Searcharxiv⌕ Search

arXiv subjects

Huan-Song Zhou

Publications and source records attributed to Huan-Song Zhou.

At least 19 recordsLinked to original sources
arXiv2025

Existence and local uniqueness of multi-spike solutions for Brézis-Nirenberg problem with prescribed mass

In this paper, we consider the following Brézis-Nirenberg problem with prescribed $ L^2$-norm (mass) constraint: \begin{equation*} \begin{cases} -Δu=|u|^{2^*-2} u +λ_ρu\quad \text { in } Ω, u>0, \quad u \in H_0^1(Ω), \quad \int_Ω u^2dx=ρ, \end{cases} \end{equation*} where $N \geqslant 6$, $2^*=2 N /(N-2)$ is the critical Sobolev exponent, $ρ>0$ is a given small constant and $λ_ρ>0$ acts as an Euler-Lagrange multiplier. For any $k\in \mathbb{R}^+$, we construct a $k$-spike solutions in some suitable bounded domain $Ω$. Our results extend those in \cite{BHG3,DGY,SZ}, where the authors obtained one or two positive solutions corresponding to the (local) minimizer or mountain pass type critical point for the energy functional of above equation. Furthermore, using blow-up analysis and local Pohozaev identities arguments, we prove that the $k$-spike solutions are locally unique. Compared to the standard Brézis-Nirenberg problem without the mass constraint, an additional difficulty arises in estimating the error caused by the differences in the Euler-Lagrange multipliers corresponding to different solutions. We overcome this difficulty by introducing novel observations and estimates related to the kernel of the linearized operators.

math.AP↗
arXiv2024

Infinitely many solutions for a Schrodinger equation with sign-changing potential and nonlinear term

We propose a new variational approach to finding multiple critical points for strongly indefinite problems without assuming the weak upper semicontinuity on the variational functionals. By this approach, we obtain the existence of infinitely many geometrically distinct solutions for a stationary periodic Schrödinger equation, in which the linear part is strongly indefinite and the nonlinear term is allowed to change sign in general ways.

math.FA↗
arXiv2022

Uniqueness of Single Peak Solutions for Coupled Nonlinear Gross-Pitaevskii Equations with Potentials

For a couple of singularly perturbed Gross-Pitaevskii equations, we first prove that the single peak solutions, if they concentrate on the same point, are unique provided that the Taylor's expansion of potentials around the concentration point is in the same order along all directions. Among other assumptions, our results indicate that the peak solutions obtained in [21,31,38] are unique. Moreover, for the radially symmetric ring-shaped potential, which attains its minimum at the spheres$Γ_i:=\{x\in\mathbb{R}^N:|x|=A_i>0\},i=1,2,\cdots,l,$ and is totally degenerate in the tangential space of $Γ_i$, we prove that the positive ground state is cylindrically symmetric and is unique up to rotations around the origin. Aa far as we know, this is the first uniqueness result for ground states under radially symmetric but non-monotonic potentials.

math.AP↗
arXiv2022

A constrained minimization problem related to two coupled pseudo-relativistic Hartree equations

We are concerned with the following constrained minimization problem: $$e(a_{1},a_{2},β) := \inf\left\{E_{a_{1},a_{2},β}(u_{1},u_{2}): \|u_{1}\|_{L^{2}(\mathbb{R}^{3})} = \|u_{2}\|_{L^{2}(\mathbb{R}^{3})} = 1\right\},$$ where $E_{a_{1},a_{2},β}$ is the energy functional associated to two coupled pseudo-relativistic Hartree equations involving three parameters $a_{1}, a_{2}, β$ and two trapping potentials $V_1(x)$ and $V_2(x)$. In this paper, we obtain the existence of minimizers of $e(a_{1},a_{2},β)$ for possible $a_{1}, a_{2}$ and $β$ under suitable conditions on the potentials, which generalizes the results of the papers [16,17,18] in different senses.

math.AP↗
arXiv2021

Asymptotic behavior of least energy solutions for a fractional Laplacian eigenvalue problem on $R^N$

We are interested in the existence and asymptotical behavior for the least energy solutions of the following fractional eigenvalue problem \begin{equation*} (P)\quad (-Δ)^{s}u+V(x)u=μu+am(x)|u|^{\frac{4s}{N}}u,\quad \int_{\mathbb{R}^{N}}|u|^{2}dx=1,\ u\in H^{s}(\mathbb{R}^{N}), \end{equation*} where $s\in(0,1)$, $μ\in\mathbb{R}$, $a>0$, $V(x)$ and $m(x)$ are $L^{\infty}(\mathbb{R}^{N})$ functions with $N\geq2$. We prove that there is a threshold $a_s^{*}>0$ such that problem $(P)$ has a least energy solution $u_{a}(x)$ for each $a\in(0,a_s^{*})$ and $u_{a}$ blows up, as $a\nearrow a_s^{*}$, at some point $x_0 \in \mathbb{R}^N$, which makes $V(x_0)$ be the minimum and $m(x_0)$ be the maximum. Moreover, the precise blowup rates for $u_a$ are obtained under suitable conditions on $V(x)$ and $m(x)$.

math.AP↗
arXiv2020

Properties of the minimizers for a constrained minimization problem arising in Kirchhoff equation

Let $a>0,b>0$ and $V(x)\geq0$ be a coercive function in $\mathbb R^2$. We study the following constrained minimization problem on a suitable weighted Sobolev space $\mathcal{H}$: \begin{equation*} e_{a}(b):=\inf\left\{E_{a}^{b}(u):u\in\mathcal{H}\ \mbox{and}\ \int_{\mathbb R^{2}}|u|^{2}dx=1\right\}, \end{equation*} where $E_{a}^{b}(u)$ is a Kirchhoff type energy functional defined on $\mathcal{H}$ by \begin{equation*} E_{a}^{b}(u)=\frac{1}{2}\int_{\mathbb R^{2}}[|\nabla u|^{2}+V(x)u^{2}]dx+\frac{b}{4}\left(\int_{\mathbb R^{2}}|\nabla u|^{2}dx\right)^{2}-\frac{a}{4}\int_{\mathbb R^{2}}|u|^{4}dx. \end{equation*} It is known that, for some $a^{\ast}>0$, $e_{a}(b)$ has no minimizer if $b=0$ and $a\geq a^{\ast}$, but $e_{a}(b)$ has always a minimizer for any $a\geq0$ if $b>0$. The aim of this paper is to investigate the limit behaviors of the minimizers of $e_{a}(b)$ as $b\rightarrow0^{+}$. Moreover, the uniqueness of the minimizers of $e_{a}(b)$ is also discussed for $b$ close to 0.

math.FA↗
arXiv2017

Blow-up behavior of ground states for a nonlinear Schrödinger system with attractive and repulsive interactions

We consider a nonlinear Schrödinger system arising in a two-component Bose-Einstein condensate (BEC) with attractive intraspecies interactions and repulsive interspecies interactions in $\mathbb{R}^2$. We get ground states of this system by solving a constrained minimization problem. For some kinds of trapping potentials, we prove that the minimization problem has a minimizer if and only if the attractive interaction strength $a_i (i=1,2)$ of each component of the BEC system is strictly less than a threshold $a^*$. %attractive intraspecies interactions satisfies $a_i< %a^*= \|Q\|_2^2,\ i=1,\,2$, where $Q$ is the unique positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$; in contrast, there is no minimizer if either $a_i > a^*$ for $i=1$ or $2$, or $a_1=a_2=a^*$. Furthermore, as $(a_1, a_2)\nearrow (a^*, a^*)$, the asymptotical behavior for the minimizers of the minimization problem is discussed. Our results show that each component of the BEC system concentrates at a global minimum of the associated trapping potential.

math-ph↗
arXiv2017

Blow-up solutions for two coupled Gross-Pitaevskii equations with attractive interactions

The paper is concerned with a system of two coupled time-independent Gross-Pitaevskii equations in $\mathbb{R}^2$, which is used to model two-component Bose-Einstein condensates with both attractive intraspecies and attractive interspecies interactions. This system is essentially an eigenvalue problem of a stationary nonlinear Schrödinger system in $\mathbb{R}^2$, solutions of the problem are obtained by seeking minimizers of the associated variational functional with constrained mass (i.e. $L^2-$norm constaints). Under certain type of trapping potentials $V_i(x)$ ($i=1,2$), the existence, non-existence and uniqueness of this kind of solutions are studied. Moreover, by establishing some delicate energy estimates, we show that each component of the solutions blows up at the same point (i.e., one of the global minima of $V_i(x)$) when the total interaction strength of intraspecies and interspecies goes to a critical value. An optimal blowing up rate for the solutions of the system is also given.

math.AP↗
arXiv2016

An Improved Fountain Theorem and its Application

The main aim of the paper is to prove a fountain theorem without assuming the $τ$-upper semicontinuity condition on the variational functional. Using this improved fountain theorem, we may deal with more general strongly indefinite elliptic problems with various sign-changing nonlinear terms. As an application, we obtain infinitely many solutions for a semilinear Schrödinger equation with strongly indefinite structure and sign-changing nonlinearity.

math.FA↗
arXiv2016

Eigenvalue problem for a p-Laplacian equation with trapping potentials

Consider the following eigenvalue problem of p-Laplacian equation \begin{equation}\label{P} -Δ_{p}u+V(x)|u|^{p-2}u=μ|u|^{p-2}u+a| u|^{s-2}u, x\in \mathbb{R}^{n}, \tag{P} \end{equation} where $a\geq0$, $p\in (1,n)$ and $μ\in\mathbb{R}$. $V(x)$ is a trapping type potential, e.g., $\inf\limits_{x \in \mathbb{R}^n}V(x)< \lim\limits_{|x|\rightarrow+\infty}V(x)$. By using constrained variational methods, we proved that there is $a^*>0$, which can be given explicitly, such that problem (\ref{P}) has a ground state $u$ with $\|u\|_{L^p}=1$ for some $μ\in \mathbb{R}$ and all $a\in [0,a^*)$, but (\ref{P}) has no this kind of ground state if $a\geq a^*$. Furthermore, by establishing some delicate energy estimates we show that the global maximum point of the ground states of problem (\ref{P}) approach to one of the global minima of $V(x)$ and blow up if $a\nearrow a^*$. The optimal rate of blowup is obtained for $V(x)$ being a polynomial type potential.

math.AP↗
arXiv2015

Properties of ground states of attractive Gross-Pitaevskii equations with multi-well potentials

We are interested in the attractive Gross-Pitaevskii (GP) equation in $\R^2$, where the external potential $V(x)$ vanishes on $m$ disjoint bounded domains $Ω_i\subset \R^2\ (i=1,2,\cdots,m)$ and $V(x)\to\infty$ as $|x|\to\infty$, that is, the union of these $Ω_i$ is the bottom of the potential well. By making some delicate estimates on the energy functional of the GP equation, we prove that when the interaction strength $a$ approaches some critical value $a^*$ the ground states concentrate and blow up at the center of the incircle of some $Ω_j$ which has the largest inradius. Moreover, under some further conditions on $V(x)$ we show that the ground states of GP equations are unique and radially symmetric at leat for almost every $a \in (0, a^*)$.

math.AP↗
arXiv2014

Energy estimates and symmetry breaking in attractive Bose-Einstein condensates with ring-shaped potentials

This paper is concerned with the properties of $L^2$-normalized minimizers of the Gross-Pitaevskii (GP) functional for a two-dimensional Bose-Einstein condensate with attractive interaction and ring-shaped potential. By establishing some delicate estimates on the least energy of the GP functional, we prove that symmetry breaking occurs for the minimizers of the GP functional as the interaction strength $a>0$ approaches a critical value $a^*$, each minimizer of the GP functional concentrates to a point on the circular bottom of the potential well and then is non-radially symmetric as $a\nearrow a^*$. However, when $a>0$ is suitably small we prove that the minimizers of the GP functional are unique, and this unique minimizer is radially symmetric.

math.FA↗
arXiv2014

Multiple solutions for a nonhomogeneous Schrödinger-Maxwell system in $R^3$

The paper considers the following nonhomogeneous Schrödinger-Maxwell system -Δu + u+λϕ(x) u =|u|^{p-1}u+g(x),\ x\in \mathbb{R}^3, -Δϕ= u^2, \ x\in \mathbb{R}^3, . \leqno{(SM)} where $λ>0$, $p\in(1,5)$ and $g(x)=g(|x|)\in L^2(\mathbb{R}^3)\setminus{0}$. There seems no any results on the existence of multiple solutions to problem (SM) for $p \in (1,3]$. In this paper, we find that there is a constant$C_p>0$ such that problem (SM) has at least two solutions for all $p\in (1,5)$ provided $\|g\|_{L^2} \leq C_p$, but only for $p\in(1,2]$ we need $λ>0$ is small. Moreover, $C_p=\frac{(p-1)}{2p}[\frac{(p+1)S^{p+1}}{2p}]^{1/(p-1)}$, where $S$ is the Sobolev constant.

math.AP↗
arXiv2013

Concentration behavior of standing waves for almost mass critical nonlinear Schrödinger equations

We study the following nonlinear Schrödinger equation $$ iu_t=-Δu+V(x)u-a|u|^qu \quad (t,x)\in \mathbb{R}^1\times \mathbb{R}^2, $$ where $a>0, \ q\in(0,2)$ and $V(x)$ is some type of trapping potentials. For any fixed $a>a^*:= \|Q\|_2^2$, where $Q$ is the unique (up to translations) positive radial solution of $Δu-u+u^3=0$ in $\mathbb{R}^2$, by directly using constrained variational method and energy estimates we present a detailed analysis of the concentration and symmetry breaking of the standing waves for the above equation as $q\nearrow 2$.

math.AP↗
arXiv2012

Schrödinger-Poisson equations with singular potentials in $R^3$

The existence and $L^{\infty}$ estimate of positive solutions are discussed for the following Schrödinger-Poisson system {ll} -Δu +(λ+\frac{1}{|y|^α})u+ϕ(x) u =|u|^{p-1}u, x=(y,z)\in \mathbb{R}^2\times\mathbb{R}, -Δϕ= u^2,\ \lim\limits_{|x|\rightarrow +\infty}ϕ(x)=0, \hfill y=(x_1,x_2) \in \mathbb{R}^2 with |y|=\sqrt{x_1^2+x_2^2}, where $λ\geqslant0$, $α\in[0,8)$ and $\max\{2,\frac{2+α}{2}\}<p<5$.

math.AP↗