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Shuangjie Peng

Publications and source records attributed to Shuangjie Peng.

At least 19 recordsLinked to original sources
arXiv2026

Quantitative analysis of ground states for the fractional logarithmic Schrödinger equation

Let $N\geq1$ and $0<s<1$. We study positive ground states of the fractional logarithmic Schrödinger equation \begin{equation*} (-Δ)^sQ=Q\log Q \quad\text{in }\mathbb{R}^N. \end{equation*} We prove that for every $N\geq1$ and $0<s<1$, the positive ground state is unique up to translations and nondegenerate. More precisely, for the linearized operator $L_Q=(-Δ)^s-1-\log Q$, it holds that \begin{equation*} \ker L_Q=\operatorname{span}\{\partial_{x_1}Q,\cdots,\partial_{x_N}Q\}. \end{equation*} A main difficulty is that the potential $-1-\log Q$ is unbounded in $\mathbb{R}^N$, which prevents a direct application of the available radial oscillation theory for fractional Schrödinger operators with bounded potentials. We overcome this difficulty by a bounded-potential approximation. Using also the fact that the associated quadratic form has Morse index one and an angular decomposition, we obtain the nondegeneracy. Based on the isolation of logarithmic ground states and the uniqueness theory for the fractional power equation, we prove uniqueness by a variational approximation with subcritical power nonlinearities. As an application, we establish sharp fractional logarithmic Sobolev inequalities and characterize all cases of equality.

math.AP↗
arXiv2026

Axial Symmetry of Normalized Solutions for Magnetic Gross-Pitaevskii Equations with Anharmonic Potentials

This paper is concerned with normalized solutions of the magnetic focusing Gross-Pitaevskii equations with anharmonic potentials in $\mathbb{R}^N$, where $N=2$ or $3$. We construct axially symmetric normalized concentrating solutions as the parameter $a>0$ approaches $a_*(N)$, where $a_*(N)\geq0$ is a critical constant depending only on $N$. We further prove that up to a constant phase (and a rotational transformation for $N=2$), normalized concentrating solutions are unique and axially symmetric as $a\to a_*(N)$. When $N=3$, we also prove that the corresponding unique normalized concentrating solution is free of vortices as $a\to a_*(3)$, even if the anharmonic potential is non-radially symmetric.

math.AP↗
arXiv2026

On the uniqueness of the critical point of $ψ_Ω$

We prove that for any bounded convex domain $Ω\subset \mathbb{R}^n$, the function \begin{equation*} ψ_Ω(ξ) = \int_{\mathbb{R}^n\setminusΩ} \frac{\mathrm{d}x}{|x-ξ|^{2n}}, \quad ξ\inΩ, \end{equation*} has exactly one critical point. This confirms an conjecture proposed by Clapp, Pistoia and Saldaña in [J. Math. Pures Appl. 205 (2026), 103783]. The proof uses a spherical coordinates representation to write $ψ_Ω$ as an integral of the distance function $ρ(ξ,ω)$. This approach is not limited to $ψ_Ω$. Instead, it provides a general framework for analyzing a broad class of functionals involving the boundary distance. We also examine non-convex domains. In particular, a single annulus exhibits a full circle of critical points, while multiple concentric annuli produce finitely many critical spheres. These examples show that the convexity hypothesis is essential for the uniqueness conclusion. The method developed here for handling spherical integrals involving the distance function is likely to be useful in other geometric and analytic contexts.

math.AP↗
arXiv2026

Quantization analysis of Moser-Trudinger equations in the Poincaré disk and applications

In this paper, we first establish the quantitative properties for positive solutions to the Moser-Trudinger equations in the two-dimensional Poincaré disk $\mathbb{B}^2$: \begin{equation*}\label{mt1} \left\{ \begin{aligned} &-Δ_{\mathbb{B}^2}u=λue^{u^2},\ x\in\mathbb{B}^2, &u\to0,\ \text{when}\ ρ(x)\to\infty, &||\nabla_{\mathbb{B}^2} u||_{L^2(\mathbb{B}^2)}^2\leq M_0, \end{aligned} \right. \end{equation*} where $0<λ<\frac{1}{4}=\inf\limits_{u\in W^{1,2}(\mathbb{B}^2)\backslash\{0\}}\frac{\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2}{\|u\|_{L^2(\mathbb{B}^2)}^2}$, $ρ(x)$ denotes the geodesic distance between $x$ and the origin and $M_0$ is a fixed large positive constant (see Theorem 1.1). Furthermore, by doing a delicate expansion for Dirichlet energy $\|\nabla_{\mathbb{B}^2}u\|_{L^2(\mathbb{B}^2)}^2$ when $λ$ approaches to $0,$ we prove that there exists $Λ^\ast>4π$ such that the Moser-Trudinger functional $F(u)=\int_{\mathbb{B}^2}\left(e^{u^2}-1\right) dV_{\mathbb{B}^2}$ under the constraint $\int_{\mathbb{B}^2}|\nabla_{\mathbb{B}^2}u|^2 dV_{\mathbb{B}^2}=Λ$ has at least one positive critical point for $Λ\in(4π,Λ^{\ast})$ up to some Möbius transformation. Finally, when $λ\rightarrow 0$, by doing a more accurate expansion for $u$ near the origin and away from the origin, applying a local Pohozaev identity around the origin and the uniqueness of the Cauchy initial value problem for ODE,Cauchy-initial uniqueness for ODE, we prove that the Moser-Trudinger equation only has one positive solution when $λ$ is close to $0.$ During the process of the proofs, we overcome some new difficulties which involves the decay properties of the positive solutions, as well as some precise expansions for the solutions both near the origin and away from the origin.

math.AP↗
arXiv2023

Existence of Boundary Layers for the supercritical Lane-Emden Systems

We consider the following supercritical problem for the Lane-Emden system: \begin{equation}\label{eq00} \begin{cases} -Δu_1=|u_2|^{p-1}u_2\ &in\ D,\\ -Δu_2=|u_1|^{q-1}u_1 \ &in\ D,\\ u_1=u_2=0\ &on\ \partial D, \end{cases} \end{equation} where $D$ is a bounded smooth domain in $\mathbb{R}^N$, $N\geq4.$ What we mean by supercritical is that the exponent pair $(p,q)\in(1,\infty)\times(1,\infty)$ satisfies $\frac1{p+1}+\frac1{q+1}<\frac{N-2}N$. We prove that for some suitable domains $D\subset\mathbb{R}^N$, there exist positive solutions with layers concentrating along one or several $k$-dimensional sub-manifolds of $\partial D$ as $$\frac1{p+1}+\frac1{q+1} \rightarrow \frac{n-2}{n},\ \ \ \ \frac{n-2}{n}<\frac1{p+1}+\frac1{q+1}<\frac{N-2}N,$$ where $n:=N-k$ with $1\leq k\leq N-3$. By transforming the original problem \eqref{eq00} into a lower $n$-dimensional weighted system, we carry out the reduction framework and apply the blow-up analysis. The properties of the ground states related to the limit problem play a crucial role in this process. The corresponding exponent pair $(p_0,q_0)$, which represents the limit pair of $(p,q)$, lies on the critical hyperbola $\frac n{p_0+1}+\frac n{q_0+1}=n-2$. It is widely recognized that the range of the smaller exponent, say $p_0$, has a profound impact on the solutions, with $p_0=\frac n{n-2}$ being a threshold. It is worth emphasizing that this paper tackles the problem by considering two different ranges of $p_0$, which is contained in $p_0>\frac n{n-2}$ and $p_0<\frac n{n-2}$ respectively. The coupling mechanisms associated with these ranges are completely distinct, necessitating different treatment approaches. This represents the main challenge overcome and the novel element of this study..

math.AP↗
arXiv2023

Sign-changing solutions to the slightly supercritical Lane-Emden system with Neumann boundary conditions

We consider the following slightly supercritical problem for the Lane-Emden system with Neumann boundary conditions: \begin{equation*} \begin{cases} -Δu_1=|u_2|^{p_ε-1}u_2,\ &in\ Ω,\\ -Δu_2=|u_1|^{q_ε-1}u_1, \ &in\ Ω,\\ \partial_νu_1=\partial_νu_2=0,\ &on\ \partialΩ\end{cases} \end{equation*} where $Ω=B_1(0)$ is the unit ball in $\mathbb{R}^n$ ($n\geq4$) centered at the origin, $p_ε=p+αε, q_ε=q+βε$ with $α,β>0$ and $\frac1{p+1}+\frac1{q+1}=\frac{n-2}n$. We show the existence and multiplicity of concentrated solutions based on the Lyapunov-Schmidt reduction argument incorporating the zero-average condition by certain symmetries. It is worth noting that we simultaneously consider two cases: $p>\frac n{n-2}$ and $p<\frac n{n-2}$. The coupling mechanisms of the system are completely different in these different cases, leading to significant changes in the behavior of the solutions. The research challenges also vary. Currently, there are very few papers that take both ranges into account when considering solution construction. Therefore, this is also the main feature and new ingredient of our work.

math.AP↗
arXiv2023

Semi-classical states for fractional Choquard equations with decaying potentials

This paper deals with the following fractional Choquard equation $$\varepsilon^{2s}(-Δ)^su +Vu=\varepsilon^{-α}(I_α*|u|^p)|u|^{p-2}u\ \ \ \mathrm{in}\ \mathbb{R}^N,$$ where $\varepsilon>0$ is a small parameter, $(-Δ)^s$ is the fractional Laplacian, $N>2s$, $s\in(0,1)$, $α\in\big((N-4s)_{+}, N\big)$, $p\in[2, \frac{N+α}{N-2s})$, $I_α$ is a Riesz potential, $V\in C\big(\mathbb{R}^N, [0, +\infty)\big)$ is an electric potential. Under some assumptions on the decay rate of $V$ and the corresponding range of $p$, we prove that the problem has a family of solutions $\{u_\varepsilon\}$ concentrating at a local minimum of $V$ as $\varepsilon\to 0$. Since the potential $V$ decays at infinity, we need to employ a type of penalized argument and implement delicate analysis on the both nonlocal terms to establish regularity, positivity and asymptotic behaviour of $u_\varepsilon$, which is totally different from the local case. As a contrast, we also develop some nonexistence results, which imply that the assumptions on $V$ and $p$ for the existence of $u_\varepsilon$ are almost optimal. To prove our main results, a general strong maximum principle and comparison function for the weak solutions of fractional Laplacian equations are established. The main methods in this paper are variational methods, penalized technique and some comparison principle developed in this paper.

math.AP↗
arXiv2023

Existence and decays of solutions for fractional Schrödinger equations with decaying potentials

We revisit the following fractional Schrödinger equation \begin{align}\label{1a} \varepsilon^{2s}(-Δ)^su +Vu=u^{p-1},\,\,\,u>0,\ \ \ \mathrm{in}\ \R^N, \end{align} where $\varepsilon>0$ is a small parameter, $(-Δ)^s$ denotes the fractional Laplacian, $s\in(0,1)$, $p\in (2, 2_s^*)$, $2_s^*=\frac {2N}{N-2s}$, $N>2s$, $V\in C\big(\R^N, [0, +\infty)\big)$ is a potential. Under various decay assumptions on $V$, we introduce a uniform penalization argument combined with a comparison principle and iteration process to detect an explicit threshold value $p_*$, such that the above problem admits positive concentration solutions if $p\in (p_*, \,2_s^*)$, while it has no positive weak solutions for $p\in (2,\,p_*)$ if $p_*>2$, where the threshold $p_*\in [2, 2^*_s)$ can be characterized explicitly by \begin{equation*}\label{qdj111} p_*=\left\{\begin{array}{l} 2+\frac {2s}{N-2s} \ \ \ \text { if } \lim\limits_{|x| \to \infty} (1+|x|^{2s})V(x)=0,\vspace{1mm} 2+\frac ω{N+2s-ω} \text { if } 0<\inf (1+|x|^ω)V(x)\le \sup (1+|x|^ω)V(x)< \infty \text { for some } ω\in [0, 2s],\vspace{1mm} 2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text { if } \inf V(x)\log(e+|x|^2)>0. \end{array}\right. \end{equation*} Moreover, corresponding to the various decay assumptions of $V(x)$, we obtain the decay properties of the solutions at infinity.

math.AP↗
arXiv2022

Qualitative analysis for Moser-Trudinger nonlinearities with a low energy

We are concerned with the Moser-Trudinger problem \begin{equation*} \begin{cases} -Δu=λue^{u^2}~~&\mbox{in}~Ω,\\[0.5mm] u>0 ~~ &{\text{in}~Ω},\\[0.5mm] u=0~~&\mbox{on}~\partial Ω, \end{cases} \end{equation*} where $Ω\subset \mathbb{R}^2$ is a smooth bounded domain and $λ>0$ is sufficiently small. Qualitative analysis for Moser-Trudinger nonlinearities has been studied in recent decades, however there is still a lot of clarity about this issue, even for a low energy. The reason is that this problem is a critical exponent for dimension two and will lose compactness. Here by using a variety of local Pohozaev identities, we qualitatively analyze the positive solutions of Moser-Trudinger problem with a low energy, which contains the Morse index, non-degeneracy, asymptotic behavior, uniqueness and symmetry of solutions. Since the fundamental solution of $-Δ$ in $Ω\subset \mathbb{R}^2$ is in logarithmic form and the corresponding bubble is exponential growth, more precise asymptotic behavior of the solutions is needed, which is of independent interest. Moreover, to obtain our results, some ODE's theory will be used to a prior estimate of the solutions and some elliptic theory in dimension two will play a crucial role.

math.AP↗
arXiv2022

The $C^0$-convergence at the Neumann boundary for Liouville equations

In this paper, we study the blow-up analysis for a sequence of solutions to the Liouville type equation with exponential Neumann boundary condition. For interior case, i.e. the blow-up point is an interior point, Li \cite{Li} gave a uniform asymptotic estimate. Later, Zhang \cite{Zhang} and Gluck \cite{Gluck} improved Li's estimate in the sense of $C^0$-convergence by using the method of moving planes or classification of solutions of the linearized version of Liouville equation. If the sequence blows up at a boundary point, Bao-Wang-Zhou \cite{Bao-Wang-Zhou} proved a similar asymptotic estimate of Li \cite{Li}. In this paper, we will prove a $C^0$-convergence result in this boundary blow-up process. Our method is different from \cite{Zhang,Gluck}.

math.AP↗
arXiv2022

Existence and non-degeneracy of positive multi-bubbling solutions to critical elliptic systems of Hamiltonian type

This paper deals with the following critical elliptic systems of Hamiltonian type, which are variants of the critical Lane-Emden systems and analogous to the prescribed curvature problem: \begin{equation*} \begin{cases} -Δu_1=K_1(y)u_2^{p},\ y\in \mathbb{R}^N,\\ -Δu_2=K_2(y)u_1^{q}, \ y\in \mathbb{R}^N,\\ u_1,u_2>0, \end{cases} \end{equation*} where $N\geq 5, p,q\in(1,\infty)$ with $\frac1{p+1}+\frac1{q+1}=\frac{N-2}N$, $K_1(y)$ and $K_2(y)$ are positive radial potentials. At first, under suitable conditions on $K_1,K_2$ and the certain range of the exponents $p,q$, we construct an unbounded sequence of non-radial positive vector solutions, whose energy can be made arbitrarily large. Moreover, we prove a type of non-degeneracy result by use of various Pohozaev identities, which is of great interest independently. The indefinite linear operator and strongly coupled nonlinearities make the Hamiltonian-type systems in stark contrast both to the systems of Gradient type and to the single critical elliptic equations in the study of the prescribed curvature problems. It is worth noting that, in higher-dimensional cases $(N\geq5)$, there have been no results on the existence of infinitely many bubbling solutions to critical elliptic systems, either of Hamiltonian or Gradient type.

math.AP↗
arXiv2022

Multi-peak semiclassical bound states for Fractional Schrödinger Equations with fast decaying potentials

We study the following fractional Schrödinger equation \begin{equation*}\label{eq0.1} \varepsilon^{2s}(-Δ)^s u + V(x)u = f(u), \,\,x\in\mathbb{R}^N, \end{equation*} where $s\in(0,1)$. Under some conditions on $f(u)$, we show that the problem has a family of solutions concentrating at any finite given local minima of $V$ provided that $V\in C(\R^N,[0,+\infty))$. All decay rates of $V$ are admissible. Especially, $V$ can be compactly supported. Different from the local case $s=1$ or the case of single-peak solutions, the nonlocal effect of the operator $(-Δ)^s$ makes the peaks of the candidate solutions affect mutually, which causes more difficulties in finding solutions with multiple bumps. The methods in this paper are penalized technique and variational method.

math.AP↗
arXiv2021

Local Uniqueness of Ground States for Rotating Bose-Einstein Condensates with Attractive Interactions

We study ground states of two-dimensional Bose-Einstein condensates with attractive interactions in a trap $V(x)$ rotating at the velocity $Ω$. It is known that there exist a critical rotational velocity $0<Ω^*:=Ω^*(V)\leq \infty$ and a critical number $0<a^*<\infty$ such that for any rotational velocity $0\le Ω<Ω^*$, ground states exist if and only if the coupling constant $a$ satisfies $a<a^*$. For a general class of traps $V(x)$, which may not be symmetric, we prove in this paper that up to a constant phase, there exists a unique ground state as $a\nearrow a^*$, where $Ω\in(0,Ω^*)$ is fixed. This result extends essentially our recent uniqueness result, where only the radially symmetric traps $V(x)$ could be handled with.

math.AP↗
arXiv2021

Non-degeneracy and existence of new solutions for the Schrödinger equations

We consider the following nonlinear problem $$ (P) \quad \quad - Δu + V(|y|)u=u^{p},\quad u>0 \quad \mbox{in} \ {\mathbb{R}}^N, \quad u \in H^1({\mathbb{R}}^N), $$ where $V(r)$ is a positive function, $1<p <\frac{N+2}{N-2}$. We show that the multi-bump solutions constructed in [20] is non-degenerate in a suitable symmetric space. We also use this non-degenerate result to construct new solutions for (P).

math.AP↗
arXiv2021

Existence and Asymptotic Behavior of Ground States for Rotating Bose-Einstein Condensates

We study ground states of two-dimensional Bose-Einstein condensates with repulsive ($a>0$) or attractive ($a<0$) interactions in a trap $V (x)$ rotating at the velocity $Ω$. It is known that there exist critical parameters $a^*>0$ and $Ω^*:=Ω^*(V(x))>0$ such that if $Ω>Ω^*$, then there is no ground state for any $a\in\R$; if $0\le Ω<Ω^*$, then ground states exist if and only if $a\in(-a^*,+\infty)$. As a completion of the existing results, in this paper, we focus on the critical case where $0<Ω=Ω^*<+\infty$ and classify the existence and nonexistence of ground states for $a\in\R$. Moreover, for a suitable class of radially symmetric traps $V(x)$, employing the inductive symmetry method, we prove that up to a constant phase, the ground states must be real-valued, unique and free of vortices as $Ω\searrow 0$, no matter whether the interactions of the condensates are repulsive or not.

math.AP↗
arXiv2020

The number of positive solutions to the Brezis-Nirenberg problem

In this paper we are concerned with the well-known Brezis-Nirenberg problem \begin{equation*} \begin{cases} -Δu= u^{\frac{N+2}{N-2}}+\varepsilon u, &{\text{in}~Ω},\\ u>0, &{\text{in}~Ω},\\ u=0, &{\text{on}~\partial Ω}. \end{cases} \end{equation*} The existence of multi-peak solutions to the above problem for small $\varepsilon>0$ was obtained by Musso and Pistoia. However, the uniqueness or the exact number of positive solutions to the above problem is still unknown. Here we focus on the local uniqueness of multi-peak solutions and the exact number of positive solutions to the above problem for small $\varepsilon>0$. By using various local Pohozaev identities and blow-up analysis, we first detect the relationship between the profile of the blow-up solutions and the Green's function of the domain $Ω$ and then obtain a type of local uniqueness results of blow-up solutions. At last we give a description of the number of positive solutions for small positive $\varepsilon$, which depends also on the Green's function.

math.AP↗
arXiv2019

Excited states on Bose-Einstein condensates with attractive interactions

We study the Bose-Einstein condensates (BEC) in two or three dimensions with attractive interactions, described by $L^{2}$ constraint Gross-Pitaevskii energy functional. First, we give the precise description of the chemical potential of the condensate $μ$ and the attractive interaction $a$. Next, for a class of degenerated trapping potential with non-isolated critical points, we obtain the existence and the local uniqueness of excited states by precise analysis of the concentrated points and the Lagrange multiplier. To our best knowledge, this is the first result concerning on excited states of BEC in Mathematics. Also, our results show that $ka_*$ are critical values in two dimension when the concentration occurs for any positive integer $k$ with some positive constant $a_*$. And we point out that our results on degenerated trapping potential with non-isolated critical points are also new even for the classical Schrödinger equations. Here our main tools are finite-dimensional reduction and various Pohozave identities. The main difficulties come from the estimates on Lagrange multiplier and the different degenerate rate along different directions at the critical points of $V(x)$.

math.AP↗