This section introduces the mathematical formulation of the inverse problem and specifies the functional framework adopted throughout the paper. We also describe the model equation, its physical motivation, and the regularity assumptions required to ensure well-posedness.

Let $\Omega\subset \mathbb{R}^N$ be a bounded domain ($N\geqslant 1$) with boundary $\partial \Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.

We consider $a,b$, $\vec{r}$ and $p$ with the following assumptions:
\begin{itemize}
    \item[{\bf (H1)}] $a,b\in W^{1,\infty}(\Omega;\mathbb{R})$ with
    \begin{align*}
        a(x)\geqslant a_\star >0\quad \text{ almost everywhere in }\Omega,
    \end{align*}
    for some $a_\star$.
    \item[{\bf (H2)}] $\vec{r}=\vec{r}(x)$, $p=p(x)$ and $y_0=y_0(x)$ are complex-valued functions such that
    \begin{align*}
        \vec{r}\in [W^{1,\infty}(\Omega)]^N,\,
        q\in L^\infty(\Omega)\text{ and }y_0\in L^2(\Omega).
    \end{align*}
\end{itemize}

Now, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:
\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \partial_t y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

Under the assumptions {\bf (H1)}, {\bf (H2)} and $f\in L^2(0,T;L^2(\Omega))$, there exists a unique weak solution of \eqref{eq:intro:01} with the following regularity:
\begin{align*}
    y\in C^0([0,T];L^2(\Omega)) \cap L^2(0,T;H_0^1(\Omega)).
\end{align*}

For completeness and self-containment,  we include in Appendix \ref{section:appendix:existence:uniqueness:results} general statements, together with their proofs, concerning the existence of weak and strong solutions of the equation \eqref{eq:intro:01}.

In this article, we will consider the following {\bf Inverse Source Problem (ISP):} Identify the unknown spatial-temporal source term $f$ in the space $L^2(0,T;L^2(\Omega))$ in \eqref{eq:intro:01} from the following final time measured output:
\begin{align*}
    u_T(x):=y(x,T),\quad x\in \Omega.
\end{align*}

Here, $y(x,T)\equiv y(x,t)\big|_{t=T}$ is a trace appropriately defined of the weak solution $y(x,t)$ of \eqref{eq:intro:01}. In addition, $u_T(x)$ represents the measured output containing a random noise.

Inverse source problems of this type are not only mathematically rich but also arise naturally in applied sciences. In many experimental settings, direct access to distributed sources is often impossible, and only partial or final-time measurements are available. For instance, in nonlinear optics one may measure the profile of a pulse at the end of a cavity, while the internal gain or loss mechanisms driving its formation remain hidden. Similarly, in chemical or biological pattern formation, one often observes the state of the system after some transient dynamics but seeks to infer the localized heterogeneities or forcing terms that generated it. This gap between accessible measurements and hidden dynamics motivates the study of reconstruction methods capable of identifying unknown excitations from indirect data.

The choice of the Ginzburg-Landau equation as the underlying model is particularly significant. It serves as a universal amplitude equation near the onset of instability in diverse physical systems, ranging from fluid dynamics and plasma physics to optics and chemical oscillations \cite{Aranson2002,levermore1996complex,chen1998numerical,bartuccelli1990possibility,mielke2002ginzburg}. Consequently, advances in the understanding of inverse problems for this equation are not limited to a single discipline but carry implications across multiple fields. Identifying unknown sources in such a model not only enriches the theory of dissipative-dispersive PDEs but also provides a mathematical framework for interpreting experiments in which only end-point information is available.

At the mathematical level, the inverse source problem we consider is severely ill-posed. Regulation strategies and variational formulations thus become essential. Our work adopts this perspective, combining the theory of weak solutions, adjoint-based gradient formulas, and Tikhonov regularization to rigorously justify reconstruction algorithms and test their performance through numerical simulations.

\subsection{Literature}

We now review some representative contributions on the complex Ginzburg-Landau equation and its variants, emphasizing their role as canonical amplitude equations and discussing prior work related to inverse or identification problems.

The complex Ginzburg-Landau (GL) equation is the canonical amplitude equation near a Hopf or Turing-Hopf bifurcation. In its cubic form, it reads
\begin{align*}
    \partial_t A=(\mu+i\omega)A + (1+c_1i)\Delta A -(1+c_3 i)|A|^2 A,
\end{align*}
while the cubic-quintic extension adds a saturating nonlinearity:
\begin{align*}
    \partial_t A=(\mu +\omega i)A +(1+c_1i)\Delta A -(1+c_3 i)|A|^2 A -(\nu +c_5i)|A|^4A,
\end{align*}
with real parameters $\mu,\nu,c_j$ ($j=1,3,5$) that encode growth, dispersion/diffusion, and nonlinear gain/loss. The cubic model is the universal normal form near onset and captures phase diffusion, plane-wave selection and Benjamin-Feir/Newell instabilities. On the other hand, the quintic term become relevant farther from threshold to model saturation and complex dissipative structures (e.g. localized pulses). See the reviews \cite{Cross1993} and \cite{Aranson2002}, which also discuss higher-order variants and parameter scalings.

The cubic and cubic-quintic complex GL equations serve as canonical models of pattern formation and nonlinear wave dynamics in nonequilibrium systems. These models describe the slow complex envelope $A(x,t)$ of oscillatory modes near stability. In hydrodynamics and Rayleigh-B\'enard convection, the cubic GL equation explains the emergence of roll patterns, travelling waves, and their stability domains \cite{Newell1969} and \cite{Cross1993}. In nonlinear optics, it governs mode-locked lasers, pulse propagation in fiber cavities, and the formation of dissipative solutions. In this case, the cubic-quintic equation extension accounts for gain saturation and quintic nonlinear effects that are essential for bounded amplitude states \cite{SotoCrespo1997}, \cite{Akhmediev2005}. Similarly, in chemical oscillations and biological excitable media, it models spiral waves, defect turbulence, and spatiotemporal chaos, highlighting its universality across disciplines \cite{Aranson2002}.

From a mathematical viewpoint, the GL equations combine parabolic smoothing (via the Laplacian operator) with dispersive phase dynamics (via imaginary coefficients), which generates a delicate interplay between diffusion, dispersion, and nonlinearity \cite{Akhmediev2005,Cross1993, levermore1996complex, mielke2002ginzburg}. This hybrid character produces a wide spectrum of phenomena: existence of plane-wave solutions, modulational (Benjamin-Feir/Newell) instabilities, turbulence, and complex coherent structures \cite{Akhmediev2005, Cross1993, Newell1969, SotoCrespo1997}. The cubic equation alone already yields chaotic attractors, defect-mediated turbulence, and spiral wave breakup \cite{Aranson2002,bartuccelli1990possibility, SotoCrespo1997}, while the quintic term stabilizes or destabilizes localized patterns depending on parameter regimes \cite{Akhmediev2005,SotoCrespo1997}. The GL equations have been used to study attractors, bifurcations, and long-time dynamics of nonlinear PDEs, serving as a testbed for methods in dissipative systems, control, and inverse problems \cite{chen1998numerical,mielke2002ginzburg,santos2019insensitizing,borzi2005analysis,junge2000synchronization}.
Recovering sources and parameters in GL-type models is crucial in practice: in optics, it enables diagnosing distributed gain/loss or saturable absorbers in cavity models from end-of-pulse measurements; in chemical/biological pattern formation, it helps infer spatially localized forcing or heterogeneities driving wave patterns; in fluid and plasma contexts, it allows one to back-out effective forcing or feedback from limited snapshots near transitions to turbulence. Because the GL framework is a universal amplitude model, successful inversion translates across disciplines, providing interpretable maps of where and how energy is injected or dissipated. From a computational perspective, adjoint-based Tikhonov schemes scale to high-dimensional discretizations and accommodate realistic noise models \cite{Cross1993}, \cite{Aranson2002}, \cite{MR2516528}.

Within this broad literature, inverse problems for GL-type models remain relatively less explored. Classical studies on inverse source problems for parabolic or Schr\"odinger-type equations \cite{hasanouglu2021introduction,hasanov2007simultaneous,GarciaOssesTapia+2013+755+779,chorfi2025identification} provide methodological foundations, yet the adaptation to GL equations, which combine diffusion, dispersion, and nonlinearity, poses unique analytical and numerical challenges.

We also note that other GL-type models, such as those with dynamic boundary conditions, have recently attracted attention in the context of controllability \cite{carreno2025local}. While these studies highlight the richness of the GL framework under more complex boundary interactions, the analysis of inverse problems in such settings remains largely open. Our contribution is thus complementary, as we address the identification of space-time dependent source terms for the standard GL equation, providing a first step toward broader inverse problem formulations in extended GL models.

It is worth emphasizing that the linear GL equation can also be expressed as a system of two coupled real-valued PDE equations, corresponding to the real and imaginary parts of the complex amplitude. In this formulation, the coupling terms are of second order and involve cross-derivative operators, which profoundly alter the analytical structure of the system. As a consequence, classical techniques from the theory of control and inverse problems for scalar parabolic equations (based on Carleman estimates and observability inequalities \cite{fursikov1996carleman}) cannot be directly applied. In particular, the presence of complex coefficients and dispersive-type interactions requires adapted Carleman weights, refined energy estimates, and the development of new stability results tailored to mixed dissipative-dispersive operators.

The present work is intended as a first step toward a comprehensive theory of inverse problems for GL-type equations, focusing here on the standard setting with Dirichlet boundary conditions.

\subsection{The input-output operator}
To formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
    \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
    \label{eq:Input-output:op}
    \Psi(f)=u_T(x),\quad x\in \Omega,\quad  f\in \mathcal{U}.
\end{align}

We emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result

Let $\Omega\subset \mathbb{R}^N$ be a bounded domain ($N\geqslant 1$) with boundary $\partial \Omega$ of class $C^2$ and $T>0$. Here and in the sequel, the function spaces refer to spaces of complex-valued functions unless otherwise specified.

Now, consider the following Ginzburg-Landau equation with Dirichlet boundary conditions:
\begin{align}
 \label{eq:intro:01}
 \begin{cases}
 \partial_t y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
 y=0&\text{ on }\partial \Omega\times (0,T),\\
 y(\cdot,0)=y_0&\text{ in }\Omega.
 \end{cases}
\end{align}

\subsection{The input-output operator}
To formalize the inverse problem, we introduce the input-output operator mapping the source term to the final-state of the system. This operator provides a compact representation of the forward model and allows for a variational reformulation of the inverse problem.

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
 \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
 \label{eq:Input-output:op}
 \Psi(f)=u_T(x),\quad x\in \Omega,\quad f\in \mathcal{U}.
\end{align}

We emphasize that, due to measurement errors in the output $u_T$, exact equality in the equation \eqref{eq:Input-output:op} cannot be satisfied in general. Moreover, we have the following result

\begin{align}
    \label{eq:Input-output:op}
    \Psi(f)=u_T(x),\quad x\in \Omega,\quad  f\in \mathcal{U}.
\end{align}

\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \pt y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

Let $\mathcal{U}\subset L^2(0,T;L^2(\Omega))$ be a non-empty set, which is supposed to be bounded, closed and convex. In addition, we define the operator $\Psi: \mathcal{U}\to L^2(\Omega)$ in the following way:
\begin{align*}
 \Psi(f)=y(\cdot,T)\quad f\in \mathcal{U},
\end{align*}
where $y=y(x,t)$ is the solution of \eqref{eq:intro:01} associated to the source term $f$.

We point out that, the {\bf (ISP)} can be formulated in terms of $\Psi$ as the following functional equation:
\begin{align}
 \label{eq:Input-output:op}
 \Psi(f)=u_T(x),\quad x\in \Omega,\quad f\in \mathcal{U}.
\end{align}

\begin{proof}
 Let $(f_k)_{k\in \mathbb{N}}$ be a bounded sequence in $L^2(\Omega\times (0,T))$. Then, by Proposition \ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\cdot,,\cdot,f_k))_{k\in \mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\Omega))$. In particular, the sequence $(Y_k)_{k\in \mathbb{N}}$ given by $Y_k:=y(\cdot,T,f_k)$ is bounded in $H_0^1(\Omega)\cap H^2(\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\Omega) \hookrightarrow L^2(\Omega)$, there exists a subsequence of $ (Y_k)_{k\in \mathbb{N}}$ which converges strongly in $L^2(\Omega)$. This implies that the input-output operator $\Psi$ is compact and the proof of Proposition \ref{Proposition:Phi:compact} is finished.
\end{proof}

In view of the Proposition \ref{Proposition:Phi:compact}, it is evident that the inverse problem {\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ by
\begin{align}
 \label{def:functional:J}
 \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}
and reformulate {\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem
\begin{align*}
 \mathcal{J}(f^\star)=\inf_{f\in \mathcal{U}} \mathcal{J}(f).
\end{align*}

Since the operator $\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \eqref{def:functional:J}. For $\epsilon>0$, we introduce the regularized functional
\begin{align}\label{regfunct}
 \mathcal{J}_\epsilon (f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx + \frac{\epsilon}{2}\int_0^T\int_\Omega |f|^2\,dx\,dt ,\quad f\in \mathcal{U}.
\end{align}

\begin{proposition}
 \label{proposition:Frechet:formula}
 Consider the assumptions {\bf (H1)} and {\bf (H2)}. Then, the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ is Fr\'echet differentiable and its gradient at each $f\in \mathcal{U}$ is given by
 \begin{align}
 \label{eq:frechet:gradient:formula}
 \mathcal{J}'(f)=\phi,\quad f\in \mathcal{U},
 \end{align}
 where $\phi$ is the unique weak solution of the following adjoint system
 \begin{align}
 \label{problem:Frechet:dif}
 \begin{cases}
 -\pt \phi -\text{div}((a(x)-ib(x))\nabla \phi) - \overline{\vec{r}(x)}\cdot \nabla \phi +(\overline{p(x)}- \overline{\text{div}(\vec{r})(x))}\phi =0&\text{ in }\Omega\times (0,T),\\
 \phi=0&\text{ on }\partial\Omega \times (0,T),\\
 \phi(\cdot,T)=y(\cdot,T;f)-u_T&\text{ in }\Omega.
 \end{cases}
 \end{align}
\end{proposition}

Using the complex identity
\begin{align*}
 \frac{1}{2}(|x-z|^2-|y-z|^2)=\Re [(x-y)\overline{(y-z)}] +\frac{1}{2} |x-y|^2,\quad \forall x,y,z\in \mathbb{C},
\end{align*}
we have
\begin{align}
 \nonumber
 \delta \mathcal{J}(f)=&\Re \int_\Omega (y(\cdot,T;f+\delta f)-y(\cdot,T;f)) \overline{y(\cdot,T;f)-u_T}\,dx \\
 \nonumber
 &+\frac{1}{2}\int_\Omega |y(\cdot,T;f+\delta f)-y(\cdot,T;f)|^2\,dx \\
 \label{eq:Frechet:for:01}
 =&-\Re \int_\Omega \delta y(\cdot,T;f)\overline{\phi (\cdot,T;f)}\,dx +\frac{1}{2}\int_\Omega |\delta y(\cdot,T;f)|^2\,dx
\end{align}
where $\phi(\cdot,\cdot,f)$ is the weak solution of \eqref{problem:Frechet:dif} (and thanks to Proposition \ref{proposition:existence:weak:solutions}, $\phi(\cdot,T,f)=y(\cdot,T;f)-u_T$ in $L^2(\Omega)$) and $\delta y$ is the solution of the following \textit{sensitivity problem}
\begin{align}
 \label{eq:sensitivity:problem}
 \begin{cases}
 \pt \delta y -\text{div}((a(x)+b(x)i)\nabla \delta y)+\vec{r}(x)\cdot \nabla \delta y + p(x)\delta y=\delta f(x,t)&\text{ in }\Omega\times (0,T),\\
 \delta y=0&\text{ on }\partial \Omega\times (0,T),\\
 \delta y(\cdot,0)=0&\text{ in }\Omega.
 \end{cases}
\end{align}

\begin{proof}
 Let us fix $f,\delta f\in \mathcal{U}$ such that $f+\delta f\in \mathcal{U}$. Then, the function $\delta \phi=\delta \phi (\cdot,\cdot,f)$ is the solution of
 \begin{align}
 \label{eq:Lipschitz:01}
 \begin{cases}
 -\pt \delta \phi -\text{div}((a(x)-ib(x))\nabla \phi) - \overline{\vec{r}(x)}\cdot \nabla \phi +(\overline{p(x)}-\overline{\text{div}(\vec{r}(x))}\phi =0&\text{ in }\Omega\times (0,T),\\
 \delta \phi =0&\text{ on }\partial \Omega \times (0,T),\\
 \delta \phi(\cdot,T)=\delta y(\cdot,T;f)&\text{ in }\Omega,
 \end{cases}
 \end{align}
 where $\delta y$ is the solution of \eqref{problem:Frechet:dif} associated to $\delta f$. We remark that, by Proposition \ref{proposition:existence:strong:solutions}, $\delta \phi$ is a strong solution of \eqref{eq:Lipschitz:01}. Thus, by Proposition \ref{proposition:Frechet:formula} we have, for some constant $C>0$:
 \begin{align*}
 |\mathcal{J}'(f+\delta f)-\mathcal{J}'(f)|= \|\delta \phi\|_{L^2(\Omega\times (0,T))}\leq C\|\delta y(\cdot,T;f)\|_{L^2(\Omega)}\leq C\|\delta f\|_{L^2(0,T;L^2(\Omega))},
 \end{align*}
 where we have used the continuity of the strong solutions applied to $\delta y$ respect to the data. This proves the assertion of the Lemma \ref{Lemma:Lipschitz:gradient}.
\end{proof}

\begin{proposition}
    \label{Proposition:Phi:compact}
    The input-output operator $\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)$ is compact.
\end{proposition}

\begin{align}
    \label{def:functional:J}
    \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}

\begin{align}
    \label{eq:intro:01}
    \begin{cases}
        \pt y -\text{div}((a(x)+ib(x))\nabla y) + \vec{r}(x)\cdot \nabla y +p(x)y=f(x,t)&\text{ in }\Omega\times (0,T),\\
        y=0&\text{ on }\partial \Omega\times (0,T),\\
        y(\cdot,0)=y_0&\text{ in }\Omega.
    \end{cases}
\end{align}

\begin{proposition}
    \label{proposition:existence:strong:solutions}
    Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\alpha$ and $\beta$ satisfy
    \begin{align}
        \label{assumption:prop:existence:02:alpha:beta}
        \alpha,\beta\in W^{1,\infty}(\Omega;\mathbb{R}),\quad \alpha(x)\geq \alpha_0 >0\text{ almost everywhere in }\Omega.
    \end{align}
    Moreover, consider \eqref{prop:existence:statement:02}, $v_0\in H_0^1(\Omega)$ and $h\in L^2(0,T;L^2(\Omega))$. Then, there exists a strong solution $v$ of \eqref{problem:existence:uniqueness} satisfying
    \begin{align*}
        v\in H^1(0,T;L^2(\Omega)) \cap C^0([0,T];H_0^1(\Omega)) \cap L^2(0,T; H^2(\Omega) \cap H_0^1(\Omega)).
    \end{align*}

    Moreover, there exists a constant $C>0$ depending only on $\alpha_0$, $\|\alpha\|_{W^{1,\infty}}$, $\||\vec{\sigma}|\|_{L^\infty}$, $\|p\|_\infty$, $\Omega$ and $T$ such that
    \begin{align*}
        \begin{split}
        &\|v\|_{L^2(0,T;H^2(\Omega))} + \|v\|_{C^0([0,T];H_0^1(\Omega))} + \|\pt v\|_{L^2(0,T;L^2(\Omega))} \\
        \leq &C\left(\|h\|_{L^2(0,T;L^2(\Omega))} + \|v_0\|_{H_0^1(\Omega)} \right).
        \end{split}
    \end{align*}
\end{proposition}

\begin{proof}
    Let $(f_k)_{k\in \mathbb{N}}$ be a bounded sequence in $L^2(\Omega\times (0,T))$. Then, by Proposition \ref{proposition:existence:strong:solutions}, the sequence of associated weak solutions $(y(\cdot,,\cdot,f_k))_{k\in \mathbb{N}}$ is bounded in $C^0([0,T];H_0^1(\Omega))$. In particular, the sequence $(Y_k)_{k\in \mathbb{N}}$ given by $Y_k:=y(\cdot,T,f_k)$ is bounded in $H_0^1(\Omega)\cap H^2(\Omega)$. Thus, using the Sobolev-Gagliardo-Nirenberg compact embedding $H_0^1(\Omega) \hookrightarrow L^2(\Omega)$, there exists a subsequence of $ (Y_k)_{k\in \mathbb{N}}$ which converges strongly in $L^2(\Omega)$. This implies that the input-output operator $\Psi$ is compact and the proof of Proposition  \ref{Proposition:Phi:compact} is finished.
\end{proof}

In view of the Proposition \ref{Proposition:Phi:compact}, it is evident that the inverse problem {\bf (ISP)} is ill-posed in the sense of Hadamard. For this reason, one needs to introduce the functional $\mathcal{J}:\mathcal{U}\to \mathbb{R}$ by
\begin{align}
    \label{def:functional:J}
    \mathcal{J}(f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx,
\end{align}
and reformulate {\bf (ISP)} in terms of the quasi-solution method, i.e., we minimize the following extremal problem
\begin{align*}
    \mathcal{J}(f^\star)=\inf_{f\in \mathcal{U}} \mathcal{J}(f).
\end{align*}

Since the operator $\Psi$ is compact, small perturbations in the data may cause large variations in the reconstructed source. Therefore, regularization becomes essential to obtain stable approximations. In this context, it is customary to consider a regularized Tikhonov version of the functional $J$ in \eqref{def:functional:J}. For $\epsilon>0$, we introduce the regularized functional
\begin{align}\label{regfunct}
    \mathcal{J}_\epsilon (f):=\frac{1}{2}\int_\Omega |\Psi(f)-u_T|^2\,dx + \frac{\epsilon}{2}\int_0^T\int_\Omega |f|^2\,dx\,dt ,\quad f\in \mathcal{U}.
\end{align}

\subsection{Outline}
The rest of the paper is as follows. In Section \ref{section:Frechet:formula}, we focus on the properties of $J$, i.e., a detailed characterization of its Fr\'echet derivative via a suitable adjoint system. In Section \ref{section:Existence:ISP}, we give sufficient conditions for the existence and uniqueness of quasi-solutions to {\bf (ISP)}. In Section \ref{section:Numerical:experiments}, we validate our theoretical results by some numerical experiments to reconstruct source terms for 1-D and 2-D case. Finally, in Section \ref{section:Summary:perspectives} we give additional comments concerning the theoretical and numerical results obtained in this article.

Take a bounded sequence $(f_k)\subset L^2(\Omega\times(0,T))$. By Proposition \ref{proposition:existence:strong:solutions}, the associated weak solutions $y(\cdot,\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\Omega))$, hence the terminal states $Y_k:=y(\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\Omega)\cap H^2(\Omega)$. Using the compact embedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\Omega)$. Therefore $\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\Omega)$ for every bounded $(f_k)$, which implies $\Psi$ is compact (Proposition~\ref{Proposition:Phi:compact}).

Take a bounded sequence $(f_k)\subset L^2(\Omega\times(0,T))$. We first use the following result.

\begin{proposition}
    \label{proposition:existence:strong:solutions}
    Let $\Omega\subset \mathbb{R}^N$ be a bounded domain with $C^2$ boundary and let $T>0$. Assume that the coefficients $\alpha$ and $\beta$ satisfy
    \begin{align}
        \label{assumption:prop:existence:02:alpha:beta}
        \alpha,\beta\in W^{1,\infty}(\Omega;\mathbb{R}),\quad \alpha(x)\geq \alpha_0 >0\text{ almost everywhere in }\Omega.
    \end{align}
    Moreover, consider \eqref{prop:existence:statement:02}, $v_0\in H_0^1(\Omega)$ and $h\in L^2(0,T;L^2(\Omega))$. Then, there exists a strong solution $v$ of \eqref{problem:existence:uniqueness} satisfying
    \begin{align*}
        v\in H^1(0,T;L^2(\Omega)) \cap C^0([0,T];H_0^1(\Omega)) \cap L^2(0,T; H^2(\Omega) \cap H_0^1(\Omega)).
    \end{align*}

    Moreover, there exists a constant $C>0$ depending only on $\alpha_0$, $\|\alpha\|_{W^{1,\infty}}$, $\||\vec{\sigma}|\|_{L^\infty}$, $\|p\|_\infty$, $\Omega$ and $T$ such that
    \begin{align*}
        \begin{split}
        &\|v\|_{L^2(0,T;H^2(\Omega))} + \|v\|_{C^0([0,T];H_0^1(\Omega))} + \|\pt v\|_{L^2(0,T;L^2(\Omega))} \\
        \leq &C\left(\|h\|_{L^2(0,T;L^2(\Omega))} + \|v_0\|_{H_0^1(\Omega)} \right).
        \end{split}
    \end{align*}
\end{proposition}

By this proposition, the associated weak solutions $y(\cdot,\cdot,f_k)$ are bounded in $C^0([0,T];H_0^1(\Omega))$, hence the terminal states $Y_k:=y(\cdot,T,f_k)$ form a bounded sequence in $H_0^1(\Omega)\cap H^2(\Omega)$. Using the compact embedding $H_0^1(\Omega)\hookrightarrow L^2(\Omega)$, extract a subsequence of $(Y_k)$ converging strongly in $L^2(\Omega)$. Therefore $\Psi(f_k)=Y_k$ has a strongly convergent subsequence in $L^2(\Omega)$ for every bounded $(f_k)$, which implies compactness. This completes the proof of the main theorem.

\label{Proposition:Phi:compact}
    The input-output operator $\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)$ is compact.,

Question:
Let \(\Omega\subset \mathbb{R}^N\) be a bounded domain with \(C^2\) boundary, let \(T>0\), and fix the coefficients and initial datum in the Dirichlet Ginzburg--Landau problem
\[
\begin{cases}
\partial_t y-\operatorname{div}((a(x)+ib(x))\nabla y)+\vec r(x)\cdot \nabla y+p(x)y=f(x,t) & \text{in } \Omega\times(0,T),\\
y=0 & \text{on } \partial\Omega\times(0,T),\\
y(\cdot,0)=y_0 & \text{in } \Omega.
\end{cases}
\]
For each source term \(f\in L^2(0,T;L^2(\Omega))\), let \(y\) denote the corresponding solution and define the input-output operator
\[
\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega),\qquad \Psi(f)=y(\cdot,T).
\]
Which statement holds for this operator?

The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is compact; equivalently, it sends every bounded subset of \(L^2(0,T;L^2(\Omega))\) into a relatively compact subset of \(L^2(\Omega)\).

Choices:
B. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is bounded but not compact; in fact, for every bounded sequence \((f_k)\subset L^2(0,T;L^2(\Omega))\), the sequence \((\Psi(f_k))\) is only guaranteed to be bounded in \(L^2(\Omega)\), not to admit a strongly convergent subsequence there.
C. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is continuous (in particular, bounded as a linear map).
D. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to H_0^1(\Omega)\) is compact; equivalently, it sends every bounded subset of \(L^2(0,T;L^2(\Omega))\) into a relatively compact subset of \(H_0^1(\Omega)\).
E. The operator \(\Psi:L^2(0,T;L^2(\Omega))\to L^2(\Omega)\) is compact only when the source set is restricted to a bounded, closed, convex subset \(\mathcal U\subset L^2(0,T;L^2(\Omega))\); on the whole space \(L^2(0,T;L^2(\Omega))\) one can assert continuity but not compactness.

upgrade from boundedness of terminal states to relative compactness via compact embedding

compactness is only obtained after embedding into L^2, not in H_0^1

domain of the operator: compactness proved on the whole bounded-sequence criterion in L^2, not only on a preset bounded closed convex subset


\begin{theorem}\label{thm:MAIN}
For any stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials, the projectivized stratum $\mathbb P\Omega\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvariety.
\end{theorem}

Let $\mu$ be a partition of $2g-2$. Denote by $\Omega\mathcal M_{g}(\mu)$ the stratum of Abelian differentials (i.e., differential one-forms) on smooth, connected, genus-$g$ complex curves whose zero and pole orders are prescribed by~$\mu$, and let $\mathbb P\Omega\mathcal{M}_{g}(\mu) = \Omega\mathcal{M}_{g}(\mu)/\mathbb C^{*}$ be the corresponding projectivized stratum.

It is a natural and meaningful question to study how the strata of differentials appear from the perspective of affine geometry—for instance, whether a given stratum can contain a positive-dimensional complete algebraic subvariety.

For a holomorphic signature~$\mu$, that is, when all entries of~$\mu$ are nonnegative, Gendron showed that the unprojectivized stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials contains no positive-dimensional complete subvariety, by applying the maximum modulus principle to shortest saddle connections; see~\cite{G20}. In this case, an alternative proof was later given by the first-named author, using the positivity of certain divisor classes on the moduli space of curves; see~\cite{C23}.

When $\mu$ is a signature of (strictly) meromorphic differentials, again by exploiting the positivity of divisor classes, the first-named author proved that both the projectivized and unprojectivized strata, $\mathbb P\Omega\mathcal M_{g}(\mu)$ and $\Omega\mathcal M_{g}(\mu)$, of strictly meromorphic differentials contain no positive-dimensional complete subvarieties; see~\cite{C19, C24}. Therefore, the remaining question is whether the projectivized strata of holomorphic differentials can contain a positive-dimensional complete subvariety.

This remaining problem is not only the most challenging case, but also significant from the viewpoint of the geometry of canonical divisors. For instance, for $\mu = (2g-2)$, the minimal stratum $\mathbb P\Omega\mathcal M_{g}(2g-2)$ parameterizes subcanonical points $z$, where $(2g-2)z$ is a canonical divisor. Harris asked whether there exist complete families of such subcanonical points (see~\cite[Proof of Corollary~5]{harris84}), a question that has remained open for four decades.

In this paper, we resolve this problem as follows.

In this paper, we resolve this problem as follows.

\begin{lemma}\label{lem:LocalHarmonic}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
For any $(X,\omega) \in M$ and any cylinder $\mathcal{C}$ in $(X,\omega)$, exactly one of the following holds:
\begin{enumerate}
\item There exists a neighborhood $U$ of $(X,\omega)$ in $M$ such that the conformal modulus of $\mathcal{C}$ remains constant on $U$ (\textbf{rigid cylinder});
\item The conformal modulus of $\mathcal{C}$ is a non-constant positive pluriharmonic function in a neighborhood of $(X,\omega)$ in $M$ (\textbf{flexible cylinder}).
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:RigidCylinder}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
Then for every $(X,\omega) \in M$, any cylinder $\mathcal{C}$ in $(X,\omega)$ has constant conformal modulus on a neighborhood of $(X,\omega)$ in $M$; equivalently, every cylinder in $(X,\omega)$ is rigid.
\end{proposition}

\begin{lemma}\label{lem:DichotomyPeriods}
Let $M$ be a positive-dimensional subvariety of the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials.
For any $(X,\omega) \in M$, and for any pair of cylinders $\mathcal{C}_{1},\mathcal{C}_{2}$ in $(X,\omega)$ with waist curves $\alpha_{1}$ and $\alpha_{2}$, exactly one of the following holds:
\begin{enumerate}
\item There exists a neighborhood $U$ of $(X,\omega)$ in $M$ such that the ratio $ \frac{\int_{\alpha_{1}}\omega}{\int_{\alpha_{2}}\omega}$ is constant on $U$ (\textbf{rigid pair of cylinders});
\item The function $ \log\left|\frac{\int_{\alpha_{1}}\omega}{\int_{\alpha_{2}}\omega}\right|$ is a non-constant pluriharmonic function in a neighborhood of $(X,\omega)$ in $M$ (\textbf{flexible pair of cylinders}).
\end{enumerate}
\end{lemma}

\begin{proposition}\label{prop:RigidPair}
Let $M$ be a positive-dimensional complete subvariety contained in the projectivized stratum $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ of holomorphic differentials. Then, for any $(X,\omega) \in M$, every pair of cylinders in $(X,\omega)$ is rigid.
\end{proposition}

\begin{proof}[Proof of Theorem~\ref{thm:MAIN}]
We first consider the case of the projectivized stratum of flat tori $\mathbb{P}\Omega\mathcal{M}_{1}(\emptyset)$, which is one-dimensional. Any holomorphic curve contained in $\mathbb{P}\Omega\mathcal{M}_{1}(\emptyset)$ must coincide with it, and hence is not compact (since flat tori degenerate to a nodal sphere). In the remainder of the proof, we will therefore assume that $\mu$ is nonempty, so that the results of Section~\ref{sec:Cylinders} apply.
\par
We assume, for the sake of contradiction, that there exists a positive-dimensional complete subvariety $M$ in $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$. In the definition of the normalized length, we use the set $MaxCyl(X,\omega)$ of the cylinders of largest width among those having the largest conformal modulus. According to Propositions~\ref{prop:RigidCylinder} and~\ref{prop:RigidPair}, these cylinders remain the ones of largest width among those with the largest conformal modulus as $(X,\omega)$ is deformed in $M$. Deformation along a closed path in $M$ can, however, permute the elements of $MaxCyl(X,\omega)$.
\par
A saddle connection $\alpha$ is called \textbf{rigid} if its normalized length $l(\alpha)$ remains constant in a neighborhood of $(X,\omega)$ in $M$, and \textbf{flexible} otherwise. Observe that for any rigid (resp. flexible) saddle connection $\alpha$ of $(X,\omega)$, there exists a neighborhood of $(X,\omega)$ in $M$ where $\alpha$ persists as a saddle connection and remains rigid (resp. flexible).
This follows from the fact that the ratio $\frac{\int_{\alpha} \omega}{\int_{\beta} \omega}$ is holomorphic in the local coordinates of $M$.
\par
By hypothesis, any translation surface in $M$ can be deformed nontrivially within $M$, so every $(X,\omega)$ in $M$ contains at least one flexible saddle connection. We define the function
$$L\colon M \to \mathbb R^{\geq 0}$$
by assigning to $(X,\omega)$ the normalized length $l(\alpha)$, where $\alpha$ is one of the shortest flexible saddle connections in $(X,\omega)$.
Since a unit-area translation surface contains only finitely many saddle connections of length at most $K$ for any $K>0$, the function $L$ is well defined.
We first show that $L$ is a continuous function on $M$.
\par
We consider a sequence of surfaces $(X_{n},\omega_{n})_{n \in \mathbb{N}}$ in $M$ converging to a surface $(X_{\infty},\omega_{\infty})$ in $M$.
In each $(X_{n},\omega_{n})$, let $\alpha_{n}$ be a flexible saddle connection such that $L(X_{n},\omega_{n}) = l(\alpha_{n})$.
Denote by $\beta_{n}$ the waist curve of a cylinder in $MaxCyl(X_{n},\omega_{n})$.
Up to passing to a subsequence of $(X_{n},\omega_{n})_{n \in \mathbb{N}}$, we may assume that
$$l(\alpha_{n})_{n \in \mathbb{N}}=\frac{|\int_{\alpha_{n}} \omega_n|}{|\int_{\beta_{n}} \omega_n|}$$
converges to some limit $\lambda$.
Let $\alpha_{\infty}$ be a flexible saddle connection such that $L(X_{\infty},\omega_{\infty}) = l(\alpha_{\infty})$.
Since $\alpha_{\infty}$ persists as a flexible saddle connection in a sufficiently small neighborhood of $(X_{\infty},\omega_{\infty})$ in $M$, we deduce that $\lambda \leq l(\alpha_{\infty})$.
It remains to prove that $\lambda \geq l(\alpha_{\infty})$.
\par
Up to passing to a subsequence again, we may assume that the ratios
$$\frac{\int_{\alpha_{n}} \omega_n}{\int_{\beta_n} \omega_n}$$
converge to some limit value $\nu$ satisfying $0 \leq |\nu| = \lambda$. Lemma~\ref{lem:NormalizedLength} already establishes that the normalized length of any saddle connection is bounded below by some positive constant. We therefore deduce that $|\nu|>0$.
\par
Since this convergence takes place in an arbitrarily small contractible neighborhood of $(X_{\infty},\omega_{\infty})$, we can mark the zeros of $\omega_{\infty}$ and, up to taking a further subsequence, require that all the saddle connections $(\alpha_{n})_{n \in \mathbb{N}}$ join the same oriented pair of zeros. The cylinders of $MaxCyl(X_{n},\omega_{n})$ can be marked within this neighborhood, so we may also assume that all the waist curves $\beta_{n}$ belong to the same cylinder. We denote by $\beta$ a waist curve of this cylinder in $(X_{\infty},\omega_{\infty})$.
\par
A subsequence of saddle connections $(\alpha_{\phi(n)})_{n \in \mathbb{N}}$ accumulates on an arc $\gamma$ satisfying $$\frac{\int_{\gamma} \omega_{\infty}}{\int_{\beta} \omega_{\infty}}=\nu,$$
which is formed by one or several oriented saddle connections of the same oriented slope. For $n$ sufficiently large, the relative homology classes $[\alpha_{\phi(n)}]$ and $[\gamma]$ coincide, so at least one saddle connection forming $\gamma$ must be flexible, which we denote by $\delta$. Then this saddle connection $\delta$ satisfies
$l(\delta) \leq |\nu| = \lambda$,
and hence $\lambda \geq l(\alpha_{\infty})$.
Consequently, we obtain $L(X_{n},\omega_{n}) \rightarrow L(X_{\infty},\omega_{\infty})$ as $n \to \infty$, thereby establishing the continuity of $L$.
\par
Since $L$ is continuous on $M$, which is compact, it attains a global minimum realized by some flexible saddle connection $\alpha$ on a translation surface $(X,\omega)$ in $M$. Because the ratio $\frac{\int_{\alpha} \omega}{\int_{\beta} \omega}$ is holomorphic in the local coordinates of $M$, its modulus can decrease again along certain local deformation paths, leading to a contradiction. Therefore, no flexible saddle connection can exist on any translation surface in $M$, which implies that no nontrivial deformation exists for any translation surface in $M$. In conclusion, we deduce that there is no positive-dimensional complete subvariety in $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$.
\end{proof}

Besides Abelian differentials, one can also study the strata of $k$-differentials with prescribed zero and pole orders, where a $k$-differential is a section of the $k$-th power of the canonical bundle. Note that a $k$-differential with pole orders at most $k-1$ (i.e., when the corresponding $\frac{1}{k}$-translation surface has finite area) can be lifted via the canonical cyclic covering construction to the $k$-th power of a holomorphic one-form; see~\cite[Section 2]{BCGGM-k}. Theorem~\ref{thm:MAIN} therefore implies the following corollary, which settles the remaining case concerning complete subvarieties in the strata of $k$-differentials; see~\cite[Remark]{C23}.

\begin{corollary}\label{cor:k-diff}
    For any stratum $\Omega^k\mathcal M_{g}(\mu)$ of $k$-differentials whose pole orders are bounded by $k-1$, the projectivized stratum $\mathbb P\Omega^k\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvarieties.
\end{corollary}

To prove Theorem~\ref{thm:MAIN}, we refine Gendron’s argument from~\cite{G20}. By applying the maximum principle to suitably chosen plurisubharmonic functions, we show that, in any complete subvariety contained in a projectivized stratum, the following quantities are rigid:
\begin{itemize}
\item the conformal moduli of cylinders (see Proposition~\ref{prop:RigidCylinder});
\item the periods of closed geodesics, up to a global scaling (see Proposition~\ref{prop:RigidPair});
\item suitably normalized lengths of saddle connections (see Section~\ref{sec:Main}).
\end{itemize}

Finally, we remark that since $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ is a quasi-projective variety, its complete algebraic subvarieties and complex analytic subvarieties   coincide by the GAGA principle. Moreover, if a complete subvariety $M$ is singular, we may work with a resolution of singularities of $M$ and pull back the family of differentials accordingly. Alternatively, one can intersect a higher-dimensional complete subvariety with ample hypersurfaces until obtaining a complete algebraic curve. Therefore, Theorem~\ref{thm:MAIN} is equivalent to showing that $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ contains no complete algebraic curves, in which case we may work with the normalization of such a curve, which is smooth. Therefore, in these senses, even when $M$ is singular, we can still speak of and make use of holomorphic local coordinates on $M$. We shall do so without further comment.

To prove Theorem~\ref{thm:MAIN}, the authors “refine Gendron’s argument from~\cite{G20}.” The strategy is to “apply[] the maximum principle to suitably chosen plurisubharmonic functions” and deduce that on any complete subvariety contained in a projectivized stratum, several geometric quantities are rigid: (1) “the conformal moduli of cylinders” (Proposition~\ref{prop:RigidCylinder}); (2) “the periods of closed geodesics, up to a global scaling” (Proposition~\ref{prop:RigidPair}); and (3) “suitably normalized lengths of saddle connections” (Section~\ref{sec:Main}). They also reduce to the curve case: since $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ is quasi-projective, complete algebraic and complex analytic subvarieties coincide by GAGA; if needed one passes to a resolution/normalization, and by intersecting with ample hypersurfaces, Theorem~\ref{thm:MAIN} is “equivalent to showing that $\mathbb{P}\Omega\mathcal{M}_{g}(\mu)$ contains no complete algebraic curves.”

\label{thm:MAIN}
For any stratum $\Omega\mathcal M_{g}(\mu)$ of holomorphic differentials, the projectivized stratum $\mathbb P\Omega\mathcal M_{g}(\mu)$ contains no positive-dimensional complete subvariety.,

Question:
Let $g$ be a genus and let $\mu$ be any holomorphic signature with total order $2g-2$ (so $\Omega\mathcal M_g(\mu)$ is the stratum of holomorphic Abelian differentials $\omega$ on smooth, connected, genus-$g$ complex curves $X$ whose zero orders are prescribed by $\mu$). Let
$$\mathbb P\Omega\mathcal M_g(\mu)=\Omega\mathcal M_g(\mu)/\mathbb C^*$$
be the projectivized stratum, obtained by identifying differentials that differ by nonzero scalar multiplication. Which statement holds for every such projectivized stratum?

The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety.

Choices:
B. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no complete subvariety of codimension one.
C. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ is not itself a positive-dimensional complete variety.
D. The projectivized stratum $\mathbb P\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety whenever $g\ge 2$.
E. The unprojectivized stratum $\Omega\mathcal M_g(\mu)$ contains no positive-dimensional complete subvariety.

absence_of_all_positive_dimensional_complete_subvarieties weakened to non-completeness of the whole stratum


\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


\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