\label{th_homo_image} The relation   $\{\langle A,B \rangle  \colon A \leftarrow\mathrel{\mkern-14mu}\leftarrow B\}$  on the Borel space of countable   groups      is  a  complete   analytic quasi-order. Thus, the relation of bi-epimorphism on the same Borel space  is  a complete      analytic equivalence relation.

The set of groups with domain $\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\omega$, using a construction of Mekler~\cite{mekler}.
 In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently,
 the fourth author of the present paper and S.\ Shelah~\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\cite{veli}. In another direction, Calderoni and Thomas~\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.

\smallskip

Throughout, when we say a structure is countable we mean that it has $\omega$ as its domain. Classes of structures we are interested in will always form a standard Borel space. For countable groups $A,B$, we write $A \leftarrow\mathrel{\mkern-14mu}\leftarrow B$ if there is a surjective homomorphism from $B$ onto $A$; equivalently, $A$ is isomorphic to a quotient of $B$. This relation is easily shown to be analytic. Our main result is that it has the maximum possible complexity. Camerlo et al. \cite{camerlo2} showed such a result for several classes of structures. Notably, the case of countable groups was left open.

\begin{abstract}
 We prove that the epimorphism relation is a complete analytic quasi-order on the space of countable groups.
 In the process we obtain the result of indepent interest showing that the epimorphism relation on pointed reflexive graph is complete.
\end{abstract}

The set of groups with domain $\omega$ admits a natural topology which makes it into a Borel space. Using the tools of invariant descriptive set theory (see~\cite{gao}), one can attempt to determine the (Borel) complexity of equivalence relations and quasi-orders of interest on this space. Friedman and Stanley~\cite{friedman_and_stanley} proved that the isomorphism relation on countable groups is Borel bireducible with the isomorphism relation between countable graphs on $\omega$, using a construction of Mekler~\cite{mekler}.
 In fact they show that also isomorphism relation on the class of $2$-nilpotent group of exponent a fixed odd prime is Borel bi-reducible with graph isomorphism Recently,
 the fourth author of the present paper and S.\ Shelah~\cite{1205} proved that the same applies to countable torsion-free abelian groups. The isomorphism relation between finitely generated groups is a universal essentially countable Borel equivalence relation~\cite{veli}. In another direction, Calderoni and Thomas~\cite{calderoni} show that the embeddability relation between countable torsion free abelian groups is a complete analytic quasi-order (and hence bi-embeddability is a complete analytic equivalence relation). Our aim is to determine the Borel complexity of a quasi-order on groups dual to that of embeddability: being an epimorphic image.

\smallskip

Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex. In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using
 a new group theoretic construction based on certain countably generated Coxeter groups.

\begin{question} Is the epimorphism relation a complete-analytic quasi-order on the Borel space of countable {\em abelian} groups?
\end{question}

\begin{remark} \label{LRrem}
Louveau and Rosendal \cite[Theorem 3.5]{LR} proved that the homomorphism relation between countable graphs is a complete analytic quasi-order. They give a Borel reduction $T \to G_T$ of their complete quasi-order $\leq_{max} $ on normal trees (see \cite[Definition 2.3 and above]{LR}) to it. One observes from the proof that for each normal tree $T$, the constructed graph $G_T$ has not isolated vertex. As they remark at the end of their proof, if $S \le_{max} T$ then $G_S$ is in fact isomorphic to an induced subgraph of $G_T$, and so in particular $G_S\preceqc_1 G_T$. Thus, their proof shows that \emph{the relation $\preceqc_1$ on countable graphs with no isolated vertices is a complete analytic quasi-order.}
\end{remark}

\begin{proposition}\label{reduction}
 There is a Borel map $F$ from the class of countable graphs without isolated vertices to the class of countable pointed reflexive graphs, such that for any countable connected graphs $\Gamma$ and $\Delta$, we have
 $$\Gamma \preceqc_1 \Delta \iff F(\Gamma) \leftarrowdbl F(\Delta). $$
 Thus, the relation $\leftarrowdbl$ between countable pointed reflexive graphs is a complete analytic quasi-order using Remark~\ref{LRrem}.
\end{proposition}

\begin{fact}[{\cite[3.1.7]{krammer}}]\label{krammer_fact} Let $(G, S)$ be a Coxeter system and let $I, J \subseteq S$. Then $G_I = \langle I \rangle_G$ and $G_J = \langle J \rangle_G$ are conjugate in $G$ if and only if
$I$ and $J$ are in the same connected component of $\mathcal{K}_S$ (cf. \ref{def_K}).
\end{fact}

Our reduction proceeds in   two steps.  In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation  on countable    pointed reflexive graphs  is analytic complete; here a pointed reflexive graph is one that has a distinguished vertex that is adajacent to each vertex.  In the second step, we Borel reduce this quasi-order to the epimorphism relation  on countable groups, using
	a new group theoretic construction based on certain countably generated Coxeter groups.

We leave open the following question.

	\begin{question}   Is      the   epimorphism relation  a complete-analytic quasi-order on  the Borel space of countable {\em abelian} groups?
\end{question}

Our reduction proceeds in two steps. In the first step, we use a result of Louveau and Rosendal \cite{LR} to show that the epimorphism relation on countable pointed reflexive graphs is analytic complete (where a pointed reflexive graph has a distinguished vertex adjacent to each vertex). In the second step, we Borel reduce this quasi-order to the epimorphism relation on countable groups, using a new group theoretic construction based on certain countably generated Coxeter groups.

Question: Let a countable group mean a group with domain \(\omega\), viewed in the standard Borel space of all countable groups. For countable groups \(A\) and \(B\), write \(A \leftarrow\!\!\leftarrow B\) if there exists a surjective homomorphism from \(B\) onto \(A\) (equivalently, \(A\) is isomorphic to a quotient of \(B\)). Let bi-epimorphism mean the symmetric relation \(A \leftarrow\!\!\leftarrow B\) and \(B \leftarrow\!\!\leftarrow A\). Which statement holds for these relations on the Borel space of countable groups? Here, “complete analytic quasi-order” means an analytic quasi-order to which every analytic quasi-order on a standard Borel space Borel reduces, and “complete analytic equivalence relation” is defined analogously among analytic equivalence relations.

The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is a complete analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is a complete analytic equivalence relation.

Choices:
B. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is an analytic quasi-order, but it is not complete analytic. Consequently, the bi-epimorphism relation on the same Borel space is analytic, though not a complete analytic equivalence relation.
C. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is an analytic quasi-order. Consequently, the bi-epimorphism relation on the same Borel space is an analytic equivalence relation.
D. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is a complete analytic equivalence relation. Consequently, the bi-epimorphism relation on the same Borel space is also a complete analytic equivalence relation.
E. The relation \(\{\langle A,B\rangle : A \leftarrow\!\!\leftarrow B\}\) on countable groups is a complete analytic quasi-order, and moreover there is a direct Borel reduction from every analytic quasi-order to epimorphism on countable groups without passing through pointed reflexive graphs.

analytic-completeness obtained via the two-step Borel reduction

dropped completeness in both the quasi-order and equivalence-relation conclusions

necessity of the intermediate pointed reflexive graph reduction step

The stem defines the epimorphism and bi-epimorphism relations and asks for the correct descriptive-set-theoretic classification, but it does not explicitly or implicitly reveal that the relation is complete analytic.

The question is not a direct restatement of a theorem from the stem itself; the student must choose among materially different claims about analyticity, completeness, and quasi-order versus equivalence-relation status.

Some reasoning is needed to compare the strength and type of the options, but the item mainly tests recall of a known theorem rather than derivation. Generative pressure is also weakened because one distractor states a weaker claim that would still be true if the correct choice is true.

Several distractors are plausible and target real confusions (analytic vs complete analytic, quasi-order vs equivalence relation, overclaiming a direct reduction). However, choice C is a weaker true statement implied by A, so it is not a clean distractor and introduces ambiguity.

Good on avoiding leakage and tautology, but only moderate as a reasoning item. Its main flaw is ambiguity: option C appears true as a weaker consequence of A, so the MCQ does not cleanly enforce a unique correct answer.


\label{thm:symmetry}
	Let $(u,v) \in \mathcal{D}^{1,p}(\mathbb{R}^n) \times \mathcal{D}^{1,p} (\mathbb{R}^n)$ be a solution to \eqref{P}. Then $(u,v)$ is radially symmetric and radially decreasing around the origin.

In this paper, we investigate qualitative properties of solutions to the following doubly critical system involving the $p$-Laplace operator:
\begin{equation}\tag{$\mathcal{S}^*$}\label{P}
\begin{cases}
\displaystyle -\Delta_p u\,=\gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \nu \alpha u^{\alpha-1} v^\beta & \text{in}\quad\R^n \vspace{0.2cm} \\
\displaystyle -\Delta_p v\,=\gamma \frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \nu \beta u^\alpha v^{\beta-1} & \text{in}\quad\R^n \vspace{0.2cm} \\
u,v > 0 &  \text{in}\quad\R^n \setminus \{0\}\\
u,v \in \mathcal D^{1,p}(\R^n),
\end{cases}
\end{equation}
where $\gamma \in [0, \Lambda_{n,p})$ with $\Lambda_{n,p} = \left[(n-p)/p\right]^p$ representing the optimal constant in Hardy's inequality for $n > p$. We note that this system is well posed once one assumes that both
$\alpha, \beta > 1$ are real parameters that satisfy
\begin{equation*}
\alpha + \beta = p^*.
\end{equation*}
Here, $p^* = np/(n-p)$ denotes the critical Sobolev exponent, $\nu > 0$ is the coupling parameter. By the previous homogeneity condition we deduce that $p^*> 2$. Thus, here and in all the paper we will assume that
$$
\frac{2n}{n+2} < p< n.
$$
We point out that $\mathcal D^{1, p}(\mathbb R^n)$ is the completion of $\mathcal C^\infty_c(\mathbb R^n)$, the space of smooth functions with compact support, with respect to the norm
$$\|u\|:=\left(\int_{\mathbb R^n}|\nabla u|^p \, dx \right)^{\frac 1p}.$$
It is well known, by standard regularity theory (see e.g.~\cite{Di,T}), it follows that solutions to \eqref{P} are of class $\mathcal C^{1,\alpha}$ far from the origin.\\

Our main objective is to prove radial symmetry of solutions to system \eqref{P}. When $\gamma = \nu = 0$, the system reduces to the standard $p$-Sobolev critical equation
\begin{equation}\label{eq:Sobolev}
	\begin{cases}
		\displaystyle -\Delta_p u\,=  u^{p^*-1}  & \text{in}\quad\R^n\\
		u > 0 &  \text{in}\quad\R^n.
	\end{cases}
\end{equation}
It is well-known that the Aubin-Talenti bubbles, defined as
\begin{equation}\label{AubinTalenti}
\mathcal{V}_{\lambda,x_0}(x) := \left[\frac{\lambda^{\frac{1}{p-1}}\sqrt[p]{n\left(\frac{n-p}{p-1}\right)^{p-1}}}{\lambda^{\frac{p}{p-1}} + |x - x_0|^{\frac{p}{p-1}}}\right]^{\frac{n-p}{p}}
\end{equation}
is a family of positive radial solutions to \eqref{eq:Sobolev}, with $\lambda > 0$, and $x_0 \in \mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \cite{GNN2}, it has been shown that any positive solution $u$ to \eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \in \mathbb{R}^n$, i.e.~$u(x)=\mathcal{V}_{\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\R^n$, developed by themselves in bounded domains in \cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \cite{CGS}, where the authors classified all the solutions to \eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \in H^1_{loc}(\R^n)$ to \eqref{eq:Sobolev} is of the form \eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.

The situation in the case $p \neq 2$ is much more involved, since the approach with Kelvin transformation is not  applicable for this equation. Under the finite energy assumption, i.e.~$u \in \mathcal{D}^{1,p}(\R^n)$, the classification of all the positive solutions to \eqref{eq:Sobolev}  has been in fact an open and challenging problem recently solved in a series of papers by Damascelli, Merch\'an, Montoro and Sciunzi~\cite{DMMS} ($2n/(n+2)<p<2$), V\'etois~\cite{Vet} ($1<p<2$) and Sciunzi~\cite{Sciu16} ($2<p<n$). In particular, in these articles it was proved that any positive solution to \eqref{eq:Sobolev} that belongs to $\mathcal D^{1,p}(\R^n)$ is of the form \eqref{AubinTalenti}. More recently, Ciraolo, Figalli and Roncoroni~\cite{CFR} used a strategy based on integral estimates to extend the classification of positive $\mathcal D^{1,p}$-solutions to a class of anisotropic $p$-Laplace-type equations in convex cones. A similar approach was used in \cite{CatMonRon} to obtain new classification results for positive, weak solutions to \eqref{eq:Sobolev} which are not a priori in $\mathcal D^{1,p}(\R^n)$. In particular, in \cite{CatMonRon}, the authors managed to obtain the complete classification of positive, weak solutions to \eqref{eq:Sobolev} in the case where $n=2$ and $1<p<2$ or $n=3$ and $3/2<p<2$. The method was recently improved by \cite{Ou} who managed to extend this result to the case where $n\geq 3$ and $p\geq (n+1)/3$ and by Vet\'ois~\cite{Vet2} for $n\geq 4$ and $p>p_n$ for some number $p_n\in (n/3,(n+1)/3)$ such that $p_n\sim n/3+ 1/n$.

Now, we focus our attention on the case $\gamma \neq 0$ and $\nu = 0$; in this setting, system \eqref{P} reduces to the Hardy-Sobolev doubly critical equation
\begin{equation}\label{eq:HardySobolev}
\begin{cases}
-\Delta_p u = \gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \text{in } \mathbb{R}^n \\
u > 0 & \text{in } \mathbb{R}^n \setminus \{0\}.
\end{cases}
\end{equation}
Let us now turn to the case $p=2$ so that $\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of  the equation \eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \eqref{eq:HardySobolev}, given by
\begin{equation}\label{eq:terraciniane}
	\mathcal{U}_\lambda(x)=\lambda^{\frac{2-n}{2}}\mathcal{U}\left(\frac{x}{\lambda}\right) \qquad \mbox{ with } \qquad \mathcal{U}(x)= \dfrac{A(n,\gamma)}{|x|^{\mu}\left(1+|x|^{2-\frac{4\mu}{n-2}}\right)^{\frac{n-2}{2}}},
\end{equation}
where
\begin{equation*}
	\mu:=\frac{n-2}{2}-\sqrt{\left( \frac{n-2}{2}\right)^2-\gamma}, \qquad A(n,\gamma):=\left(\frac{n(n-2-2\mu)^2}{n-2}\right)^{\frac{n-2}{4}},
\end{equation*}
and $\lambda>0$ is a scaling factor. Obviously, $\mathcal{U}_\lambda=\mathcal{V}_{\lambda,0}$ if $\gamma=0$.

The case of $p$-Lapalce operator was firstly treated in \cite{abdellaoui}, where the authors studied the existence of solutions to \eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \cite{OSV} the authors proved that each solution to \eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution  it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \emph{radial} solutions as well as  the asymptotic behavior are proved showing in particular that, a \emph{radial} solution  \eqref{eq:HardySobolev}, satisfy the following properties:
\begin{equation*}
    \begin{split}
        &\mathcal{U}_{p,\lambda}(x)= \lambda^\frac{ p-n}{p} \mathcal{U}_p\left(\frac{x}{\lambda}\right) \quad \text{ with } \quad \mathcal U_p(x)= u(r)\\
        &\lim_{r\to0^+} r^{\mu_1} \mathcal U_p(r)=C_1,\qquad \lim_{r\to+\infty}r^{\mu_2} \mathcal U_p(r)=C_2,\\
        &\lim_{r\to0^+} r^{\mu_1+1} |\mathcal U_p'(r)|=C_1\mu_1,\qquad \lim_{r\to+\infty}r^{\mu_2+1}|\mathcal{U}_p'(r)|=C_2\mu_2,
    \end{split}
\end{equation*}
for some positive constants $C_1, C_2$. {Obviously, $\mathcal{U}_{p,\lambda}=\mathcal{U}_{\lambda}$ and $\mathcal{U}_{p}=\mathcal{U}$ if $p=2$.} Here and in the sequel $\mu_1, \mu_2\in [0, +\infty)$, $\mu_1<\mu_2$ are two roots of the equation
\begin{equation*}
\mu^{p-2}\left[(p-1)\mu^2-(n-p)\mu\right]+\gamma=0.
\end{equation*}
We remark (for later use) that
\begin{equation*}\label{relgammaj}
0\leqslant \mu_1<\frac{n-p}{p}<\mu_2\leqslant \frac{n-p}{p-1}.
\end{equation*}
Note that when $p=2$ then $\mu_{1}=\tau$ and $\mu_2=n-2-\tau$. Instead, when $p\neq 2$ but $\gamma=0$ then $\mu_1=0$ and $\mu_2=\frac{n-p}{p-1}$.\\

We now turn our attention to the study of \eqref{P}. Nonlinear Schrödinger problems, particularly those of Gross--Pitaevskii type, seems to have strong connections to various physical phenomena. These kind of systems appear naturally in the Hartree--Fock theory for double condensates specifically in binary mixtures of Bose Einstein condensates occupying different hyperfine states while spatially overlapping (see \cite{Esry} and \cite{Frantz} for comprehensive details).
The solitary-wave solutions to coupled Gross--Pitaevski equations satisfy the following system:
\begin{equation}\label{eq:pBSsystem}
	\begin{cases}
		-\Delta u + V(x) u = \mu u^{2q-1} + \nu u^{q-1} v^{q} & \text{in } \mathbb{R}^n \\
		-\Delta v + V(x) v = \mu v^{2q-1} + \nu u^{q}v^{q-1} & \text{in } \mathbb{R}^n,
	\end{cases}
\end{equation}
where $V$ represents the potential of the system and $1 < q \leq \frac{2^*}{2}$. This formulation is commonly referred to as the Bose--Einstein condensate system. For subcritical regimes, we refer to \cite{ambrosetti}, \cite{bartsch2}, \cite{lin1}, \cite{sirakov}, and \cite{nicola2} for existence and multiplicity results under various assumptions on $V$ and $\nu$.

In the critical case where $q = \frac{2^*}{2}$, if $V$ is a non-zero constant, system \eqref{eq:pBSsystem} admits only the trivial solution $(0,0)$, that is a consequence of the Pohozaev identity. When $V=0$, \cite{Wang} demonstrated the uniqueness of ground states under specific parameter conditions for generalized systems. Conversely, \cite{pistoia} examined the competitive setting ($\nu < 0$), establishing the existence of infinitely many fully nontrivial solutions that are not conformally equivalent.

In \cite{ELS}, the authors treated the case of Hardy-type potential $V = -\frac{\gamma}{|x|^2}$. With this choice of $V$, the authors were able to study problem \eqref{P} when $p=2$. What they have proved in \cite{ELS} is that any $(u,v) \in \mathcal{D}^{1,2}(\mathbb{R}^n) \times \mathcal{D}^{1,2}(\mathbb{R}^n)$ solution to \eqref{P} is of synchronized type:
\begin{equation*}
(u,v)=\left(c_1 \, \mathcal{U}_{\lambda_0}, c_2 \, \mathcal{U}_{\lambda_0}\right),
\end{equation*}
where $\mathcal{U}_\lambda$ is given in \eqref{eq:terraciniane}, $\lambda_0>0$ and $c_1,c_2$ are any positive constants satisfying the system
\begin{equation*}
		\left\{\begin{array}{ll}
			\displaystyle{c_1^{2^*-2}+\nu \alpha c_1^{\alpha-2}c_2^\beta}=1  & \vspace{.3cm}\\
			\displaystyle{c_2^{2^*-2}+\nu \beta c_1^{\alpha}c_2^{\beta-2}=1}. &
		\end{array}\right.
\end{equation*}
This result was new also in the case  $\gamma=0$.  Under that assumption the explicit solutions in \eqref{eq:terraciniane} reduce to those of \eqref{AubinTalenti}.\\

Doubly critical problems have received significant attention in recent years. The pioneering work of \cite{AFP} investigated general Hardy-Sobolev type systems making use of variational techniques. In cooperative regimes ($\nu > 0$), they established the existence of ground and bound states contingent on parameters $\alpha$, $\beta$, dimension $n$, and a potential function $h$ in the coupling term. We also refer to \cite{ChenZou3, EduRafaAle} for further existence results in the critical regime.

The primary objective of this paper is to show that any positive solutions, with finite energy, to problem \eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \cite{ELS}.

Thus our main result is given by the following:

Our main objective is to prove radial symmetry of solutions to system \eqref{P}. When $\gamma = \nu = 0$, the system reduces to the standard $p$-Sobolev critical equation
\begin{equation}\label{eq:Sobolev}
 \begin{cases}
 \displaystyle -\Delta_p u\,= u^{p^*-1} & \text{in}\quad\R^n\\
 u > 0 & \text{in}\quad\R^n.
 \end{cases}
\end{equation}
It is well-known that the Aubin-Talenti bubbles, defined as
\begin{equation}\label{AubinTalenti}
\mathcal{V}_{\lambda,x_0}(x) := \left[\frac{\lambda^{\frac{1}{p-1}}\sqrt[p]{n\left(\frac{n-p}{p-1}\right)^{p-1}}}{\lambda^{\frac{p}{p-1}} + |x - x_0|^{\frac{p}{p-1}}}\right]^{\frac{n-p}{p}}
\end{equation}
is a family of positive radial solutions to \eqref{eq:Sobolev}, with $\lambda > 0$, and $x_0 \in \mathbb{R}^n$. These functions achieve equality in the Sobolev inequality in $\mathbb{R}^n$. The situation in case $p=2$ is fully understood. In fact, in \cite{GNN2}, it has been shown that any positive solution $u$ to \eqref{eq:Sobolev} that satisfies the decay assumption $u(x) = \mathcal{O}(1/|x|^m)$ at infinity for $m > 0$ is radially symmetric and decreasing around some point $x_0 \in \mathbb{R}^n$, i.e.~$u(x)=\mathcal{V}_{\lambda,x_0}(x)$ (with $p=2$). The authors used a refined version of the moving planes technique taking into account the lack of compactness in the whole space $\R^n$, developed by themselves in bounded domains in \cite{GNN}. The problem was completely solved in the seminal paper by Caffarelli, Gidas and Spruck \cite{CGS}, where the authors classified all the solutions to \eqref{eq:Sobolev} with $p=2$ and without an apriori finite energy assumption. Exploiting the Kelvin transformation, they showed that the moving planes procedure can start. Finally, they proved that any solution $u \in H^1_{loc}(\R^n)$ to \eqref{eq:Sobolev} is of the form \eqref{AubinTalenti}, hence solutions are unique up to rescaling and translations.

Now, we focus our attention on the case $\gamma \neq 0$ and $\nu = 0$; in this setting, system \eqref{P} reduces to the Hardy-Sobolev doubly critical equation
\begin{equation}\label{eq:HardySobolev}
\begin{cases}
-\Delta_p u = \gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \text{in } \mathbb{R}^n \\
u > 0 & \text{in } \mathbb{R}^n \setminus \{0\}.
\end{cases}
\end{equation}
Let us now turn to the case $p=2$ so that $\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \eqref{eq:HardySobolev}, given by
\begin{equation}\label{eq:terraciniane}
 \mathcal{U}_\lambda(x)=\lambda^{\frac{2-n}{2}}\mathcal{U}\left(\frac{x}{\lambda}\right) \qquad \mbox{ with } \qquad \mathcal{U}(x)= \dfrac{A(n,\gamma)}{|x|^{\mu}\left(1+|x|^{2-\frac{4\mu}{n-2}}\right)^{\frac{n-2}{2}}},
\end{equation}
where
\begin{equation*}
 \mu:=\frac{n-2}{2}-\sqrt{\left( \frac{n-2}{2}\right)^2-\gamma}, \qquad A(n,\gamma):=\left(\frac{n(n-2-2\mu)^2}{n-2}\right)^{\frac{n-2}{4}},
\end{equation*}
and $\lambda>0$ is a scaling factor. Obviously, $\mathcal{U}_\lambda=\mathcal{V}_{\lambda,0}$ if $\gamma=0$.

The case of $p$-Lapalce operator was firstly treated in \cite{abdellaoui}, where the authors studied the existence of solutions to \eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \cite{OSV} the authors proved that each solution to \eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \emph{radial} solution \eqref{eq:HardySobolev}, satisfy the following properties:
\begin{equation*}
 \begin{split}
 &\mathcal{U}_{p,\lambda}(x)= \lambda^\frac{ p-n}{p} \mathcal{U}_p\left(\frac{x}{\lambda}\right) \quad \text{ with } \quad \mathcal U_p(x)= u(r)\\
 &\lim_{r\to0^+} r^{\mu_1} \mathcal U_p(r)=C_1,\qquad \lim_{r\to+\infty}r^{\mu_2} \mathcal U_p(r)=C_2,\\
 &\lim_{r\to0^+} r^{\mu_1+1} |\mathcal U_p'(r)|=C_1\mu_1,\qquad \lim_{r\to+\infty}r^{\mu_2+1}|\mathcal{U}_p'(r)|=C_2\mu_2,
 \end{split}
\end{equation*}
for some positive constants $C_1, C_2$. {Obviously, $\mathcal{U}_{p,\lambda}=\mathcal{U}_{\lambda}$ and $\mathcal{U}_{p}=\mathcal{U}$ if $p=2$.} Here and in the sequel $\mu_1, \mu_2\in [0, +\infty)$, $\mu_1<\mu_2$ are two roots of the equation
\begin{equation*}
\mu^{p-2}\left[(p-1)\mu^2-(n-p)\mu\right]+\gamma=0.
\end{equation*}
We remark (for later use) that
\begin{equation*}\label{relgammaj}
0\leqslant \mu_1<\frac{n-p}{p}<\mu_2\leqslant \frac{n-p}{p-1}.
\end{equation*}
Note that when $p=2$ then $\mu_{1}=\tau$ and $\mu_2=n-2-\tau$. Instead, when $p\neq 2$ but $\gamma=0$ then $\mu_1=0$ and $\mu_2=\frac{n-p}{p-1}$.\\

The primary objective of this paper is to show that any positive solutions, with finite energy, to problem \eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \cite{ELS}.

Thus our main result is given by the following:

Now, we focus our attention on the case $\gamma \neq 0$ and $\nu = 0$; in this setting, system \eqref{P} reduces to the Hardy-Sobolev doubly critical equation
\begin{equation}\label{eq:HardySobolev}
\begin{cases}
-\Delta_p u = \gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1} & \text{in } \mathbb{R}^n \\
u > 0 & \text{in } \mathbb{R}^n \setminus \{0\}.
\end{cases}
\end{equation}
Let us now turn to the case $p=2$ so that $\Lambda_{n,2}$ is the best constant in the Hardy-Sobolev inequality for $p=2$. The main contribution in the semilinear case is due to Terracini in \cite{terracini}, where the author, by means of variational arguments and the concentration compactness principle, showed the existence of solutions to a more general version of the equation \eqref{eq:HardySobolev}. Making use of the Kelvin transformation and of the moving plane technique, she was able to prove that any positive solution with finite energy to \eqref{eq:HardySobolev} is radially symmetric about the origin. Finally, thanks to a detailed ODE's analysis, she gave a complete classification of the solutions to \eqref{eq:HardySobolev}, given by
\begin{equation}\label{eq:terraciniane}
 \mathcal{U}_\lambda(x)=\lambda^{\frac{2-n}{2}}\mathcal{U}\left(\frac{x}{\lambda}\right) \qquad \mbox{ with } \qquad \mathcal{U}(x)= \dfrac{A(n,\gamma)}{|x|^{\mu}\left(1+|x|^{2-\frac{4\mu}{n-2}}\right)^{\frac{n-2}{2}}},
\end{equation}
where
\begin{equation*}
 \mu:=\frac{n-2}{2}-\sqrt{\left( \frac{n-2}{2}\right)^2-\gamma}, \qquad A(n,\gamma):=\left(\frac{n(n-2-2\mu)^2}{n-2}\right)^{\frac{n-2}{4}},
\end{equation*}
and $\lambda>0$ is a scaling factor. Obviously, $\mathcal{U}_\lambda=\mathcal{V}_{\lambda,0}$ if $\gamma=0$.

The case of $p$-Lapalce operator was firstly treated in \cite{abdellaoui}, where the authors studied the existence of solutions to \eqref{eq:HardySobolev} and a very fine ODE's analysis was carried out. The radial symmetry of the solutions there was an assumption. In \cite{OSV} the authors proved that each solution to \eqref{eq:HardySobolev} with finite energy is radially symmetric (and radially decreasing) about the origin. Having in force the radial symmetry of the solution it is easy to derive the ordinary differential equation associated to the PDE and fulfilled by the solution $u = u(r)$ and, hence, to apply the results in \cite{abdellaoui}. In particular, they proved also uniqueness up to scaling of the \emph{radial} solutions as well as the asymptotic behavior are proved showing in particular that, a \emph{radial} solution \eqref{eq:HardySobolev}, satisfy the following properties:
\begin{equation*}
 \begin{split}
 &\mathcal{U}_{p,\lambda}(x)= \lambda^\frac{ p-n}{p} \mathcal{U}_p\left(\frac{x}{\lambda}\right) \quad \text{ with } \quad \mathcal U_p(x)= u(r)\\
 &\lim_{r\to0^+} r^{\mu_1} \mathcal U_p(r)=C_1,\qquad \lim_{r\to+\infty}r^{\mu_2} \mathcal U_p(r)=C_2,\\
 &\lim_{r\to0^+} r^{\mu_1+1} |\mathcal U_p'(r)|=C_1\mu_1,\qquad \lim_{r\to+\infty}r^{\mu_2+1}|\mathcal{U}_p'(r)|=C_2\mu_2,
 \end{split}
\end{equation*}
for some positive constants $C_1, C_2$. {Obviously, $\mathcal{U}_{p,\lambda}=\mathcal{U}_{\lambda}$ and $\mathcal{U}_{p}=\mathcal{U}$ if $p=2$.} Here and in the sequel $\mu_1, \mu_2\in [0, +\infty)$, $\mu_1<\mu_2$ are two roots of the equation
\begin{equation*}
\mu^{p-2}\left[(p-1)\mu^2-(n-p)\mu\right]+\gamma=0.
\end{equation*}
We remark (for later use) that
\begin{equation*}\label{relgammaj}
0\le \mu_1<\frac{n-p}{p}<\mu_2\le \frac{n-p}{p-1}.
\end{equation*}
Note that when $p=2$ then $\mu_{1}=\tau$ and $\mu_2=n-2-\tau$. Instead, when $p\neq 2$ but $\gamma=0$ then $\mu_1=0$ and $\mu_2=\frac{n-p}{p-1}$.\\

The primary objective of this paper is to show that any positive solutions, with finite energy, to problem \eqref{P} is radially symmetric and radially decreasing around the origin, in the same spirit of \cite{ELS}.

\begin{rem}
This result holds (and is new) also in the case $\gamma=0$.
\end{rem}

Now, we state the upper and lower estimates of the solutions and the upper one of their gradient, contained in \cite{KL20}, in our case.
\begin{thm} \label{thm:asymptEst}$($Theorem $1.1$, \cite{KL20}$)$
 Let $(u,v) \in \mathcal{D}^{1,p}(\R^n)\times \mathcal{D}^{1,p}(\R^n)$ be a weak solution of \eqref{P}
 with $1<p<n$, $0 \leq \gamma < \Lambda_{n,p}$, $\alpha+\beta=p^*$. Then there exist positive constants $0 < R_0< 1 < R_1$ depending on $n, p, \gamma, \nu, \alpha,\beta$, $u$ and $v$, such that
 \begin{equation}\label{eq:estat0}
 \frac{c_1^u}{|x|^{\mu_1}} \leq u(x) \leq \frac{C_1^u}{|x|^{\mu_1}},\quad \frac{c_1^v}{|x|^{\mu_1}} \leq v(x) \leq \frac{C_1^v}{|x|^{\mu_1}} \qquad x \in B_{R_0}\setminus\{0\},
 \end{equation}
 and
 \begin{equation}\label{eq:estatInf}
 \frac{c_2^u}{|x|^{\mu_2}} \leq u(x) \leq \frac{C_2^u}{|x|^{\mu_2}},\quad \frac{c_2^v}{|x|^{\mu_2}} \leq v(x) \leq \frac{C_2^v}{|x|^{\mu_2}} \qquad x \in (B_{R_1})^c,
 \end{equation}

\begin{thm}\label{stimagradiente1}$($Theorem $1.3$, \cite{KL20}$)$
 Let $(u,v) \in \mathcal{D}^{1,p}(\R^n)\times \mathcal{D}^{1,p}(\R^n)$ be a weak solution of \eqref{P}
 with $1<p<n$, $0 \leq \gamma < \Lambda_{n,p}$ and $\alpha+\beta=p^*$. Then there exist positive constants $\tilde C^u$, $\tilde C^v$ depending on $n,p, \gamma, \nu, \alpha,\beta$ and $u$ or $v$, respectively, such that
 \begin{equation}\label{eq:estat0grada}
 |\nabla u(x)| \leq \frac{\tilde C^u}{|x|^{\mu_1+1}}, \quad |\nabla v(x)| \leq \frac{\tilde C^v}{|x|^{\mu_1+1}} \qquad x \in B_{R_0},
 \end{equation}
 and
 \begin{equation}\label{eq:estatInfgrada}
 |\nabla u(x)| \leq \frac{\tilde C^u}{|x|^{\mu_2+1}}, \quad |\nabla v(x)| \leq \frac{\tilde C^v}{|x|^{\mu_2+1}} \qquad x \in (B_{R_1})^c,
 \end{equation}
where $\mu_1, \mu_2$ are roots of \eqref{eq:algebraicHardy} as in Theorem \ref{thm:asymptEst}, and $0 < R_0< 1 < R_1$ are constants depending on $n, p, \gamma,\nu, \alpha,\beta$, $u$ and $v$.
\end{thm}

\begin{thm}\label{stimagradiente2}
 Let $(u,v) \in \mathcal{D}^{1,p}(\R^n)\times \mathcal{D}^{1,p}(\R^n)$ be a weak solution of \eqref{P}
 with $1<p<n$, $0 \leq \gamma < \Lambda_{n,p}$ and $\alpha+\beta=p^*$. Then there exist positive constants $\tilde c^u$, $\tilde c^v$ depending on $n,p, \gamma, \nu, \alpha,\beta$ and $u$ or $v$, respectively, such that
 \begin{equation}\label{eq:estat0gradb}
 |\nabla u(x)| \geq \frac{\tilde c^u}{|x|^{\mu_1+1}}, \quad |\nabla v(x)| \geq \frac{\tilde c^v}{|x|^{\mu_1+1}} \qquad x \in B_{R_2},
 \end{equation}
 and
 \begin{equation}\label{eq:estatInfgradb}
 |\nabla u(x)| \geq \frac{\tilde C^u}{|x|^{\mu_2+1}}, \quad |\nabla v(x)| \geq \frac{\tilde C^v}{|x|^{\mu_2+1}} \qquad x \in (B_{R_3})^c,
 \end{equation}
where $\mu_1, \mu_2$ are roots of \eqref{eq:algebraicHardy} as in Theorem \ref{thm:asymptEst}, and $0 < R_2< 1 < R_3$ are constants depending on $n, p, \gamma,\nu, \alpha,\beta$, $u$ and $v$.
\end{thm}

\begin{rem}
This result holds (and is new)  also in the case $\gamma=0$.
\end{rem}

The proof of the radial symmetry of the solutions is based on a fine adaptation of the moving plane procedure for quasilinear elliptic problems. This procedure goes back to the seminal papers of Alexandrov and Serrin \cite{A, BN, GNN, serrin}. The situation in the quasilinear case is much more involved, and in the same spirit of \cite{OSV, Sciu16} we first recover asymptotic estimates for the solutions and its gradient. Part of this estimate are due to \cite{KL20}. The missing part is proved in this paper, and it was inspired by \cite{EMSV}.
Regarding elliptic systems, the moving plane technique was adapted by Troy in \cite{troy}, where the cooperative case is analyzed. The procedure was also applied for semilinear systems in the half-space in \cite{Dan} and in the whole space by Busca and Sirakov in \cite{busca}. There are several results on the qualitative properties of solutions for cooperative elliptic systems; in particular, we refer to \cite{esposito, ELS} for a complete analysis in the semilinear case and to \cite{BEMV, EMM} for the quasilinear case.

The proof of the radial symmetry of the solutions is “based on a fine adaptation of the moving plane procedure for quasilinear elliptic problems,” going back to Alexandrov and Serrin. Since “the situation in the quasilinear case is much more involved,” the approach (in the spirit of \cite{OSV, Sciu16}) is to “first recover asymptotic estimates for the solutions and its gradient”; part of these estimates are due to \cite{KL20}, while “the missing part is proved in this paper,” inspired by \cite{EMSV}. For systems, it relies on the moving plane technique as adapted to elliptic systems (starting with Troy \cite{troy} in the cooperative case, and subsequent adaptations in \cite{Dan, busca}).,

\label{thm:symmetry}
	Let $(u,v) \in \mathcal{D}^{1,p}(\mathbb{R}^n) \times \mathcal{D}^{1,p} (\mathbb{R}^n)$ be a solution to
\begin{equation}\tag{$\mathcal{S}^*$}\label{P}
\begin{cases}
\displaystyle -\Delta_p u\,=\gamma \frac{u^{p-1}}{|x|^p} + u^{p^*-1}+ \nu \alpha u^{\alpha-1} v^\beta & \text{in}\quad\R^n \vspace{0.2cm} \\
\displaystyle -\Delta_p v\,=\gamma \frac{v^{p-1}}{|x|^p} + v^{p^*-1}+ \nu \beta u^\alpha v^{\beta-1} & \text{in}\quad\R^n \vspace{0.2cm} \\
u,v > 0 &  \text{in}\quad\R^n \setminus \{0\}\\
u,v \in \mathcal D^{1,p}(\R^n),
\end{cases}
\end{equation}
Then $(u,v)$ is radially symmetric and radially decreasing around the origin.,

Question:
Let $1<p<n$, let $p^*:=\frac{np}{n-p}$, and let $\Delta_p w:=\operatorname{div}(|\nabla w|^{p-2}\nabla w)$ be the $p$-Laplace operator. Consider a pair $(u,v)\in \mathcal D^{1,p}(\mathbb R^n)\times \mathcal D^{1,p}(\mathbb R^n)$, where $\mathcal D^{1,p}(\mathbb R^n)$ is the usual homogeneous Sobolev space, such that
\[
\begin{cases}
-\Delta_p u=\gamma \dfrac{u^{p-1}}{|x|^p}+u^{p^*-1}+\nu\alpha u^{\alpha-1}v^{\beta} & \text{in }\mathbb R^n,\\[0.4em]
-\Delta_p v=\gamma \dfrac{v^{p-1}}{|x|^p}+v^{p^*-1}+\nu\beta u^{\alpha}v^{\beta-1} & \text{in }\mathbb R^n,\\[0.4em]
u,v>0 & \text{in }\mathbb R^n\setminus\{0\}.
\end{cases}
\]
Which statement holds for every such solution pair $(u,v)$?

Both components are radially symmetric about the origin and radially decreasing; equivalently, there exist functions $U,V:(0,\infty)\to(0,\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\neq 0$, and $U$ and $V$ decrease as $|x|$ increases.

Choices:
B. Both components are radially symmetric and radially decreasing with respect to the same center $x_0\in\mathbb R^n$; equivalently, there exist $x_0\in\mathbb R^n$ and functions $U,V:(0,\infty)\to(0,\infty)$ such that $u(x)=U(|x-x_0|)$ and $v(x)=V(|x-x_0|)$ for all $x\neq x_0$, and $U$ and $V$ decrease as $|x-x_0|$ increases.
C. Both components are radially symmetric about the origin; equivalently, there exist functions $U,V:(0,\infty)\to(0,\infty)$ such that $u(x)=U(|x|)$ and $v(x)=V(|x|)$ for all $x\neq 0$.
D. There exist positive constants $R_0<1<R_1$ such that both components are radially symmetric and radially decreasing on $B_{R_0}\setminus\{0\}$ and on $(B_{R_1})^c$, respectively; equivalently, each of $u$ and $v$ depends only on $|x|$ near the origin and near infinity, but no global symmetry conclusion is forced on all of $\mathbb R^n\setminus\{0\}$.
E. Both components are radially symmetric about the origin, and there exists a constant $C>0$, depending only on $n,p,\gamma,\nu,\alpha,\beta$, such that $|\nabla u(x)|+|\nabla v(x)|\le C\big(|x|^{-\mu_1-1}+|x|^{-\mu_2-1}\big)$ for every $x\in\mathbb R^n\setminus\{0\}$; in particular, both are radially decreasing.

forced center at the origin due to the Hardy singularity

dropped radial monotonicity while keeping symmetry about the origin

global moving-plane conclusion replaced by only local symmetry near 0 and infinity

solution-dependent asymptotic/gradient constants made uniform and global

The stem does not state the symmetry/monotonicity conclusion, nor does it directly signal which option is correct. The Hardy singularity at the origin is relevant context, but it is not an explicit giveaway.

The item is close to a theorem-recall question: it asks for the exact qualitative conclusion satisfied by every solution pair. However, it is not a pure restatement, since the options vary in center of symmetry, monotonicity, locality vs. globality, and extra quantitative estimates.

Some reasoning is required to distinguish the strongest valid conclusion from nearby alternatives, especially between origin-centered symmetry, arbitrary-center symmetry, and symmetry without monotonicity. Still, for someone who knows the result, the task is mainly recognition rather than deep derivation.

The distractors are plausible and mathematically targeted: one weakens the true claim, one shifts the symmetry center, one reduces global to local symmetry, and one adds an unjustified uniform gradient bound. These align with realistic failure modes.

A solid theorem-understanding MCQ with strong distractors and little answer leakage, though it leans more toward theorem recognition than genuinely generative reasoning.


\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, $n\geq 1$, with smooth boundary $\partial\Omega$, and
$\Omega_e := \mathbb{R}^n\setminus \overline\Omega$ be the exterior of $\Omega$.
Let $m\geq 0$ be an integer and $s\in \mathbb{R}^+\setminus \mathbb{Z}$, where $\mathbb{R}^+:=(0,\infty)$ and $\mathbb{Z}$ denotes the set of all integers in $\mathbb{R}$.
We consider inverse problems for the nonlinear fractional Schr\"odinger equation (NLFSE):
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\
u = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the fractional Laplacian $(-\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\Delta)^su: = \mathcal{F}^{-1}(|\cdot|^{2s}\mathcal{F}(u))$. Here $\mathcal{F} (u)$ represents the Fourier transform of $u$.
The local nonlinear perturbation $\mathbf{P} (u)$ is defined as
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
where $K\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form
$$
P_k(x,D):=\sum_{\sigma: \,|\sigma|\leq m}a_{\sigma,k}(x)D^\sigma,\quad k=1,\ldots,K-1,
$$
with scalar coefficients $a_{\sigma,k}$ depending only on $x$. Here $\sigma=(\sigma_1,\ldots,\sigma_n)$ is multi-index of length $|\sigma|=\sigma_1+\ldots+\sigma_n=\ell$ for $\ell\leq m$, and therefore $D^\sigma u := \frac{\partial^{\sigma_1}}{\partial x_1^{\sigma_1}}\ldots\frac{\partial^{\sigma_n}}{\partial x_n^{\sigma_n}} u$. \\

We show in Theorem \ref{thm_wellposdness} that \eqref{eq_1_IBVP} is well-posed in $H^s(\mathbb{R}^n)$ for any data $f$ in a small ball $\mathcal{X}_{\varepsilon_0}(\Omega_e)$ in $C^\infty_c(\Omega_e)$, where for any open set $O\subset \mathbb R^n$ we define
\[
\mathcal{X}_{\varepsilon_0}(O):=\{f\in C^\infty_c(O):\,\|f\|_{C^\infty_c(O)}\le\varepsilon_0\}.
\]
This well-posedness result for \eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map
$$
\Lambda_{\mathbf{P}}: H^{s}(\Omega_e)\rightarrow (H^{-s}(\Omega_e))^*,\quad f\mapsto (-\Delta)^s u_f|_{\Omega_e}.
$$

In this framework, we ask the following inverse problem: Does the map $\Lambda_{\mathbf{P}}$ uniquely determine the nonlinearity $\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\ref{thm_main}:

\begin{theorem}\label{thm_main}(Unique determination)
Let $\Omega\subset\R^n$ be a bounded open set with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$, $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.
\end{theorem}

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem, known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Because the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.
\begin{prop}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then the DN-map $\Lambda_\mathbf{P}$ is well-defined and bounded.
\end{prop}
\begin{proof}
We first show that $\Lambda_\mathbf{P}$ only depends on the equivalence classes. Let $\phi,\,\psi\in \widetilde{H}^s(\Omega)$. Since $u_f$ and $u_{f+\phi}$ both solve problem \eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \eqref{eq_1_IBVP} implies $u_f = u_{f+\phi}$.
By the linearity of $B_\mathbf{P}$ in the second component, we have
\[
B_\mathbf{P}(u_{f+\phi},v+\psi) = B_\mathbf{P}(u_f, v+\psi) = B_\mathbf{P}(u_f, v)+ B_\mathbf{P}(u_f, \psi).
\]
Using the fact that $\psi = 0$ in $\Omega_e$ and $u_f$ solves \eqref{eq_1_IBVP}, we get $B_\mathbf{P}(u_f,\psi) = 0$. This proves that $\LA \Lambda_\mathbf{P}(f+\phi), v+\psi \RA =\LA \Lambda_\mathbf{P}f,v\RA $, and thus $\Lambda_\mathbf{P}$ is well-defined.

\subsection{Linearization}\label{sec_linearization}
For $f=(f_1,\ldots,f_K)$ with $f_\ell\in C^\infty_c(\Omega_e)$, $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$, Theorem \ref{thm_wellposdness} yields that, for sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ with $|\varepsilon|<\varepsilon_0$, there exists a unique solution $u_\varepsilon \in H^s(\R^n)\cap C^s(\R^n)$ of the following problem
\begin{equation}\label{eq_3_linearization}
\begin{cases}
[(-\Delta)^s + q(x)]u_\varepsilon =- \mathbf{P}(u_\varepsilon) &\hbox{ in } \Omega,\\
u_\varepsilon = \sum_{\ell=1}^K \varepsilon_\ell f_\ell=:\varepsilon\cdot f &\hbox{ in } \Omega_e.
\end{cases}
\end{equation}
Moreover, we have
\begin{align}\label{eq_3_linear_u_est}
\|u_\varepsilon\|_{H^{s}(\R^n)}+
\|u_\varepsilon\|_{C^{s}(\R^n)}\le C\|\varepsilon\cdot f\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|\sum_{\ell=1}^K\|f_\ell\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|,
\end{align}
where the constant $C > 0$ is independent of $u_\varepsilon$ and $f_\ell$. We will expand the solution $u_\varepsilon$ in terms of the small parameter $\varepsilon$. Let $\alpha=(\ell_1,\ldots,\ell_K)\in \mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \ref{prop_wellposedness_linear} and \ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\alpha}\in H^s(\R^n)\cap C^{s}(\R^n)$ to the linear fractional Schr\"odinger equation
\begin{equation}
\label{eq_3_linear_w_3}
\begin{cases}
[(-\Delta)^s + q(x)]w_{\alpha} =- \mathcal{T}_{\alpha} &\hbox{ in } \Omega,\\
w_{\alpha} = D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0} &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the inhomogeneous term is
\begin{equation}\label{eq_3_def_T}
\begin{aligned}
\mathcal{T}_{\alpha} :=
\sum^{|\alpha|-1}_{\ell=1} \sum_{\substack{ \beta_1,\ldots, \beta_\ell\in \mathbb{N}^K_0\setminus \{0,\cdots,0\} : \\ \beta:= \sum_{j=1}^\ell\beta_j < \alpha }}\binom{\alpha}{\beta} \binom{\beta}{\beta_1,\cdots, \beta_\ell}\bigg( \prod_{j=1}^\ell w_{\beta_j } \bigg) P_\ell(x,D)w_{\alpha-\beta}.
\end{aligned}
\end{equation}
Note here $\mathcal{T}_{\alpha} = 0$ when $|\alpha| = 0,\, 1$.
Furthermore, the solution satisfies the stability estimate
\begin{equation}\label{eq_3_linear_w3_est}
\begin{aligned}
\|w_\alpha\|_{C^{s}(\R^n)}+ \|w_\alpha\|_{H^{s}(\R^n)}
\le
\begin{cases}
C\|D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0}\|_{C^\infty_c(\Omega_e)}&|\alpha|=1,\\
C\|\mathcal{T}_\alpha\|_{L^\infty(\Omega)},&|\alpha|\geq 2.
\end{cases}
\end{aligned}
\end{equation}
Here we denote
$$
\binom{\alpha}{\beta} = \frac{\alpha !}{\beta !(\alpha-\beta)!} \quad \hbox{and} \quad
\binom{\beta}{\beta_1, \cdots, \beta_k}= \frac{\beta!}{\beta_1! \cdots\beta_k!}.
$$
Observe that the definition of $\mathcal T_\alpha$ only requires knowledge of those $w_\beta$ with $\beta<\alpha$, and that in turn $w_\alpha$ can be computed from $\mathcal T_\alpha$ as the solution of \eqref{eq_3_linear_w_3}. Thus the definitions of $\mathcal T_\alpha$ and $w_\alpha$ are not circular.

By using the derivatives given above, and the fact $u_\varepsilon=0$ when $\varepsilon=0$, we obtain the following lemma through direct computation.
\begin{lemma}\label{lamma_DN_higher}
Let $f=(f_1,\ldots,f_K)$, where $f_\ell\in C^\infty_c(\Omega_e)$ satisfies $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$. For sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ such that $|\varepsilon|<\varepsilon_0$, we have for $|\alpha|\ge 1$
that
\begin{align*}
\LA \frac{\p^{|\alpha|}}{\p \varepsilon^\alpha} \Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA = \int_{\R^n} (-\Delta)^{\frac{s}{2}}w_\alpha (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)w_\alpha\, g \dx+
\int_{\Omega}\mathcal{T}_{\alpha}g \dx, \quad\hbox{for }g\in H^s(\R^n).
\end{align*}
Notice that $\mathcal T_\alpha \equiv 0$ when $|\alpha|=1$.
\end{lemma}
\begin{proof}
We first show the first-order derivative of the DN-map.
For any $g\in H^s(\R^n)$, by the definition of $\Lambda_\mathbf{P}$, taking the derivatives on the bilinear form yields that
\begin{align*}
\LA \p _{\varepsilon_\ell}\big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g\RA =\p_{\varepsilon_{\ell}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g) = \int_{\R^n}(-\Delta)^{\frac{s}{2}}v_\ell (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)v_\ell\, g \dx.
\end{align*}
Note that all nonlinear terms vanish as $u_\varepsilon=0$ when $\varepsilon=0$.
Similarly, let $\alpha=e_{\ell_1}+e_{\ell_2}$, for second-order partial derivatives of $\Lambda_\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\mathbf{P}$ remain after taking derivatives, we get
\begin{align*}
\LA \p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA &=\p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g)\\
&=\int_{\R^n} (-\Delta)^{\frac{s}{2}}w_{\alpha} (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x) w_{\alpha}\, g \dx\\
&\quad
+\int_{\Omega} \LC v_{\ell_1}P_1(x,D)v_{\ell_2} + v_{\ell_2}P_1(x,D)v_{\ell_1} \RC g \dx.
\end{align*}
Following a similar argument, we can also derive the integral representation of
$\frac{\p^{|\alpha|}}{\p \varepsilon^\alpha}
\Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f)$ for $|\alpha|\geq 3$.
\end{proof}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem,  known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Besides the mathematical interest, the fractional Calder\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \cite{CL24,V09}. A novel application is that of the recent paper \cite{LNOS25}, which considers an inverse problem of fractional Calder\'on type related to quantum field theory. Many aspects of the fractional Calder\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \cite{RS20a} and anisotropic background metrics \cite{GLX17}, reconstruction for a single measurement \cite{GRSU20} and by monotonicity methods \cite{ HL20, HL19}, the stability and instability results of \cite{ BCR25,RS18, RS20b}, general anisotropic metrics \cite{Fei24, FGKRSU25, FGKU21},
and boundary reconstruction \cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \cite{FKU24, FL24} and poly-parabolic operators \cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \cite{S17} and to the recent book \cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\'on problem has been studied also in its conductivity formulation, as shown in \cite{Cov20a} and the subsequent works \cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \cite{CdHS25}, and the fractional wave equation \cite{KLW22, LZ25}, among many others \cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\'on problems have also attracted substantial attention \cite{BCR25, BR25, CGRU23, R23}.

The two distinctive features of our equation \eqref{eq_1_IBVP}  include the \textit{higher-order nonlocal leading operator} and \textit{general local nonlinear perturbations} of the fractional Schr\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\"odinger equation. As such, it differs substantially from the perturbations considered in \cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\"odinger equation were studied.

The consideration of the higher-order fractional Laplace operator $(-\Delta)^s$ with $s\in \mathbb R^+\setminus\mathbb Z$ in this paper is inspired by the work \cite{CMRU22}, which investigated the unique determination of a general linear term.
Indeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\"odinger type, and \cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian.

Meanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \cite{Isakov93} is to perform first-order linearization of the DN-map.
Lately, the higher-order linearization method \cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.
The interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\"odinger equation has been studied in \cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \cite{J25}. The fractional Schr\"odinger \cite{KMS23, KMSS25, lai_global_2018}, wave \cite{LTZ24} and several other related equations \cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case.

There is a major difference between the cases $s\in (0,1)$ and $s\in \mathbb{R}^+\setminus\mathbb{Z}$ in $(-\Delta)^s$. For the nonlinear Schr\"odinger equation with $s\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\infty$ bounded solution to the linear Schr\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.

\subsection{Outline of the article}

The remaining part of the paper is organized as follows. In Section~\ref{sec:pre} we survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \ref{thm_wellposdness}. In Section~\ref{sec_linearization} we show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the  inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\ref{sec:proof of theorem}, where we prove our main result, Theorem \ref{thm_main}. After the acknowledgments of Section~\ref{sec:ack}, we included an Appendix dedicated to computations.

To prove Theorem~\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they “require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.”

For the inverse problem, in Section~\ref{sec_linearization} they “show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\ref{sec:proof of theorem} they “prove our main result, Theorem~\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.

To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\"odinger equation to suit the current setting, and establish the following theorem.

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\\nu = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}

For the inverse problem, they then show that the solution to the nonlinear Schr\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.

\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.,

Question:
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set with smooth boundary, and let $\Omega_e:=\mathbb{R}^n\setminus \overline{\Omega}$. Fix an integer $m\ge 0$, an integer $K\ge 2$, and $s\in \mathbb{R}^+\setminus \mathbb{Z}$ such that $\lfloor s\rfloor>\max\{m,n/2\}$. For $j=1,2$, assume $q_j\in L^\infty(\Omega)$ with $q_j\ge 0$ in $\Omega$, and let $\mathbf P_j$ be nonlinearities of the form
\[
\mathbf P_j(u)=uP^{(j)}_1(x,D)u+u^2P^{(j)}_2(x,D)u+\cdots+u^{K-1}P^{(j)}_{K-1}(x,D)u,
\]
where
\[
P^{(j)}_k(x,D)=\sum_{|\sigma|\le m} a^{(j)}_{\sigma,k}(x)D^\sigma,
\qquad a^{(j)}_{\sigma,k}\in C(\overline\Omega),
\]
for every $k=1,\dots,K-1$. Let $W_1,W_2\subset \Omega_e$ be arbitrary nonempty bounded open sets, and define
\[
\mathcal X_{\varepsilon_0}(W_1):=\{f\in C_c^\infty(W_1):\ \|f\|_{C_c^\infty(W_1)}\le \varepsilon_0\}.
\]
For each $j$, let $\Lambda_{\mathbf P_j}$ be the Dirichlet-to-Neumann map for the exterior value problem
\[
(-\Delta)^s u+q_j(x)u+\mathbf P_j(u)=0\ \text{in }\Omega,\qquad u=f\ \text{in }\Omega_e,
\]
namely $\Lambda_{\mathbf P_j}(f)=(-\Delta)^s u_f|_{\Omega_e}$. Suppose that the partial DN data agree in the sense that
\[
\Lambda_{\mathbf P_1}f|_{W_2}=\Lambda_{\mathbf P_2}f|_{W_2}\qquad \text{for all }f\in \mathcal X_{\varepsilon_0}(W_1).
\]
Which statement is guaranteed for every such pair of nonlinear equations?

One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|\le m$ and every $k=1,\dots,K-1$, the coefficients satisfy $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; equivalently, the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide in $\Omega$.

Choices:
B. One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|< m$ and every $k=1,\dots,K-1$, the coefficients satisfy $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; hence the lower-order parts of the operators $P_k^{(j)}(x,D)$ agree, though the coefficients of order exactly $m$ need not be determined.
C. One must have $q_1=q_2$ in $\Omega$.
D. One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|\le m$ there exists an index $k=k(\sigma)\in\{1,\dots,K-1\}$ such that $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; in particular, each differential order is represented by some matching coefficient, although not necessarily for every $k$.
E. There exists a sufficiently small $\varepsilon_*>0$, depending only on $\Omega$, $n$, $m$, $s$, and the a priori bounds for the coefficients, such that if the partial DN maps agree on $W_2$ for all $f\in \mathcal X_{\varepsilon_*}(W_1)$, then one must have $q_1=q_2$ in $\Omega$ and $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$ for all $0\le |\sigma|\le m$ and all $k=1,\dots,K-1$.

full recovery up to top differential order $m$

dropped identification of all nonlinear coefficients $a_{\sigma,k}$

quantifier structure over $(\sigma,k)$ in coefficient recovery

dependence on the given small-data radius $\varepsilon_0$ versus a newly asserted uniform threshold

The stem does not state the conclusion explicitly and does not reveal the correct option by wording. It gives the full hypotheses, but the answer is not leaked directly.

This is largely a theorem-recall item: the stem reproduces a full technical hypothesis set and asks for the guaranteed conclusion. The alternatives introduce quantifier/order variations, so it is not a pure verbatim restatement, but it is still close to one.

Some reasoning is needed to distinguish full recovery from weaker or tampered conclusions, especially among B, C, D, and E. However, the item mainly tests recognition of the exact theorem conclusion rather than substantial generative mathematical reasoning.

The distractors are strong and mathematically plausible: one weakens the conclusion, one drops top-order recovery, one alters the quantifier structure, and one changes the smallness/uniform-threshold claim. These reflect realistic failure modes.

A solid theorem-recognition MCQ with high-quality distractors and no major answer leakage, but it is still fairly close to a restatement of a specific uniqueness theorem rather than a deeply generative reasoning task.


\label{bc}

We note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an

\begin{theorem}\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.
\end{theorem}

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

This theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \cite[p.~104]{andrews book} and Schur's 1926 theorem \cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \cite{alder} for an account of some of the results in the late 20th century.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

\subjclass[2020]{Primary 11P84, 05A17}
 \keywords{Euler's theorem, Partitions, Glaisher's bijection}
\maketitle
\pagenumbering{arabic}
\pagestyle{headings}
\begin{abstract}
Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,
\begin{align*}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align*}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

On the other hand, we have
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
We now employ the following Euler's identity \cite[p.~19, (2.2.5)]{andrews book}
\begin{align}\label{euler identity}
\frac{1}{(t;q)_\infty}=\sum_{m=0}^\infty\frac{t^m}{(q,q)_m},
\end{align}
(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \eqref{before p19} so as to obtain
\begin{align}\label{series}
\sum_{n=0}^\infty C(n)q^{n}&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}}\sum_{m=0}^\infty\frac{q^{2nm+2m}}{(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{1}{2}\left(1+(-1)^n\right)\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\nonumber\\
&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\sum_{n=0}^\infty\frac{q^{n(m+1)}}{(q;q)_n}+\sum_{n=0}^\infty\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\right\}.
\end{align}
Upon invoking \eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \eqref{series}, we conclude that
\begin{align}\label{3.4}
\sum_{n=0}^\infty C(n)q^{n}&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\frac{1}{(q^{m+1};q)_\infty}+\frac{1}{(-q^{m+1};q)_\infty}\right\}\nonumber\\
&=\frac{1}{2}(-q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(-q;q)_m}+\frac{1}{2}(q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q;q)_m}\nonumber\\
&=\frac{1}{2}\sum_{m=0}^\infty q^{2m}(-q^{m+1};q)_\infty+\frac{1}{2}(1-q),
\end{align}
where the last step follows upon again using \eqref{euler identity} with letting $t=q^2$.

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct.
\end{quote}
In other words, a partition $\lambda=(\lambda_1,\ldots, \lambda_{\ell})$ counted by $D(n)$ satisfies
$$
\lambda_1>\lambda_2> \cdots > \lambda_{\ell-1}\ge \lambda_{\ell} \text{ if $\ell>1$}
$$
or
$\lambda=(\lambda_1)$ has only one part.

Let $M \le 2N$ be an even number, which can be written as
$$
M=2^k a \text{ for some odd integer $a$ and some $ k\ge 1$}.
$$
We apply Glaisher's bijection $\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since
$
2^k a \le 2N,
$
$$
a \le N.
$$
Also, if $M\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that
\begin{equation}
2^k\le N/a. \label{power1}
\end{equation}
On the other hand, if $N< M\le 2N$, then
\begin{equation}
2^k> N/a, \label{power2}
\end{equation}
and $M$ can repeat.
Suppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\phi$ to $f$ copies of $M$, we obtain
$$
 f 2^{k} \text{ copies of $a$}.
$$
By writing $f$ as a binary expansion,
$$
f 2^k =(f_0 \,2^{0}+f_1 \, 2^{1} +\cdots ) 2^k,
$$
where $f_j$ is either $0$ or $1$ for $j\ge 0$. By \eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e.,
\begin{equation}
f_j \, 2^{j+k} >N/a. \label{power3}
\end{equation}
It follows from \eqref{power1} and \eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

 \vspace{2mm}$A(6)$\newline  &  \vspace{2mm} $B(6)$ & \vspace{2mm} $C(7)$& \vspace{2mm} $D(7)$ \\

 $6$ \newline 5+1 \newline 4+2\newline 3+2+1   & 5+1 \newline 3+3 \newline 3+1+1+1 \newline 1+1+1+1+1+1  & 6+1\newline 4+3 \newline 4+2+1 \newline 2+2+2+1  & 0+0+7\newline 0+0+6+1\newline 0+0+5+2\newline 0+0+4+3\newline 0+0+4+2+1\newline 1+1+5\newline 1+1+2+3\newline 2+2+3
\end{tabular}
\end{center}

We conclude the introduction with the following remark.
\begin{remark}
In their recent paper \cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\infty$ to the generating functions in question.
\end{remark}

This paper is organised as follows. In section \ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \ref{cd}, we prove that $C(n)=\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \ref{main theorem} in Section \ref{bijective proofs}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\ref{bc} they "provide the brief proof that $B(n)=C(n+1)$"; in Section~\ref{cd} they "prove that $C(n)=\frac{1}{2}D(n)$"; and in Section~\ref{bijective proofs} they "provide bijective proofs" of the assertions in Theorem~\ref{main theorem}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an
and also that
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Comparing \eqref{bn} and \eqref{cn}, we see that $B(n)=C(n+1)$.

\section{Proof that $C(n)=\frac{1}{2}D(n)$}\label{cd}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
They then prove that $C(n)=\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:
\label{bijective proofs}

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ count

\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,

Question: For each positive integer n, let A(n) denote the number of partitions of n into distinct parts, let B(n) denote the number of partitions of n into odd parts, let C(n) denote the number of partitions of n whose largest part is even and for which every part not exceeding half of the largest part occurs at most once, and let D(n) denote the number of partitions of n into nonnegative parts in which the smallest part appears exactly twice and no other part is repeated. Which statement holds for every positive integer n?

Choices:
B. A(n)=B(n)=C(n)=\frac{1}{2}D(n).
C. A(n)=B(n)=C(n+1).
D. A(n)=B(n)=C(n+1)=D(n+1).
E. A(n)=B(n)=C(n)+1=\frac{1}{2}D(n+1).

The stem only defines the four partition-counting functions and asks which identity holds universally. It does not reveal the correct shift by n+1 or the factor of 1/2 in D(n+1), so there is no meaningful answer leakage.

This is not a direct restatement of a theorem from the stem. The question requires comparing several closely related candidate identities involving shifts and scaling factors, rather than recognizing a verbatim claim.

Identifying the correct statement requires substantial reasoning or prior knowledge of nontrivial partition identities. The options differ in subtle but important ways (index shifts, omission of the D-term factor, stronger vs. weaker equalities), creating real generative pressure.

The distractors are plausible and mathematically structured: they reflect common failure modes such as missing an index shift, dropping a factor of 1/2, or selecting a weaker true statement instead of the strongest correct one. They are distinct and well-aligned with likely reasoning errors.

High-quality MCQ. It avoids answer leakage and tautology, uses strong distractors, and genuinely tests mathematical reasoning about subtle partition identities.