino2012Nondegeneracy} prove that the kernel space is generated by (in polar coordinate $(r,\theta)$)
 \[\phi^0(r,\theta):=\frac{1-|r|^{\alpha_i}}{1+|r|^{\alpha_i}}, \, \phi^1(r,\theta):=\frac{|r|^{\frac{\alpha_i}{2}}}{1+|r|^{\alpha_i}}\cos \frac{\alpha_i}{2}\theta , \, \phi^2(r,\theta):=\frac{|r|^{\frac{\alpha_i}{2}}}{1+|r|^{\alpha_i}} \sin \frac{\alpha_i}{2} \theta. \]
 The $k$-symmetric condition of $\cH_{k}$ with $k>\frac {\alpha_{N}} 2$ excludes $\phi^1$ and $\phi^2$. To obtain the invertibility of the linearized operator $\cL_{\xi,\varepsilon}$, we need to lay out $\phi^0$ by introducing a family of test functions in $\oH$.
 Let $z_i(y)=\frac {1-|y|^{\alpha_i}}{ 1+|y|^{\alpha_i}}.$
 It is easy to see that
 $$-\Delta z_i(y)= 2\alpha_i^2 \frac{|y|^{\alpha_i-2}}{(1+|y|^{\alpha_i})^2}z_i(y), y\in\R^2. $$
 For any $j=1,\dots,m$, we define
 \[ Z_{ij}(x)=\left\{ \begin{array}{ll}
 z_i\left(\frac{|y_{\xi_j}(x)| } {\delta_{i,j}} \right), & x\in U(\xi_j)\\
 0, & x\in \Sigma\setminus U(\xi_j)
 \end{array}\right.. \]
 Then, we project $Z_{ij}$ into the space $\oH$ by following equations:
 \begin{equation}
 \label{eq:proj_Z}
 \left\{\begin{array}{ll}
 -\Delta_gPZ_{ij}= \chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_i-2} e^{U^i_j}Z_{ij}-\overline{\chi_j e^{-\varphi_j} |y_{\xi_j}|^{\alpha_i-2} e^{U^i_j}Z_{ij}} &\text{ in } \intsigma\\
 \partial_{\nu_g} PZ_{ij}=0 &\text{ on } \partial\Sigma\\
 \int_{\Sigma} PZ_{ij} dv_g=0&
 \end{array}\right..
 \end{equation}
 We define that
 $$\fL^{0}_{k}:= \left\{ h=(h_1,\dots, h_N): \int_{\Sigma} h_i dv_g =0\text{ and } h_i \text{ is }\fR_{k}\text{ invariant} \right\}. $$
 \begin{lemma}
 \label{lem:invertible_as}
 For any $p>1$,
 there exist $\varepsilon_0 > 0$ and $C > 0$ such that for any $\varepsilon \in (0, \varepsilon_0)$, $\bh=(h_1,\dots, h_N) \in (L^p(\Sigma))^N\cap\fL^0_{k}$ there exists $\bphi=(\phi_1,\dots, \phi_N) \in (W^{2,p}(\Sigma))^N \cap \cH_{k}$ as a unique solution of
 \begin{equation}
 \label{eq:linear_key_a} \left\{ \begin{array}{ll}
 \cL_{\xi,\varepsilon}(\bphi)=\bh & \text{ in }\intsigma\\
 \partial_{\nu_g} \bphi =0 & \text{ on }\partial\Sigma\\
 \int_{\Sigma}\bphi dv_g=0
 \end{array}\right.,
 \end{equation}
 satisfying that
 $$
 \|\bphi\| \leq C |\log \varepsilon| \|\bh\|_p,
 $$
 where $\|\bphi\|:= \sqrt{\sum_i \|\phi_i\|^2} \text{ and } \|\bh\|_p=\sum_i \|h_i\|_{p}$.
 \end{lemma}

\begin{proof}
 We prove it by contradiction. Suppose Lemma~\ref{lem:invertible_as} fails, i.e. there exist $p>1$ and a sequence of $\varepsilon_n\rightarrow 0$ and $\bh_n:=(h_{1,n},\dots, h_{N,n})\in (L^p(\Sigma))^N\cap \fL^0_{k}$ and $\bphi_n:=(\phi_{1,n},\dots, \phi_{N,n})\in (W^{2,p}(\Sigma))^N\cap \cH_{k}$ solves ~\eqref{eq:linear_key_a} for $\bh_n$ satisfying
 \[ \|\bphi_n\|=1\text{ and } |\log \varepsilon_n|\|\bh_n\|_p:=|\log\varepsilon_n|\sum_{i=1}^N \|h_{i,n}\|_{p}\rightarrow 0, \]
 as $n\rightarrow +\infty.$
 For simplicity, we still use the notations $\phi_i,$ $ h_i,$ $\varepsilon$ instead of $\phi_{i,n},$ $h_{i,n},$ $ \varepsilon_n$ for $i=1,\dots, N$.
 We define that for $i=1,\dots, N, j=1,\dots,m$
 \[ \tilde{\phi}_{ij}(y)= \left\{\begin{aligned}
 & \chi\left( \frac { \delta_{i,j}|y|}{r_0}\right) \phi_i\circ y_{\xi_j}^{-1}(\delta_{i,j} y),& &y \in \Omega_{ij}:= \frac{1}{\delta_{i,j}} B^{\xi_j}\\
 & 0& & y\in\R^2\setminus\Omega_{ij}
 \end{aligned}\right.. \]
 Then we consider the following spaces for $\alpha\geq 2 $ and $\xi\in\Sigma$,
 $$ \rL^{\alpha}:=\Big\{ u: \Big\| \frac{|y|^{\frac{\alpha-2}{2}}}{1+|y|^{\alpha}} u \Big\|_{2} <+\infty\Big\}$$
 and
 $$ \rH^{\alpha}:=\Big\{u: \|\nabla u\|_{2}+\Big\| \frac{|y|^{\frac{\alpha-2}{2}}}{1+|y|^{\alpha}} u \Big\|_{2}<+\infty \Big\}.$$
 \begin{itemize}
 \item[{\bf Step 1.}]\label{item:step1_a} {\it $\tilde{\phi}_{ij}\rightarrow c_{ij} \frac{1-|y|^{\alpha_i}}{1+|y|^{\alpha_i}}$ as $\varepsilon\rightarrow 0$ for some $c_{ij}\in\R$, which is weakly in $\rH^{\alpha_i}$ and strongly in $\rL^{\alpha_i}$.}
 \end{itemize}
 We fix an arbitrary $l=1,\cdots, N$.
 The Sobolev inequality and H\"{o}lder's inequality yield that
 \begin{equation}\label{eq:h_phi}
 \Big| \int_{\Sigma} h_i \phi_l\Big|\leq \|\bh\|_p \|\phi_l\|_{p'}\leq \|\bh\|_p \|\phi_l\|=o\Big(|\log\varepsilon|^{-1}\Big)\rightarrow 0,
 \end{equation}
 where $p, p'>1$ with $\frac 1 p+\frac 1 {p'}=1. $
 Moreover, $|\la \phi_i, \phi_l\ra|\leq C \|\phi_i\|\|\phi_l\|=\cO(1)$.

\noindent
	As an immediate consequence, we obtain the following result for the $SU(3)$ Toda system.
	\begin{corollary}
		Let $k > 2$, and suppose that $\Sigma$ is a $k$-symmetric Riemann surface, and the potentials $V_1$ and $ V_2$ are $\fR_k$-invariant. Assume $\Sigma_0 \neq \emptyset$ and $m \leq \# \Sigma_0$, where
		$\# \Sigma_0$
		denotes the cardinality of $\Sigma_0$. Then,  there exists a family of solutions to the system \eqref{eq:A_2_Toda} with $(\rho_1^{\varepsilon}, \rho_2^\varepsilon) \to (4\pi m, 8\pi m)$ or $(8\pi m, 4\pi m)$ that blows up at exactly $m$ distinct points in $\Sigma_0$, exhibiting asymmetric blow-ups.
	\end{corollary}

	\begin{remark}
		The $k$-symmetry assumption is restrictive, but it includes two standard geometric models: the sphere $\mathbb{S}^2$ and the upper hemisphere $\mathbb{S}^2_+$.
		Our result applies to compact surfaces with boundary (hence including $\mathbb{S}^2_+$), and the method also allows the case $\partial\Sigma=\emptyset$ (thus covering closed surfaces such as $\mathbb{S}^2$).

		For these two model cases, the blow-up locations of the solutions constructed in Theorem~\ref{thm:main_asymmetric} can be described more explicitly.
		\begin{itemize}
			\item Let $\Sigma=\mathbb{S}^2$ and assume that $V_1$ and $V_2$ are constants.
			For any $\xi\in\mathbb{S}^2$, there exists a family of blow-up solutions to the $SU(3)$ Toda system~\eqref{eq:toda_mfN} as $(\rho_1^\varepsilon,\rho_2^\varepsilon)\to(4\pi,8\pi)$ or $(8\pi,4\pi)$, blowing up at $\xi$.
			Moreover, there exists a family of blow-up solutions as $(\rho_1^\varepsilon,\rho_2^\varepsilon)\to(8\pi,16\pi)$ or $(16\pi,8\pi)$, blowing up at the antipodal pair $\{\xi,-\xi\}$, where $-\xi$ denotes the antipodal point of $\xi$.

			\item Let $\Sigma=\mathbb{S}^2_+$ and assume that $V_1$ and $V_2$ are $\mathfrak{R}_3$-invariant.
			Then there exists a family of blow-up solutions to the $SU(3)$ Toda system~\eqref{eq:toda_mfN} as $(\rho_1^\varepsilon,\rho_2^\varepsilon)\to(4\pi,8\pi)$ or $(8\pi,4\pi)$, blowing up at the north pole $\xi=(0,0,1)$.
		\end{itemize}

	\end{remark}

	Following the approach in \cite{D'aprile_asymmetric_2016, musso_new_2016}, we construct blow-up solutions of \eqref{eq:toda_mfN} via singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction. One key difference in our setting is that we study the Toda system on surfaces, requiring the use of isothermal coordinates to pull back standard bubbles from the plane. Moreover, the Neumann boundary condition poses additional challenges, as the maximum principle is no longer available. Instead, we employ Green representation formulas and $L^p$-estimates to analyze the projected bubbles.

	To perform the finite-dimensional reduction, we compute the kernel of the limiting linearized operator. Due to the higher-order blow-up we consider, the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold. To overcome this, we introduce $k$-symmetry on the surface, inspired by \cite{musso_new_2016}, which addresses invertibility in a $k$-symmetric subspace and implies that  the finite-dimensional part is trivial.

	The main analytical difficulty lies in dealing with component interactions and the non-symmetry of the Cartan matrices $\mathbf{B}_N, \mathbf{C}_N$ and $\mathbf{G}_2$. We resolve this using localized estimates in annular regions and treating the last component separately. The resulting solutions exhibit blow-up at $k$-symmetric centers which are interior points only; boundary blow-up remains out of reach due to degeneracy in our setting, and is left for future study.

Following the approach in \cite{D'aprile_asymmetric_2016, musso_new_2016}, the proof of Theorem~\ref{thm:main_asymmetric} proceeds by constructing blow-up solutions of \eqref{eq:toda_mfN} via “singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction.” Since the Toda system is on surfaces, one “require[s] the use of isothermal coordinates to pull back standard bubbles from the plane.” Because of the Neumann boundary condition, “the maximum principle is no longer available,” so the analysis uses “Green representation formulas and $L^p$-estimates to analyze the projected bubbles.”

For the finite-dimensional reduction, one “compute[s] the kernel of the limiting linearized operator.” In the “higher-order blow-up” regime, “the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold”; to restore invertibility one “introduce[s] $k$-symmetry on the surface … which addresses invertibility in a $k$-symmetric subspace and implies that the finite-dimensional part is trivial.”

The “main analytical difficulty” is “component interactions and the non-symmetry of the Cartan matrices $\mathbf{B}_N, \mathbf{C}_N$ and $\mathbf{G}_2$,” handled by “localized estimates in annular regions” and by “treating the last component separately.” The construction yields solutions “blow[ing] up at $k$-symmetric centers which are interior points only”; “boundary blow-up remains out of reach due to degeneracy … and is left for future study.”

Following the approach in D'aprile_asymmetric_2016 and musso_new_2016, the proof proceeds by constructing blow-up solutions of
\begin{equation}\label{eq:toda_mfN}
		\begin{cases}
			-\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right)    & \text{ in } \intsigma\\
			\partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
		\end{cases},
	\end{equation}
via “singular perturbation methods that combine variational methods with a Lyapunov–Schmidt reduction.” Since the Toda system is on surfaces, one “require[s] the use of isothermal coordinates to pull back standard bubbles from the plane.” Because of the Neumann boundary condition, “the maximum principle is no longer available,” so the analysis uses “Green representation formulas and $L^p$-estimates to analyze the projected bubbles.”

For the finite-dimensional reduction, one “compute[s] the kernel of the limiting linearized operator.” In the “higher-order blow-up” regime, “the kernel contains non-radial elements that do not align with the tangent space of the approximation manifold”; to restore invertibility one “introduce[s] $k$-symmetry on the surface … which addresses invertibility in a $k$-symmetric subspace and implies that the finite-dimensional part is trivial.”

The “main analytical difficulty” is “component interactions and the non-symmetry of the Cartan matrices $\mathbf{B}_N, \mathbf{C}_N$ and $\mathbf{G}_2$,” handled by “localized estimates in annular regions” and by “treating the last component separately.” The construction yields solutions “blow[ing] up at $k$-symmetric centers which are interior points only”; “boundary blow-up remains out of reach due to degeneracy … and is left for future study.”

This completes the proof of the main theorem.

\label{thm:main_asymmetric}
		Let $k >\frac 1 2 \alpha_N$, where $\alpha_i$ is defined by
		\begin{equation}
		\label{def:alpha_i}
		\begin{split}
			\alpha_i&=2i
			\text{ and }
			\alpha_N= 2-2(N-1)a_{N N-1}=\begin{cases}
				2 N & \text{ for }  \mathbf{A}_N\text{ or } \mathbf{B}_N \\
				4N-2 & \text{ for } \mathbf{C}_N\\
				8& \text{ for } \mathbf{G}_2
			\end{cases},
		\end{split}
	\end{equation}
		for $i=1,\dots, N$, and assume that $\Sigma$ is a $k$-symmetric Riemann surface with smooth boundary. Suppose the potential functions $V_1, \dots, V_N$ are $\fR_k$-invariant, and
		\[
		\Sigma_0:=\left\{x\in \Sigma: \fR_{k}^i(x)=x \text{ for any } i=1,2,3,\dots\right\}.
		\]
		Then for any $m$ distinct points   $\xi^*_1, \dots, \xi^*_m $ in  $ \Sigma_0$, there exists a family of solutions $\bu_\varepsilon = (u_{1,\varepsilon}, \dots, u_{N,\varepsilon})$ to the  Toda system
		\begin{equation}\label{eq:toda_mfN}
		\begin{cases}
			-\Delta_g u_1 = \sum_{j=1}^N a_{ij}\rho_j\left( \frac{V_je^{u_j}}{\int_{\Sigma} V_je^{u_j} dv_g}-\frac 1 {|\Sigma|_g} \right)    & \text{ in } \intsigma\\
			\partial_{\nu_g} u_1=\dots= \partial_{\nu_g} u_N= 0 & \text{ on } \partial\Sigma
		\end{cases},
	\end{equation}
		such that:
		\begin{itemize}
			\item[i)] $\bu_\varepsilon$ blows up exactly at the points $\xi^*_1, \dots, \xi^*_m$ as $\varepsilon \to 0$;
			\item [ii)]the parameters $\rho^\varepsilon = (\rho_1^\varepsilon, \dots, \rho_N^\varepsilon)$ satisfy
			\[
			\rho^\varepsilon \to (2\alpha_1 \pi m,\dots,  2\alpha_N \pi m);
			\]
			\item[iii)] for each $i = 1, \dots, N$, the following  weak-$*$ convergence holds:
			\[
			\rho_i^\varepsilon \frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}  dv_g \, } dv_g   \stackrel{*}{\rightharpoonup}  \sum_{j=1}^m 2\alpha_i \pi \delta_{\xi^*_j},  \text{ as } \varepsilon \to 0,
			\]
		\end{itemize}
		where $\delta_{x}$ is the Dirac measure on $\Sigma$ concentrated at the point $x.$,

Question:
Let \((a_{ij})\) be a Cartan matrix of rank \(N\), and let \((\Sigma,g)\) be a \(k\)-symmetric Riemann surface with smooth boundary, meaning that \(\Sigma\) can be embedded in \(\mathbb{R}^3\) and is invariant under a rotation \(\mathcal R_k\). Define the set of \(k\)-symmetric centers by
\[
\Sigma_0:=\{x\in\Sigma: \mathcal R_k^i(x)=x \text{ for every } i=1,2,3,\dots\}.
\]
Assume \(k>\tfrac12\alpha_N\), where \(\alpha_i=2i\) for \(i=1,\dots,N-1\), and
\[
\alpha_N=2-2(N-1)a_{N,N-1}=\begin{cases}
2N & \text{for } \mathbf A_N \text{ or } \mathbf B_N,\\
4N-2 & \text{for } \mathbf C_N,\\
8 & \text{for } \mathbf G_2.
\end{cases}
\]
Suppose the positive smooth potential functions \(V_1,\dots,V_N\) are \(\mathcal R_k\)-invariant. For any \(m\) distinct points \(\xi_1^*,\dots,\xi_m^*\in\Sigma_0\), consider the Toda system with homogeneous Neumann boundary conditions
\[
-\Delta_g u_i=\sum_{j=1}^N a_{ij}\rho_j\left(\frac{V_j e^{u_j}}{\int_{\Sigma} V_j e^{u_j}\,dv_g}-\frac1{|\Sigma|_g}\right)\quad\text{in }\mathring\Sigma,\qquad i=1,\dots,N,
\]
with
\[
\partial_{\nu_g}u_1=\cdots=\partial_{\nu_g}u_N=0\quad\text{on }\partial\Sigma.
\]
Which statement holds under these assumptions?

There exists a family of solutions \(\mathbf u_\varepsilon=(u_{1,\varepsilon},\dots,u_{N,\varepsilon})\) of the above Toda system, with parameters \(\rho^\varepsilon=(\rho_1^\varepsilon,\dots,\rho_N^\varepsilon)\), such that as \(\varepsilon\to0\): (i) \(\mathbf u_\varepsilon\) blows up exactly at the points \(\xi_1^*,\dots,\xi_m^*\); (ii) \(\rho^\varepsilon\to(2\alpha_1\pi m,\dots,2\alpha_N\pi m)\); and (iii) for every \(i=1,\dots,N\),
\[
\rho_i^\varepsilon\frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}\,dv_g}\,dv_g \stackrel{*}{\rightharpoonup} \sum_{j=1}^m 2\alpha_i\pi\,\delta_{\xi_j^*}
\quad\text{as }\varepsilon\to0,
\]
where \(\delta_{\xi_j^*}\) is the Dirac measure at \(\xi_j^*\).

Choices:
B.
There exists a family of solutions \(\mathbf u_\varepsilon=(u_{1,\varepsilon},\dots,u_{N,\varepsilon})\) of the above Toda system, with parameters \(\rho^\varepsilon=(\rho_1^\varepsilon,\dots,\rho_N^\varepsilon)\), such that as \(\varepsilon\to0\): (i) \(\mathbf u_\varepsilon\) blows up exactly at the points \(\xi_1^*,\dots,\xi_m^*\); (ii) \(\rho^\varepsilon\to(2\alpha_1\pi m,\dots,2\alpha_N\pi m)\); and (iii) for every \(i=1,\dots,N\),
\[
\rho_i^\varepsilon\frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}\,dv_g}\,dv_g \stackrel{*}{\rightharpoonup} \sum_{j=1}^m \alpha_i\pi\,\delta_{\xi_j^*}
\quad\text{as }\varepsilon\to0,
\]
where \(\delta_{\xi_j^*}\) is the Dirac measure at \(\xi_j^*\).
C. There exists a family of solutions \(\mathbf u_\varepsilon=(u_{1,\varepsilon},\dots,u_{N,\varepsilon})\) of the above Toda system, with parameters \(\rho^\varepsilon=(\rho_1^\varepsilon,\dots,\rho_N^\varepsilon)\), such that as \(\varepsilon\to0\): (i) \(\mathbf u_\varepsilon\) blows up exactly at the points \(\xi_1^*,\dots,\xi_m^*\); and (ii) \(\rho^\varepsilon\to(2\alpha_1\pi m,\dots,2\alpha_N\pi m)\).
D.
There exists a family of solutions \(\mathbf u_\varepsilon=(u_{1,\varepsilon},\dots,u_{N,\varepsilon})\) of the above Toda system for any \(m\) distinct points \(\xi_1^*,\dots,\xi_m^*\in\Sigma\), with parameters \(\rho^\varepsilon=(\rho_1^\varepsilon,\dots,\rho_N^\varepsilon)\), such that as \(\varepsilon\to0\): (i) \(\mathbf u_\varepsilon\) blows up exactly at the points \(\xi_1^*,\dots,\xi_m^*\); (ii) \(\rho^\varepsilon\to(2\alpha_1\pi m,\dots,2\alpha_N\pi m)\); and (iii) for every \(i=1,\dots,N\),
\[
\rho_i^\varepsilon\frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}\,dv_g}\,dv_g \stackrel{*}{\rightharpoonup} \sum_{j=1}^m 2\alpha_i\pi\,\delta_{\xi_j^*}
\quad\text{as }\varepsilon\to0.
\]
E.
There exists a family of solutions \(\mathbf u_\varepsilon=(u_{1,\varepsilon},\dots,u_{N,\varepsilon})\) of the above Toda system, with parameters \(\rho^\varepsilon=(\rho_1^\varepsilon,\dots,\rho_N^\varepsilon)\), such that as \(\varepsilon\to0\): (i) \(\mathbf u_\varepsilon\) blows up exactly at the points \(\xi_1^*,\dots,\xi_m^*\); (ii) \(\rho^\varepsilon=(2\alpha_1\pi m,\dots,2\alpha_N\pi m)\) for all sufficiently small \(\varepsilon\); and (iii) for every \(i=1,\dots,N\),
\[
\rho_i^\varepsilon\frac{V_i e^{u_{i,\varepsilon}}}{\int_\Sigma V_i e^{u_{i,\varepsilon}}\,dv_g}\,dv_g \stackrel{*}{\rightharpoonup} \sum_{j=1}^m 2\alpha_i\pi\,\delta_{\xi_j^*}
\quad\text{as }\varepsilon\to0.
\]

exact local mass coefficient at each blow-up point

weak-* convergence of the normalized measures

restriction to k-symmetric centers in \Sigma_0 (interior fixed points only)

asymptotic convergence of parameters versus eventual exact equality


\label{Preliminaries}

In this section, we recall some facts on the hyperbolic space. There are several models of hyperbolic space, such as  the Poincar\'e half space model and the Poincar\'e ball mod

\begin{theorem}\label{EPS} Assume that $p=\frac{2n}{n-2k}$ and $n\geq2k+2$.
    Then there is a positive function $f_0\in H\left(\mathbb{B}^n\right)$ which achieves the equality
    for  the inequality \eqref{BPS}. Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\in \mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\in \mathbb{B}^n$ and radially decreasing about the geodesic distance from P.
\end{theorem}

The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the  Sobolev and the Hardy inequalities on the upper half space $\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\leq \frac{2n}{n-2k}$ and $\gamma=\frac{\left(n-2k\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\in C_{c}^{\infty}\left(\mathbb{R}^{n}_{+}\right)$, there holds
\begin{equation}\label{HHSM}
\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,p} \left(\int_{\mathbb{R}^{n}_{+}}x_1^\gamma|u|^{p}dx\right)^{\frac{2}{p}}.
\end{equation}

The first order case ($k=1$) was established by Maz'ya \cite{Mazya}, while the higher order case was proved by Lu and Yang \cite{LY19}. In particular, when $p=\frac{2n}{n-2k}$ (corresponding to $\gamma=0$), inequality \eqref{HHSM} is referred to as the critical HSM inequality:
\begin{equation}\label{AHSM}
\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,\frac{2n}{n-2k}} \left(\int_{\mathbb{R}^{n}_{+}}|u|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}.
\end{equation}

 Lu and Yang \cite{LY19} developed Fourier analysis  techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \cite{FlynnLuYang} and \cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.)  In fact, they derived the following  Poincar\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces  $\mathbb{B}^n$: for $n>2k$ and $2<p\leq \frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, there holds

\begin{equation}\label{BPS}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\geq C_{n,k,p}\left(\int_{\mathbb{B}^n}|f|^{p}dV\right)^{\frac{2}{p}},
\end{equation}
where $P_k$ is the GJMS operator on the hyperbolic space  $\mathbb{B}^n$. The precise definition of this operator, along with relevant background on hyperbolic space, is provided in Section \ref{Preliminaries}.
By establishing the equivalence of the Poincar\'e-Sobolev inequalities on $\mathbb{B}^n$ and the HSM inequality on half spaces, they derived the higher order HSM inequality. This approach provides a technically powerful route to the HSM inequality, allowing the higher order HSM inequality to be obtained in a unified and indirect manner via its hyperbolic counterpart.
Motivated by this equivalence, in this article,  we first study the existence of the extremal functions for the Poincar\'e-Sobolev inequalities on the hyperbolic space   $\mathbb{B}^n$ and use this framework to establish the existence of extremal functions for the corresponding HSM inequalities.
\medskip

The $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ reads as follows:
\begin{equation}\label{BSobolev}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV\geq S_{n,k}\left(\int_{\mathbb{B}^n}|f|^{\frac{2n}{n-2k}}dV\right)^{\frac{n-2k}{n}},
\end{equation}
where $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, $1\leq k <\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev  inequality on $\mathbb{R}^n$.
(see Hebey \cite{Hebey} and Liu \cite{Liu}). In \cite{LY19}, Lu and Yang  gave an alternative way to prove \eqref{BSobolev}. In fact, let $g(x)=\big(\frac{2}{1-|x|^2}\big)^{\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \cite{LY19}) $$P_k(f)=\big(\frac{2}{1-|x|^2}\big)^{-\frac{n}{2}-k}(-\Delta)^k\big((\frac{2}{1-|x|^2})^{\frac{n}{2}-k}f(x)\big),$$
we can see that the $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ is equivalent to the following $k$-th order  Sobolev inequality on the unit ball of $\mathbb{R}^n$:
\begin{equation}\label{BallSobolev}
\int_{\mathbb{B}^n}(-\Delta)^k g\cdot gdx\geq S_{n,k}\left(\int_{\mathbb{B}^n}|g|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}.
\end{equation}
Obviously, the $k$-th order Sobolev inequality on the unit ball of $\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\mathbb{R}^n$ (see Talenti \cite{Talenti} and Aubin \cite{Aubin} and also Proposition 1.43 of \cite{Willem} for the first order, and  Lieb \cite{Lieb} forthe  high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ does not admit any extremal function.
\medskip

On the other hand, we have the following $k$-th Poincar\'e inequality on the hyperbolic space $\mathbb{B}^n$:
\begin{equation}\label{Bpoincare}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV\geq \prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\left(\int_{\mathbb{B}^n}|f|^{2}dV\right),
\end{equation}
where $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, and the constant $\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained.    Combining $k$-th order Sobolev inequality and $k$-th order Poincar\'e inequality on the hyperbolic space $\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.

For the first order critical HSM inequality, Hebey \cite{Hebey} showed that the best constant $C_{n,1,\frac{2n}{n-2}}$ is strictly smaller than the Sobolev constant $S_{n,1}$ for $n\geq 4$. Tertikas and Tintarev  \cite{TT} further proved that the existence of the extremal function for $n\geq 4$. On the other hand, Benguria, Frank and Loss \cite{BFL} demonstrated that the best constant $C_{n,1,\frac{2n}{n-2}}$ is the same as the Sobolev constant $S_{n,1}$ and the best constant $C_{n,1,\frac{2n}{n-2}}$ is not achieved when  $n=3$. For the high order critical case, Lu and Yang    proved in \cite {LY22}  that the best constant $C_{n,k,\frac{2n}{n-k}}$ for the kth-order HSM inequality coincides with the $k$-th Sobolev contant $S_{n,k}$ if $n=2k+1$ and the best constant $C_{n,k,\frac{2n}{n-k}}$ is strictly smaller than $k$-th Sobolev contant $S_{n,k}$ if $n\geq 2k+2$. However, the attainability of the best constant $C_{n,k,\frac{2n}{n-2k}}$ remains open for $n\ge 2k+2$ when $k\ge 2$. In \cite{TT}, the authors employed a clever transformation that reduces the HSM inequality to a weighted Sobolev inequality on the upper half-space, and by applying the concentration compactness principle, they established the existence of extremal functions in the first order case $k=1$. Due to the increased complexity of higher order operators, such a transformation is no longer available in the higher order setting.
\medskip

Our first main result concerns the existence of extremal functions for the critical higher order Poincar\'e-Sobolev inequality.
We define the Hilbert space $H\left(\mathbb{B}^n\right)$ by the completion of $C_c^{\infty}\left(\mathbb{B}^n\right)$ under the norm
$$\|f\|_{H\left(\mathbb{B}^n\right)}=\left(\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\right)^{\frac{1}{2}}.$$

It is noted that $\|f\|_{H\left(\mathbb{B}^n\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.

The classical Hardy-Sobolev-Maz'ya (HSM) inequality provides a refinement of both the Sobolev and the Hardy inequalities on the upper half space $\mathbb{R}^n_+$. It reads as follows: for $n>2k$, $2<p\leq \frac{2n}{n-2k}$ and $\gamma=\frac{\left(n-2k\right)p}{2}-n$, there exists some positive constant $C_{n,k,p}$ such that for each $u\in C_{c}^{\infty}\left(\mathbb{R}^{n}_{+}\right)$, there holds
\begin{equation}\label{HHSM}
\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\geq C_{n,k,p} \left(\int_{\mathbb{R}^{n}_{+}}x_1^\gamma|u|^{p}dx\right)^{\frac{2}{p}}.
\end{equation}

Lu and Yang \cite{LY19} developed Fourier analysis techniques on the hyperbolic space which is a noncompact complete Riemannian manifold to establish the HSM inequalities for higher order derivatives on half spaces. (see also Flynn et al \cite{FlynnLuYang} and \cite{LuYang-AIM} for the other cases of symmetric spaces of rank one.) In fact, they derived the following Poincar\'e-Sobolev inequalities for the GJMS operators on hyperbolic spaces $\mathbb{B}^n$: for $n>2k$ and $2<p\leq \frac{2n}{n-2k}$, there exists some constant $C_{n,k,p}$ such that for each $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, there holds

The $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ reads as follows:
\begin{equation}\label{BSobolev}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV\geq S_{n,k}\left(\int_{\mathbb{B}^n}|f|^{\frac{2n}{n-2k}}dV\right)^{\frac{n-2k}{n}},
\end{equation}
where $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, $1\leq k <\frac{n}{2}$, $S_{n,k}$ is the sharp constant of $k$-th order Sobolev inequality on $\mathbb{R}^n$.
(see Hebey \cite{Hebey} and Liu \cite{Liu}). In \cite{LY19}, Lu and Yang gave an alternative way to prove \eqref{BSobolev}. In fact, let $g(x)=\big(\frac{2}{1-|x|^2}\big)^{\frac{n}{2}-k}f(x)$, using the conformal transformation laws (see Lemma 5.1 of \cite{LY19}) $$P_k(f)=\big(\frac{2}{1-|x|^2}\big)^{-\frac{n}{2}-k}(-\Delta)^k\big((\frac{2}{1-|x|^2})^{\frac{n}{2}-k}f(x)\big),$$
we can see that the $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ is equivalent to the following $k$-th order Sobolev inequality on the unit ball of $\mathbb{R}^n$:
\begin{equation}\label{BallSobolev}
\int_{\mathbb{B}^n}(-\Delta)^k g\cdot gdx\geq S_{n,k}\left(\int_{\mathbb{B}^n}|g|^{\frac{2n}{n-2k}}dx\right)^{\frac{n-2k}{n}}.
\end{equation}
Obviously, the $k$-th order Sobolev inequality on the unit ball of $\mathbb{R}^n$ has no extremals using translation, dilation and classification of extremals of Sobolev inequality in $\mathbb{R}^n$ (see Talenti \cite{Talenti} and Aubin \cite{Aubin} and also Proposition 1.43 of \cite{Willem} for the first order, and Lieb \cite{Lieb} forthe high-order case). Hence the $k$-th order Sobolev inequality on the hyperbolic space $\mathbb{B}^n$ does not admit any extremal function.
\medskip

On the other hand, we have the following $k$-th Poincar\'e inequality on the hyperbolic space $\mathbb{B}^n$:
\begin{equation}\label{Bpoincare}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV\geq \prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\left(\int_{\mathbb{B}^n}|f|^{2}dV\right),
\end{equation}
where $f\in C_{c}^{\infty}\left(\mathbb{B}^n\right)$, and the constant $\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ is sharp. We need to note that the sharp constants cannot be attained. Combining $k$-th order Sobolev inequality and $k$-th order Poincar\'e inequality on the hyperbolic space $\mathbb{B}^n$, there must exist some refinement of both of these two kinds of sharp inequalities. The refinement is simply demonstrated by the high-order Poincare-Sobolev inequality on the hyperbolic space \eqref{BPS}. However, the problem for the existence of extremal of high-order Poincare-Sobolev inequality on the hyperbolic space still remains open.

Our first main result concerns the existence of extremal functions for the critical higher order Poincar\'e-Sobolev inequality.
We define the Hilbert space $H\left(\mathbb{B}^n\right)$ by the completion of $C_c^{\infty}\left(\mathbb{B}^n\right)$ under the norm
$$\|f\|_{H\left(\mathbb{B}^n\right)}=\left(\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\right)^{\frac{1}{2}}.$$

It is noted that $\|f\|_{H\left(\mathbb{B}^n\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.

\begin{equation}\label{BPS}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\geq C_{n,k,p}\left(\int_{\mathbb{B}^n}|f|^{p}dV\right)^{\frac{2}{p}},
\end{equation}

It is noted that $\|f\|_{H\left(\mathbb{B}^n\right)}$ is indeed a norm by using the Poincare-Sobolev inequality.

\begin{theorem}\label{EHMS}
 Assume $n\geq2k+2$. Then there is a positive function $f_0\in \widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ which achieves the equality in inequality
 \eqref{AHSM},
where $\widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ is the completion of $C_c^{\infty}\left(\mathbb{R}^{n}_{+}\right)$ under the norm
$$\|u\|_{\widetilde{H}\left(\mathbb{R}^{n}_{+}\right)}=\left(\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\right)^{\frac{1}{2}}.$$

\begin{theorem}\label{E-subHPS}
 Assume that $2<p<\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\in H\left(\mathbb{B}^n\right)$ which achieves the equality in the inequality \eqref{BPS},
\end{theorem}

\begin{theorem}\label{E-subHHSM}
Assume that $2<p<\frac{2n}{n-2k}$ and $n>2k$.
 Then there is a positive function $f_0\in \widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ which achieves the equality in inequality \eqref{HHSM}.

\begin{corollary}\label{EL}
Let $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$. If either $2k+2\leq n$, $p=\frac{2n}{n-2k}$ or $2k<n$, $2<p<\frac{2n}{n-2k}$, then there exists at least one positive solution u in $H\left(\mathbb{B}^n\right)$ to the equation \eqref{sublambdaBN}, which is radially symmetry and monotone decreasing about some point $P\in \mathbb{B}^n$.
\end{corollary}

From Lemma \ref{kernel-decreasing} (presented later), we know that $G^{-1}_k\left(x,y\right)$ is symmetric and decreasing about the distance function $\rho\left(T_{x}\left(y\right)\right)$. From the Symmetrization Lemma \ref{becknersymm} in \cite{Beckner93}, we can pick up radially decreasing maximizing sequence $\{h_m\}$ for the Hardy-Littlewood-Sobolev inequality \eqref{BHLS}. By the direct equivalence of the inequality \eqref{BHLS} and \eqref{G2}, we know that $\{h_m\}$ is also the radially decreasing maximizing sequence for inequality \eqref{G2}, that is,
$$\int_{\mathbb{B}^n}|h_m|^{\frac{2n}{n+2k}}dV=1,\ \ \lim\limits_{m\rightarrow +\infty} \int_{\mathbb{B}^n}|G_k^{-\frac{1}{2}}\left(h_m\right)|^{2}dV=C_{n,k,\frac{2n}{n-2k}}^{-1}.$$
In the following Lemma \ref{relationshipsequence}, we proved the relationship between the extremizing sequence of \eqref{G1} and the extremizing sequence of \eqref{G2}. Pick $f_m=G_{k}^{-1}\left(h_m\right)$, then $f_m\left(x\right)$ is a radially decreasing extremizing sequence for inequality \eqref{G1}. Up to some constants, we have
$$\lim_{m\rightarrow\infty}\|f_m\|^2_{H(\mathbb{B}^n)}=C_{n,k,\frac{2n}{n-2k}},\ \ \ \int_{\mathbb{B}^n}|f_m|^{\frac{2n}{n-2k}}dV=1.$$
\end{proof}

\section{The proof of Theorem \ref{E-subHPS}}
We now start to prove that the high-order subcritical Poincar\'e-Sobolev inequality on Hyperbolic space $\mathbb{B}^n$:
$$\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\geq C_{n,k,q}\left(\int_{\mathbb{B}^n}|f|^{q}dV\right)^{\frac{2}{q}}$$
for $2<q<\frac{2n}{n-2k}$
admits an extremal function. Assume that $\{f_m\}$ is a minimizing sequence for the subcritical Poincar\'e-Sobolev inequality on hyperbolic space $\mathbb{B}^n$, that is
$$\int_{\mathbb{B}^n}|f_m|^{q}dV=1,\ \ \lim\limits_{m\rightarrow +\infty}\left(\int_{\mathbb{B}^n}P_k\left(f_m\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f_m|^2dV\right)=C_{n,k,q}.$$
Similar to the proof of the existence of Poincar\'e-Sobolev inequality on hyperbolic space $\mathbb{B}^n$ (Theorem \ref{EPS}), by Green representative formula and Riesz rearrangement inequality, we can assume that the minimizing sequence $\{f_m\}$ is radially decreasing about origin.
In order to prove the existence of extremal function, we only need to prove that
$f_m\rightarrow f_0$ strongly in $L^{q}\left(\mathbb{B}^n\right)$. Since $f_m$ is radially decreasing, by Helly theorem, we know that there exists $f_0\in L^{p}\left(\mathbb{B}^n\right)$ such that $f_m$ converges to $f_0$ for almost $x\in \mathbb{B}^n$. Through Poincar\'e-Sobolev inequality, we see that
$f_m$ is also bounded in $L^{\frac{2n}{n-2k}}\left(\mathbb{B}^n\right)$. This together with the Vitali-convergence theorem yields that
$$\lim\limits_{R\rightarrow 1}\lim\limits_{m\rightarrow +\infty}\int_{B^n\left(0,R\right)}|f_m|^qdV=\lim\limits_{R\rightarrow 1}\int_{B^n\left(0,R\right)}|f_0|^qdV=\int_{\mathbb{B}^n}|f_0|^qdV.$$
In order to prove that $f_m$ strongly converges to $f_0$ in $L^q\left(\mathbb{B}^n\right)$, it remains to show that
$$\lim\limits_{R\rightarrow 1}\lim\limits_{m\rightarrow +\infty}\int_{\mathbb{B}^n\setminus B^n\left(0,R\right)}|f_m|^qdV=0,$$
that is,
$$\lim\limits_{R\rightarrow 1}\lim\limits_{m\rightarrow +\infty}\int_{B^n\setminus B^n\left(0,R\right)}|f_m|^q\left(\frac{2}{1-|x|^2}\right)^ndx=0.$$
From the previous radial Lemma \ref{radial}, we see that there exists $D_{n,k,q_0}>0$ such that
$f_m\left(R\right)\leq \left(1-R\right)^{\frac{n-1}{q_0}}$, where $q_0$ is chosen as $q_0<q$. Then it is not difficult to check that
\begin{equation}\begin{split}
&\lim\limits_{R\rightarrow 1}\lim\limits_{m\rightarrow +\infty}\int_{B^n\setminus B^n\left(0,R\right)}|f_m|^q\left(\frac{2}{1-|x|^2}\right)^ndx\\
\leq &D_{n,k,q_0}\lim\limits_{R\rightarrow 1}\lim\limits_{m\rightarrow +\infty}\int_{B^n\setminus B^n\left(0,R\right)}\left(1-R\right)^{\frac{q\left(n-1\right)}{q_0}}\left(\frac{2}{1-|x|^2}\right)^ndx=0.
\end{split}\end{equation}

The proof relies on developing the Lions' type of the concentration-compactness principle \cite{Lions841, Lions1} for differential inequalities on the hyperbolic space. The concentration-compactness principle for differential inequalities can be divided into the first and the second differential concentration-compactness. One of the key points is to prove the minimizing sequence $\{u_k\}$ for higher order Poincar\'e-Sobolev inequality is tight. In order to achieve this, we must exclude the vanishing  and dichotomy phenomenon. The loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon. In order to overcome these difficulties, we need to find the radially minimizing sequence which decays at infinity of hyperbolic space and help to exclude the vanishing phenomenon. However, P\"{o}lya-Szeg\"{o} rearrangement inequality is invalid for high-order differential inequality. Applying the equivalence of Poincar\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula, one can find the radially minimizing sequence for Poincar\'e-Sobolev inequality. To exclude  the dichotomy, one cannot follow the same line of Lions' proof in \cite{Lions1}. In fact, recall the proof of excluding dichotomy for minimizing sequence $w_k$ of $L^p$ Sobolev inequality involving high order derivatives: $$I=\inf\{\int_{\mathbb{R}^n}|\nabla^{m}w|^pdx,\ \ \int_{\mathbb{R}^n}|w|^{\frac{np}{n-mp}}dx=1\}.$$ Lions constructed a suitable concentration-function $$\rho_k=\sum_{j=0}^{m}|D^jw_k|^{q_j},\  q_j=\frac{np}{n-(m-j)p}$$
and prove that for any $\epsilon>0$, there exists $y_k\in \mathbb{R}^n$ and $R_{\epsilon}>0$ such that $\int_{|x-y_k|\geq R}\rho_k(x)dx<\epsilon$ for any $R>R_{\epsilon}$. This is achieved by scaling cut-off function and
$$\int_{R_0<|x-y_k|<R_k}\sum_{j=0}^{m-1}|D^jw_k|^{q_j}dx=o_k(1).$$ However, for higher order Poincar\'e-Sobolev inequality in hyperbolic space, it is impossible to introduce the replacement of concentration-function $\rho_k$ of $\mathbb{R}^n$. Moreover, the loss of scaling cut-off function can also add extra difficulty. In order to overcome this difficulty, we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the  equivalent integral inequalities (It is differen from the equivalence of extremals), allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\'e-Sobolev inequality. Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\delta$-measure using $C_{n,k,\frac{2n}{n-k}}<S_{n,k}$.

\medskip

Through the equivalence, combining Theorem \ref{EPS}, we immediately obtain the existence of the extremal function for the  high order critical Hardy-Sobolev-Maz'ya inequality \eqref{AHSM} on the upper half space.

\begin{theorem}\label{EHMS}
 Assume $n\geq2k+2$.  Then there is a positive function $f_0\in \widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ which achieves the equality in inequality
 \eqref{AHSM},
where $\widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ is the completion of $C_c^{\infty}\left(\mathbb{R}^{n}_{+}\right)$ under the norm
$$\|u\|_{\widetilde{H}\left(\mathbb{R}^{n}_{+}\right)}=\left(\int_{\mathbb{R}^{n}_{+}}|\nabla^{k}u|^2dx-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{R}^{n}_{+}}\frac{u^2}{x_1^{2k}}dx\right)^{\frac{1}{2}}.$$

\end{theorem}

For $n=2k+1$, according to Lemma 4.3 of \cite{LY22}, we can deduce the nonexistence of the extremal function for critical inequality \eqref{BPS} and \eqref{HHSM}. Different from the critical case, the subcritical inequalities have an extremal function for all $n>2k$. We have the following results:

\begin{theorem}\label{E-subHPS}
  Assume that $2<p<\frac{2n}{n-2k}$ and $n>2k$. Then there is a positive function $f_0\in H\left(\mathbb{B}^n\right)$ which achieves the equality in the inequality \eqref{BPS},
\end{theorem}

\begin{theorem}\label{E-subHHSM}
Assume that $2<p<\frac{2n}{n-2k}$ and $n>2k$.
 Then there is a positive function $f_0\in \widetilde{H}\left(\mathbb{R}^{n}_{+}\right)$ which achieves the equality in   inequality \eqref{HHSM}.

\end{theorem}

The Poincar\'e-Sobolev inequalities  are highly linked with the  Brezis-Nirenberg problem \cite{Brezis} on the entire hyperbolic space $\mathbb{B}^n$:
\begin{equation}\label{sublambdaBN}
P_{k}\left(f\right)-\alpha f=|f|^{p-2}f.
\end{equation}
For $k=1$, Mancini and Sandeep \cite{MS} showed that the entire solution exists either when  $n\geq4$, $p=\frac{2n}{n-2}$, $0<\alpha\leq\frac{1}{4}$ or when $n\geq3$, $1<p<\frac{2n}{n-2}$ and $\alpha\leq\frac{1}{4}$. Bhakta et al. \cite{BGKM25} proved the quantitative stability for the equation \eqref{sublambdaBN} when $k=1$.
 For $k\geq 2$, Li et al. \cite{LLY} established the existence of the nontrivial solution  when $n\geq 2k+2$, $p=\frac{2n}{n-2k}$, $\alpha<\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$. It is not hard to calculate that the corresponding Euler-Lagrange equation of the inequality \eqref{BPS} is the  equation \eqref{sublambdaBN} with $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$, $2<p\leq\frac{2n}{n-2k}$.  A straightforward corollary of Theorem \ref{EPS} and Theorem \ref{E-subHPS} is as follows:

\begin{corollary}\label{EL}
Let $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$. If either $2k+2\leq n$, $p=\frac{2n}{n-2k}$ or $2k<n$, $2<p<\frac{2n}{n-2k}$,  then there exists at least one positive solution u in $H\left(\mathbb{B}^n\right)$ to the equation \eqref{sublambdaBN},  which is radially symmetry  and monotone decreasing about some point $P\in \mathbb{B}^n$.
\end{corollary}

\begin{remark}
   It is the first time to derive the existence of positive solutions for high-order Brezis-Nirenberg problem for $\alpha=\prod\limits_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}$ on the whole hyperbolic space.
\end{remark}

The proof of Theorem~\ref{EPS} is sketched as follows.

1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”

2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\{u_k\}$ for higher order Poincar\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”

3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\"{o}lya-Szeg\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”

4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\rho_k$ of $\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\'e-Sobolev inequality.”

5. **Compactness and exclusion of concentration to a $\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\delta$-measure using $C_{n,k,\frac{2n}{n-k}}<S_{n,k}$.”

These steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in Theorem~\ref{EPS}.

The proof of the main theorem is sketched as follows.

1. **Concentration--compactness framework on hyperbolic space.** The proof “relies on developing the Lions' type of the concentration-compactness principle … for differential inequalities on the hyperbolic space,” split into “the first and the second differential concentration-compactness.”

2. **Tightness of a minimizing sequence and exclusion of vanishing/dichotomy.** A “key point is to prove the minimizing sequence $\{u_k\}$ for higher order Poincar\'e-Sobolev inequality is tight,” which requires excluding “the vanishing and dichotomy phenomenon.” The “loss of invariance of dilation causes much challenge to exclude the vanishing phenomenon.”

3. **Radially minimizing sequence to rule out vanishing (via integral-equation equivalence and rearrangement tools).** To overcome the failure of direct rearrangement (“P\"{o}lya-Szeg\"{o} rearrangement inequality is invalid for high-order differential inequality”), the sketch uses “the equivalence of Poincar\'e-Sobolev inequality and Hardy-Littlewood-Sobolev inequality on the hyperbolic space, Riesz's rearrangement inequality on hyperbolic space and Green's representative formula,” to “find the radially minimizing sequence for Poincar\'e-Sobolev inequality” which “decays at infinity of hyperbolic space” and “help[s] to exclude the vanishing phenomenon.”

4. **Ruling out dichotomy via a quantitative link to equivalent integral inequalities.** Since “to exclude the dichotomy, one cannot follow the same line of Lions' proof,” and it is “impossible to introduce the replacement of concentration-function $\rho_k$ of $\mathbb{R}^n$” (and the “loss of scaling cut-off function” adds difficulty), the approach is: “we found a quantitative relationship between the minimizing sequences for the Poincare-Sobolev inequalities and those for the equivalent integral inequalities … allowing us to apply the concentration-compactness techniques for equivalent integral inequalities to rule out dichotomy of the minimizing sequences for Poincar\'e-Sobolev inequality.”

5. **Compactness and exclusion of concentration to a $\delta$-measure (second Lions procedure).** “Once we prove that the minimizing sequence is compact, we can follow the standard second Lions' concentration-compactness procedure to exclude the minimizing sequence converging $\delta$-measure using $C_{n,k,\frac{2n}{n-k}}<S_{n,k}$.”

These steps yield existence of an extremal, and the sketch indicates that the construction produces a “radially minimizing sequence,” matching the stated radial symmetry/monotone decrease conclusion in the main theorem.

\label{EPS} Assume that $p=\frac{2n}{n-2k}$ and $n\geq2k+2$.
Then there is a positive function $f_0\in H\left(\mathbb{B}^n\right)$ which achieves equality in the inequality
\begin{equation}\label{BPS}
\int_{\mathbb{B}^n}P_k\left(f\right)fdV-\prod_{i=1}^{k}\frac{\left(2i-1\right)^2}{4}\int_{\mathbb{B}^n}|f|^2dV\geq C_{n,k,p}\left(\int_{\mathbb{B}^n}|f|^{p}dV\right)^{\frac{2}{p}},
\end{equation}
Furthermore, the extremal function $f_0$ must be radially symmetric and monotone decreasing about some point $P\in \mathbb{B}^n$, that is, $f_0$ is a constant on the geodesic sphere centered at $P\in \mathbb{B}^n$ and radially decreasing about the geodesic distance from $P$.

Question:
Let $p=\frac{2n}{n-2k}$ and assume $n\ge 2k+2$. On the hyperbolic space $\mathbb{B}^n$ with volume element $dV$, let $P_k$ denote the $k$th-order GJMS operator, and let $H(\mathbb{B}^n)$ be the completion of $C_c^{\infty}(\mathbb{B}^n)$ under the norm
$$
\|f\|_{H(\mathbb{B}^n)}=\left(\int_{\mathbb{B}^n}P_k(f)f\,dV-\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f|^2\,dV\right)^{1/2}.
$$
Which statement holds for the critical higher-order Poincar\'e-Sobolev inequality
$$
\int_{\mathbb{B}^n}P_k(f)f\,dV-\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f|^2\,dV\ge C_{n,k,p}\left(\int_{\mathbb{B}^n}|f|^{p}\,dV\right)^{2/p}?
$$

There exists a positive function $f_0\in H(\mathbb{B}^n)$ that attains equality in this inequality. Moreover, this extremal function is radially symmetric and monotone decreasing about some point $P\in\mathbb{B}^n$; equivalently, $f_0$ is constant on each geodesic sphere centered at $P$ and decreases with the geodesic distance from $P$.

Choices:
B. There exists a positive function $f_0\in H(\mathbb{B}^n)$ that attains equality in this inequality. Moreover, every extremal function is radially symmetric and monotone decreasing about the origin; equivalently, $f_0$ is constant on each geodesic sphere centered at $0\in\mathbb{B}^n$ and decreases with the geodesic distance from the origin.
C. There exists a positive function $f_0\in H(\mathbb{B}^n)$ that attains equality in this inequality.
D. For every minimizing sequence $\{f_m\}\subset H(\mathbb{B}^n)$ for this inequality, there exists a point $P\in\mathbb{B}^n$ such that, after passing to a subsequence, $f_m$ converges strongly in $H(\mathbb{B}^n)$ to a positive function $f_0$ attaining equality. Moreover, $f_0$ is radially symmetric and monotone decreasing about $P$.
E. There exists a positive function $f_0\in H(\mathbb{B}^n)$ that attains equality in this inequality, and the best constant satisfies $C_{n,k,p}=S_{n,k}$. Moreover, the extremal function is radially symmetric and monotone decreasing about some point $P\in\mathbb{B}^n$.


Question:
Let \(\mathbb{B}^n\) be the unit ball model of hyperbolic space with hyperbolic volume element \(dV\), and let \(P_k\) denote the \(k\)-th GJMS operator on \(\mathbb{B}^n\). Define the Hilbert space \(H(\mathbb{B}^n)\) as the completion of \(C_c^{\infty}(\mathbb{B}^n)\) under the norm
\[
\|f\|_{H(\mathbb{B}^n)}=
\left(
\int_{\mathbb{B}^n}P_k(f)f\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f|^2\,dV
\right)^{1/2}.
\]
Assume that
\[
p=\frac{2n}{n-2k}\quad\text{and}\quad n\ge 2k+2.
\]
Consider the critical higher-order Poincaré-Sobolev inequality
\[
\int_{\mathbb{B}^n}P_k(f)f\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f|^2\,dV
\ge
C_{n,k,p}\left(\int_{\mathbb{B}^n}|f|^{p}\,dV\right)^{2/p}.
\]
Which existence statement holds under these assumptions?

There exists a positive function \(f_0\in H(\mathbb{B}^n)\) such that
\[
\int_{\mathbb{B}^n}P_k(f_0)f_0\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f_0|^2\,dV
=
C_{n,k,p}\left(\int_{\mathbb{B}^n}|f_0|^{p}\,dV\right)^{2/p},
\]
so \(f_0\) achieves equality in the critical higher-order Poincaré-Sobolev inequality. Moreover, \(f_0\) is radially symmetric and monotone decreasing about some point \(P\in\mathbb{B}^n\); equivalently, \(f_0\) is constant on each geodesic sphere centered at \(P\) and decreases with the geodesic distance from \(P\).

Choices:
B.
There exists a positive function \(f_0\in H(\mathbb{B}^n)\) such that
\[
\int_{\mathbb{B}^n}P_k(f_0)f_0\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f_0|^2\,dV
=
C_{n,k,p}\left(\int_{\mathbb{B}^n}|f_0|^{p}\,dV\right)^{2/p},
\]
so \(f_0\) achieves equality in the critical higher-order Poincare9-Sobolev inequality. Moreover, every extremal function is radially symmetric and monotone decreasing about the origin.
C.
There exists a positive function \(f_0\in H(\mathbb{B}^n)\) such that
\[
\int_{\mathbb{B}^n}P_k(f_0)f_0\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f_0|^2\,dV
=
C_{n,k,p}\left(\int_{\mathbb{B}^n}|f_0|^{p}\,dV\right)^{2/p},
\]
so \(f_0\) achieves equality in the critical higher-order Poincare9-Sobolev inequality.
D. For every minimizing sequence \(\{f_m\}\subset H(\mathbb{B}^n)\) for the critical higher-order Poincare9-Sobolev inequality, there exists a subsequence converging strongly in \(H(\mathbb{B}^n)\) to a positive function \(f_0\) that achieves equality, and \(f_0\) is radially symmetric and monotone decreasing about some point \(P\in\mathbb{B}^n\).
E.
There exists a positive function \(f_0\in H(\mathbb{B}^n)\) such that
\[
\int_{\mathbb{B}^n}P_k(f_0)f_0\,dV-
\prod_{i=1}^{k}\frac{(2i-1)^2}{4}\int_{\mathbb{B}^n}|f_0|^2\,dV
=
S_{n,k}\left(\int_{\mathbb{B}^n}|f_0|^{p}\,dV\right)^{2/p},
\]
where \(S_{n,k}\) is the sharp Euclidean \(k\)-th order Sobolev constant. Moreover, \(f_0\) is radially symmetric and monotone decreasing about some point \(P\in\mathbb{B}^n\).


A capset is a subset $C \subset \F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \cite{Tao},\cite{Matrix}) is the following: What is the largest size of a capset in $\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \cite[Proposition 2.2]{Tyrrell}) that $c=\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \le c$ (\cite{Nature}) and $c \le 2.756$ (\cite{Jordan}). The exact value of $a(n)$ is known for $n \le 6$ (\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\geq 512$ (\cite{Nature}).

\begin{Definition}
 A \emph{capset} is a subset of $\mathbb{F}_3^n$ containing no three distinct points on a line. A capset is \emph{complete} if it is not a subset of a larger capset.
 \end{Definition}

The main purpose of this paper is to exhibit some constructions of complete capsets and, in particular, construct the smallest known complete capsets.

If $q=3^m$ is a power of $3$, choosing a basis for $\F_q/\F_3$ induces an
identification of $\F_q^d$ with $\F_3^{dm}$ for any integer $d$. We will
define some subsets of $\F_q^d$ by algebraic equations and view them as
capsets in $\F_3^{dm}$ to study their properties.

We can also consider subsets of $\F_q^d$ with no three points on an $\F_q$-line.
We call such a set a cap (to distinguish it from a capset under the above
identification). It is clear that a cap is a capset but not conversely.
We can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every
point $P$ in the complement of $C$, there exists an $\F_q$-line through $P$
which meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\F_q^2, q >3$. This follows from the lemma below.

\begin{abstract}
Capsets are subsets of $\F_3^n$ with no three points on a line and a capset is complete if it is not a subset of a larger capset. We study some new constructions of capsets via algebraic equations over extensions of $\F_3$. In particular we construct the smallest known complete capsets with size proportional to the best known lower bound.
\end{abstract}

A capset is a subset $C \subset \F_3^n$ with no three points on a line. A question that has attracted a lot of interest (see e.g. \cite{Tao},\cite{Matrix}) is the following: What is the largest size of a capset in $\F_3^n$ as a function of $n$? If we denote this value by $a(n)$ then it can be shown (see \cite[Proposition 2.2]{Tyrrell}) that $c=\lim a(n)^{1/n}$ exists. Moreover, it is known that $2.2202 \le c$ (\cite{Nature}) and $c \le 2.756$ (\cite{Jordan}). The exact value of $a(n)$ is known for $n \le 6$ (\cite{capsetKnownSizes1, capetKnownSizes2, capsetKnownSizes3, capsetKnownSizes4}). It is also known that $a(8)\geq 512$ (\cite{Nature}).

We can also consider subsets of $\F_q^d$ with no three points on an $\F_q$-line.
We call such a set a cap (to distinguish it from a capset under the above
identification). It is clear that a cap is a capset but not conversely.
We can also consider the analogous notion of complete caps. A cap $C$ is complete as a cap if $C$ is not contained in a larger cap. That is, for every
point $P$ in the complement of $C$, there exists an $\F_q$-line through $P$
which meets $C$ in two points. We remark that a cap can be complete as a cap and yet not be complete as a capset. For example, a conic in the plane $\F_q^2, q >3$. This follows from the lemma below.

\begin{proof}
The $\F_3$ lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement of the capset, so if $N(N-1)/2 < 3^n -N$, there is a point in the complement not on any line through two points in the capset, hence the capset is not complete.
\end{proof}
In \cite{Csajbok}, subsets of $\F_p^n$ with $p$ prime no three points in arithmetic progression are studied. This condition is equivalent to the capset condition when $p=3$. They prove a lower bound similar, but slightly weaker, to Lemma \ref{lem:lower} in this more general situation. They also ask if this lower bound is best possible up to a multiplicative constant (loc. cit., Problem 5.1). We give a positive answer to this question in the case $p=3$ (see Corollary \ref{cor:upper}).

\begin{Corollary}
\label{cor:upper}
For every $n$, there exists a complete capset in $\F_3^n$ of size $O(\sqrt{3^n})$. \end{Corollary}

\begin{proof}
Assume first that $n=2m$ even and start with the construction given by Theorem \ref{thm:two-conics} of a capset of size $2(3^m -1) = O(\sqrt{3^n})$ in $\F_{3^m}^2$. If $m$ is odd, this is already complete.
When $m$ is even, we can only add points of the form $(0,b), b$ non-square. So we need a complete capset in $\F_{3^m}$ comprised of non-squares. Viewing $\F_{3^m}$ as $\F_{3^{m/2}}(\sqrt{d})$ and selecting $\lambda$ a non-square in $\F_{3^m}$, we can take elements of the form $\lambda(\pm 1+x\sqrt{d})^2, x \in \F_{3^{m/2}}^*$ which will be a capset as in the theorem, up to an affine transformation, since $(\pm 1+x\sqrt{d})^2 = (1+dx^2) \pm 2x\sqrt{d}$. Iterating this construction gives the result for $n$ even.

To construct small complete capsets in $\F_{3^n}$ when $n$ is odd, write $n=2m+1$, take the complete capset $C \subset \F_{3^{2m}}$ constructed above and consider $C \times \{0,1\} \subset \F_{3^n}$. It is straightfoward that this is a capset. To see it is complete, consider a pointi $P$ in its complement with last coordinate $c \in \F_3$ and let $P_0 \in \F_{3^{2m}}$ be the point corresponding to the other coordinates. If $c \in \{0,1\}$, then $P_0 \not\in C$ and there is a line in the subspace of points whose last coordinate is $c$ through $P$ meeting $C \times \{c\}$ in two points. If $c=2$ and $P_0 \in C$, the vertical line through $P$ meets the capset in two points. If $c=2$ and $P_0 \not\in C$, there is a line through $P_0$ meeting $C$ in $Q_0,Q_1$, say. Then $(Q_0,0), (Q_1,1)$ are in $C \times \{0,1\}$ and are collinear with $P$.
\end{proof}

\begin{Theorem}
\label{thm:quadric}
Let $\lambda$ be a non-square in $\F_q, q=3^m$ and $Q \subset \F_q^3$ be given by
$$Q = \{(x,y,x^2-\lambda y^2) \mid x,y \in \F_q\}.$$
Then $Q$ is a complete capset.
\end{Theorem}

\begin{Lemma}
\label{lem:lower}
If there exists a complete capset in $\F_3^n$ with $N$ points, then
$N(N+1)/2 \ge 3^n$.
\end{Lemma}

\begin{proof}
The $\F_3$ lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement of the capset, so if $N(N-1)/2 < 3^n -N$, there is a point in the complement not on any line through two points in the capset, hence the capset is not complete.
\end{proof}
In \cite{Csajbok}, subsets of $\F_p^n$ with $p$ prime no three points in arithmetic progression are studied. This condition is equivalent to the capset condition when $p=3$. They prove a lower bound similar, but slightly weaker, to Lemma \ref{lem:lower} in this more general situation. They also ask if this lower bound is best possible up to a multiplicative constant (loc. cit., Problem 5.1). We give a positive answer to this question in the case $p=3$ (see Corollary \ref{cor:upper}).

\subsection*{Acknowledgements}
The second author was supported by the Marsden Fund administered by the Royal
Society of New Zealand. He would also like to thank J. Sheekey and G. Van de Voorde for helpful discussions.

The argument is a counting/completeness contradiction: the $\F_3$-lines through the $N(N-1)/2$ pairs of points of the capset have their third point in the complement. Thus these lines can cover at most $N(N-1)/2$ points of the complement. If $N(N-1)/2 < 3^n-N$, then there is a point in the complement not lying on any line through two capset points, so it cannot be added to form a 3-term progression; hence the capset is not complete. Therefore completeness forces $N(N-1)/2 \ge 3^n-N$, i.e. $N(N+1)/2\ge 3^n$.

Question: Let \(C \subset \mathbb{F}_3^n\) be a complete capset, where a capset means a subset of \(\mathbb{F}_3^n\) containing no three distinct points on a line, and complete means that \(C\) is not contained in any larger capset. If \(|C|=N\), which quantitative estimate must hold?

\(\dfrac{N(N+1)}{2} \ge 3^n\).

Choices:
B. \(\dfrac{N(N-1)}{2} \ge 3^n\).
C. \(\dfrac{N(N+1)}{2} \ge 3^n-N\).
D. \(\dfrac{N(N-1)}{2} \ge 3^n-N+1\).
E. For every point \(P\in \mathbb{F}_3^n\setminus C\), there exists a unique line through \(P\) meeting \(C\) in two points; consequently \(\dfrac{N(N-1)}{2} \ge 3^n-N\).

final algebraic rearrangement from complement count to total count

dropped the final rewrite using \(3^n-N\) to obtain a weaker bound

strict-vs-nonstrict boundary in the contradiction inequality

existence of a covering line for each complement point replaced by uniqueness