arXiv subjects

Chunhua Wang

Publications and source records attributed to Chunhua Wang.

At least 19 recordsLinked to original sources

arXiv2026

Existence and Uniqueness of Normalized Multi-peak Solutions for Coupled Nonlinear Schrödinger Systems

We consider the following two-component coupled nonlinear Schrödinger (CNLS) system: \[ \begin{cases} -Δu +(P(x) + λ) u=μ_1 u^3+βu v^2, & \text{in } \mathbb{R}^N,\\ -Δv +(Q(x) + λ) v =μ_2 v^3+βvu^2, & \text{in } \mathbb{R}^N \end{cases} \] with the mass constraint $\int_{\mathbb{R}^N} (u^2+v^2)\,dx = ρ^2$ for $N=2,3$, where $ρ>0$ is a parameter. By employing the Lyapunov-Schmidt reduction and local Pohozaev identities, we establish the existence and local uniqueness of normalized multi-peak solutions: the result holds for sufficiently small $ρ$ when $N=3$, and for $ρ$ approaching a critical threshold when $N=2$. The main difficulty lies in that the mass constraint involves interactions among all concentration points, while a more refined characterization of such normalized solutions further requires sharp order estimates. In this work, we have discovered some new phenomena that differ from those of solutions without mass constraint and single-peak solutions.

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↗

arXiv2026

Quantitative properties of the Hardy-type mean field equation

In this paper, we consider the following Hardy-type mean field equation \[ \left\{ {\begin{array}{*{20}{c}} { - Δu-\frac{1}{(1-|x|^2)^2} u = λe^u}, & {\rm in} \ \ B_1,\\ {\ \ \ \ u = 0,} &\ {\rm on}\ \partial B_1, \end{array}} \right. \] \[\] where $λ>0$ is small and $B_1$ is the standard unit disc of $\mathbb{R}^2$. Applying the moving plane method of hyperbolic space and the accurate expansion of heat kernel on hyperbolic space, we establish the radial symmetry and Brezis-Merle lemma for solutions of Hardy-type mean field equation. Meanwhile, we also derive the quantitative results for solutions of Hardy-type mean field equation, which improves significantly the compactness results for classical mean-field equation obtained by Brezis-Merle and Li-Shafrir. Furthermore, applying the local Pohozaev identity from scaling, blow-up analysis and a contradiction argument, we prove that the solutions are unique when $λ$ is sufficiently close to 0.

math.AP↗

arXiv2025

Normalized vector solutions of nonlinear Schrödinger systems

Given $μ>0$ we look for solutions $ λ\in\mathbb{R}$ and $v_1,\dots,v_k\in H^1(\mathbb{R}^N)$ of the system \[ \begin{cases} \displaystyle -Δv_i+ λv_i+V_i(x)v_i = \sum_{\substack{j=1}}^kβ_{ij} v_iv_j^2 &\text{ in } \mathbb{R}^N, \text{ } i=1,\dots,k,\newline \displaystyle \int_{\mathbb{R}^N} \left(v_1^2+\dots+v_k^2 \right)\mathrm{d} x = μ, \end{cases}\] where $N=1,2,3$, $V_i:\mathbb R^N\to \mathbb R$ and $β_{ij}\in\mathbb{R}$ satisfy $β_{ij}=β_{ji}$ and $β_{ii}>0$. Under suitable assumptions on the $β_{ij}$'s, given a non-degenerate critical point $ξ_0$ of a suitable linear combination of the potentials $V_i$, we build solutions whose components concentrate at $ξ_0$ as the prescribed global mass $μ$ is either large (when $N=1$) or small (when $N=3$) or it approaches some critical threshold (when $N=2$).

math.AP↗

arXiv2025

Brouwer degree for Chern-Simons Higgs models on finite graphs

Let $G=(V, E)$ be a finite connected graph, where $V$ denotes the set of vertices and $E$ denotes the set of edges. We revisit the following Chern-Simons Higgs model, \begin{equation*} Δu=λ\mathrm{e}^u\left(\mathrm{e}^u-1\right)+f \ \text {in} \ V, \end{equation*} where $Δ$ is the graph Laplacian, $λ$ is a real number and $f$ is a function defined on $V$. Firstly, when $λ\int_V f \mathrm{d} μ\neq 0$, we find that the odevity of the number of vertices in the graph affects the number of solutions. Then by calculating the topological degree and using the relationship between the degree and the critical group of a related functional, we obtain the existence of multiple solutions. Also we study the existence of solutions when $λ\int_V f \mathrm{d} μ=0$. These findings extend the work of Huang et al. [Comm Math Phys 377:613-621 (2020)], Hou and Sun [Calc Var 61:139 (2022)] and Li et al. [Calc Var 63:81 (2024)]. Similarly, for the generalized Chern-Simons Higgs model, we obtain the same results. Moreover, this method is also applied to the Chern-Simons Higgs system, yielding partial results for the existence of multiple solutions. To our knowledge, this is the first instance where it has been concluded that an equation on graphs can have at least three distinct solutions. We think that our results will be valuable for studying the multiplicity of solutions to analogous equations on graphs.

math.AP↗

arXiv2024

A new type of bubble solutions for a critical fractional Schrödinger equation

We consider the following critical fractional Schrödinger equation \begin{equation*} (-Δ)^s u+V(|y'|,y'')u = u^{2_s^*-1},\quad u>0,\quad y =(y',y'') \in \mathbb{R}^3\times\mathbb{R}^{N-3}, \end{equation*} where $N\geq 3,s\in(0,1)$, $2_s^*=\frac{2N}{N-2s}$ is the fractional critical Sobolev exponent and $V(|y'|,y'')$ is a bounded non-negative function in $\mathbb{R}^3\times\mathbb{R}^{N-3}$. If $r^{2s}V(r,y'')$ has a stable critical point $(r_0,y_0'')$ with $r_0>0$ and $V(r_0,y_0'')>0$, by using a finite-dimensional reduction method and various local Pohozaev identities, we prove that the problem above has a new type of infinitely many solutions which concentrate at points lying on the top and the bottom of a cylinder. And the concentration points of the bubble solutions include saddle points of the function $r^{2s}V(r,y'')$. We have to overcome some difficulties caused by the non-localness of the fractional Laplacian.

math.AP↗

arXiv2022

Existence and local uniqueness of multi-peak solutions for the Chern-Simons-Schrödinger system

In the present paper, we consider the Chern-Simons-Schrödinger system \begin{equation} \left\{ \begin{aligned} &-\varepsilon^{2}Δu+V(x)u+(A_{0}+A_{1}^{2}+A_{2}^{2})u=|u|^{p-2}u,\,\,\,\,x\in \mathbb{R}^2,\\ &\partial_1 A_0 = A_2 u^2,\ \partial_{2}A_{0}=-A_{1}u^{2},\\ &\partial_{1}A_{2}-\partial_{2}A_{1}=-\frac{1}{2}|u|^{2},\ \partial_{1}A_{1}+\partial_{2}A_{2}=0,\\ \end{aligned} \right. \end{equation} where $p>2,$ $\varepsilon>0$ is a parameter and $V:\mathbb{R}^{2}\rightarrow\mathbb{R}$ is a bounded continuous function. Under some mild assumptions on $V(x)$, we show the existence and local uniqueness of positive multi-peak solutions. Our methods mainly use the finite dimensional reduction method, various local Pohozaev identities, blow-up analysis and the maximum principle. Because of the nonlocal terms involved by $A_{0},A_{1}$ and $A_{2},$ we have to obtain a series of new and technical estimates.

math.AP↗

arXiv2022

Existence, Local uniqueness and periodicity of bubbling solutions for a critical nonlinear elliptic equation

We revisit the following nonlinear critical elliptic equation \begin{equation*} -Δu+Q(y)u=u^{\frac{N+2}{N-2}},\;\;\; u>0\;\;\;\hbox{ in } \mathbb{R}^N, \end{equation*} where $N\geq 5.$ There seems to be no results about the periodicity of bubbling solutions. Here we try to investigate some related problems. Assuming that $Q(y)$ is periodic in $y_1$ with period 1 and has a local minimum at 0 satisfying $Q(0)=0,$ we prove the existence and local uniqueness of infinitely many bubbling solutions of the problem above. This local uniqueness result implies that some bubbling solutions preserve the symmetry of the potential function $Q(y),$ i.e. the bubbling solution whose blow-up set is $\{(jL,0,...,0):j=0,\pm 1, \pm 2,..., \pm m\}$ must be periodic in $y_{1}$ provided that $L$ is large enough, where $m$ is the number of the bubbles which is large enough but independent of $L.$ Moreover, we also show a non-existence of this bubbling solutions for the problem above if the local minimum of $Q(y)$ does not equal to zero.

math.AP↗

arXiv2022

Nonstandard solutions for a perturbed nonlinear Schrödinger system with small coupling coefficients\protect\thanks{A perturbed nonlinear Schrödinger system

In this paper, we consider the following weakly coupled nonlinear Schrödinger system \begin{equation*} \left\{ \begin{array}{ll} -ε^{2}Δu_1 + V_1(x)u_1 = |u_1|^{2p - 2}u_1 + β|u_1|^{p - 2}|u_2|^pu_1, & x\in \mathbb{R}^N,\\ -ε^{2}Δu_2 + V_2(x)u_2 = |u_2|^{2p - 2}u_2 + β|u_2|^{p - 2}|u_1|^pu_2, & x\in \mathbb{R}^N, \end{array} \right. \end{equation*} where $ε>0$, $β\in\mathbb{R}$ is a coupling constant, $2p\in (2,2^*)$ with $2^* = \frac{2N}{N - 2}$ if $N\geq 3$ and $+\infty$ if $N = 1,2$, $V_1$ and $V_2$ belong to $C(\mathbb{R}^N,[0,\infty))$. When $p\ge 2$ and $β>0$ is suitably small, we show that the problem has a family of nonstandard solutions $\{w_ε = (u^1_ε,u^2_ε):0<ε<ε_{0}\}$ concentrating synchronously at the common local minimum of $V_1$ and $V_2$. All decay rates of $V_i(i=1,2)$ are admissible and we can allow that $β>0$ is close to $0$ in this paper. Moreover, the location of concentration points is given by local Pohozaev identities. Our proofs are based on variational methods and the penalized technique.

math.AP↗

arXiv2022

A new type of bubble solutions for a Schrödinger equation with critical growth

In this paper, we investigate the following critical elliptic equation $$ -Δu+V(y)u=u^{\frac{N+2}{N-2}},\,\,u>0,\,\,\text{in}\,\R^{N},\,\,u\in H^{1}(\R^{N}), $$ where $V(y)$ is a bounded non-negative function in $\R^{N}.$ Assuming that $V(y)=V(|\hat{y}|,y^{*}),y=(\hat{y},y^{*})\in \R^{4}\times \R^{N-4}$ and gluing together bubbles with different concentration rates, we obtain new solutions provided that $N\geq 7,$ whose concentrating points are close to the point $(r_{0},y^{*}_{0})$ which is a stable critical point of the function $r^{2}V(r,y^{*})$ satisfying $r_{0}>0$ and $V(r_{0},y^{*}_{0})>0.$ In order to construct such new bubble solutions for the above problem, we first prove a non-degenerate result for the positive multi-bubbling solutions constructed in \cite{PWY-18-JFA} by some local Pohozaev identities, which is of great interest independently. Moreover, we give an example which satisfies the assumptions we impose.

math.AP↗

arXiv2021

Existence and local uniqueness of normalized peak solutions for a Schrodinger-Newton system

In this paper, we investigate the existence and local uniqueness of normalized peak solutions for a Schrödinger-Newton system under the assumption that the trapping potential is degenerate and has non-isolated critical points. First we investigate the existence and local uniqueness of normalized single-peak solutions for the Schrödinger-Newton system. Precisely, we give the precise description of the chemical potential $μ$ and the attractive interaction $a$. Then we apply the finite dimensional reduction method to obtain the existence of single-peak solutions. Furthermore, using various local Pohozaev identities, blow-up analysis and the maximum principle, we prove the local uniqueness of single-peak solutions by precise analysis of the concentrated points and the Lagrange multiplier. Finally, we also prove the nonexistence of multi-peak solutions for the Schrödinger-Newton system, which is markedly different from the corresponding Schrödinger equation. The nonlocal term results in this difference. The main difficulties come from the estimates on Lagrange multiplier, the different degenerate rates along different directions at the critical point of $P(x)$ and some complicated estimates involved by the nonlocal term. To our best knowledge, it may be the first time to study the existence and local uniqueness of solutions with prescribed $L^{2}$-norm for the Schrödinger-Newton system.

math.AP↗

arXiv2021

Concentrated solutions to fractional Schrödinger equations with prescribed $L^2$-norm

We investigate the existence and local uniqueness of normalized $k$-peak solutions for the fractional Schrödinger equations with attractive interactions with a class of degenerated trapping potential with non-isolated critical points. Precisely, applying the finite dimensional reduction method, we first obtain the existence of $k$-peak concentrated solutions and especially describe the relationship between the chemical potential $μ$ and the attractive interaction $a$. Second, after precise analysis of the concentrated points and the Lagrange multiplier, we prove the local uniqueness of the $k$-peak solutions with prescribed $L^2$-norm, by use of the local Pohozaev identities, the blow-up analysis and the maximum principle associated to the nonlocal operator $(-Δ)^s$. To our best knowledge, there is few results on the excited normalized solutions of the fractional Schrödinger equations before this present work. The main difficulty lies in the non-local property of the operator $(-Δ)^s$. First, it makes the standard comparison argument in the ODE theory invalid to use in our analysis. Second, because of the algebraic decay involving the approximate solutions, the estimates, on the Lagrange multiplier for example, would become more subtle. Moreover,when studying the corresponding harmonic extension problem, several local Pohozaev identities are constructed and we have to estimate several kinds of integrals that never appear in the classic local Schrödinger problems. In addition, throughout our discussion, we need to distinct the different cases of $p$, which are called respectively that the mass-subcritical, the mass-critical, and the mass-supercritical case, due to the mass-constraint condition. Another difficulty comes from the influence of the different degenerate rates along different directions at the critical points of the potential.

math.AP↗

arXiv2019

Large number of bubble solutions for a perturbed fractional Laplacian equation

This paper deals with the following nonlinear perturbed fractional Laplacian equation $$(-Δ)^s u = K(|y'|,y'')u^{\frac{N+2s}{N-2s}\pmε},\,\,u>0,\,\,u\in D^{1,s}(\mathbb{R}^N),$$ where $0 0$ is a small parameter and $K(y)$ is nonnegative and bounded. By combining a finite reduction argument and local Pohozaev type of identities, we prove that if $N\geq 4,\max\{\frac{N+1-\sqrt{N^{2}-2N+9}}{4},\frac{3-\sqrt{N^{2}-6N+13}}{2}\} 0$ and $K(r_0, y_0'')>0,$ then the above problem has large number of bubble solutions if $ε>0$ is small enough. Also there exist solutions whose functional energy is in the order $ε^{-\frac{N-2s-2}{(N-2s)^{2}}}$. Here, instead of estimating directly the derivatives of the reduced functional, we apply some local Pohozaev identities to locate the concentration points of the bubble solutions. Moreover, the concentration points of the bubble solutions include a saddle point of $K(y)$.

math.AP↗

arXiv2017

Multi-peak positive solutions to a class of Kirchhoff equations

In the present paper, we consider the nonlocal Kirchhoff problem \begin{eqnarray*} -\left(ε^2a+εb\int_{\mathbb{R}^{3}}|\nabla u|^{2}\right)Δu+V(x)u=u^{p},\,\,\,u>0 & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b>0$, $1 0$ is a parameter. Under some mild assumptions on the function $V$, we obtain multi-peak solutions for $ε$ sufficiently small by Lyapunov-Schmidt reduction method. Even though many results on single peak solutions to singularly perturbed Kirchhoff problems have been derived in the literature by various methods, there exist no results on multi-peak solutions before this paper, due to some difficulties caused by the nonlocal term $\left(\int_{\mathbb{R}^3}|\nabla u|^2\right)Δu$. A remarkable new feature of this problem is that the corresponding unperturbed problem turns out to be a system of partial differential equations, but not a single Kirchhoff equation, which is quite different from most of elliptic singular perturbation problems.

math.AP↗

arXiv2017

Uniqueness of positive solutions with Concentration for the Schrödinger-Newton problem

We are concerned with the following Schrödinger-Newton problem \begin{equation} -\varepsilon^2Δu+V(x)u=\frac{1}{8π\varepsilon^2} \big(\int_{\mathbb R^3}\frac{u^2(ξ)}{|x-ξ|}dξ\big)u,~x\in \mathbb R^3. \end{equation} For $\varepsilon$ small enough, we show the uniqueness of positive solutions concentrating at the nondegenerate critical points of $V(x)$. The main tools are a local Pohozaev type of identity, blow-up analysis and the maximum principle. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from those of Schrödinger equations.

math.AP↗

arXiv2016

Uniqueness and nondegeneracy of positive solutions to a class of Kirchhoff equations in $\mathbb{R}^3$

In this paper, we establish a type of uniqueness and nondegeneracy results for positive solutions to the following nonlocal Kirchhoff equations \begin{eqnarray*} -\left(a+b\int_{\mathbb{R}^{3}}|\nabla u|^{2}\text{d} x\right)Δu+u=|u|^{p-1}u & & \text{in }\mathbb{R}^{3}, \end{eqnarray*} where $a,b$ are positive constants and $1<p<5$. Before this paper, it seems that there have no this type of results even on positive ground states solutions to Kirchhoff type equations, much less on general positive solutions. To overcome the difficulty brought by the nonlocality, some new observation on Kirchhoff equations is found, and some related theories on classical Schrödinger equations are applied.

math.AP↗

arXiv2015

Solutions for a nonlocal elliptic equation involving critical growth and Hardy potential

In this paper, by an approximating argument, we obtain infinitely many solutions for the following Hardy-Sobolev fractional equation with critical growth \begin{equation*}\label{0.1} \left\{% \begin{array}{ll} (-Δ)^{s} u-\ds\frac{μu}{|x|^{2s}}=|u|^{2^*_s-2}u+au, & \hbox{$\text{in}~ Ω$},\vspace{0.1cm} u=0,\,\, &\hbox{$\text{on}~\partial Ω$}, \\ \end{array}% \right. \end{equation*} provided $N>6s$, $μ\geq0$, $0< s<1$, $2^*_s=\frac{2N}{N-2s}$, $a>0$ is a constant and $Ω$ is an open bounded domain in $\R^N$ which contains the origin.

math.AP↗

arXiv2015

Modules of the toroidal Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$

Highest weight modules of the double affine Lie algebra $\widehat{\widehat{\mathfrak{sl}}}_{2}$ are studied under a new triangular decomposition. Singular vectors of Verma modules are determined using a similar condition with horizontal affine Lie subalgebras, and highest weight modules are described under the condition that $c_1>0$ and $c_2=0$.

math.QA↗