arXiv subjects
Publications and source records attributed to Xuexiu Zhong.
We consider the existence and multiplicity of solutions for a class of quasi-linear Schrödinger equations which include the modified nonlinear Schrödinger equations. A new perturbation approach is used to treat the sub-cubic nonlinearity.
We consider the following singularly perturbed Kirchhoff type equations $$-\varepsilon^2 M\left(\varepsilon^{2-N}\int_{\R^N}|\nabla u|^2 dx\right)Δu +V(x)u=|u|^{p-2}u~\hbox{in}~\R^N, u\in H^1(\R^N),N\geq 1,$$ where $M\in C([0,\infty))$ and $V\in C(\R^N)$ are given functions. Under very mild assumptions on $M$, we prove the existence of single-peak or multi-peak solution $u_\varepsilon$ for above problem, concentrating around topologically stable critical points of $V$, by a direct corresponding argument. This gives an affirmative answer to an open problem raised by Figueiredo et al. in 2014 [ARMA,213].
In this paper, we consider the Hamiltonian elliptic system in dimension two\begin{equation}\label{1.5}\aligned \left\{ \begin{array}{lll} -ε^2Δu+V(x)u=g(v)\ & \text{in}\quad \mathbb{R}^2,\\ -ε^2Δv+V(x)v=f(u)\ & \text{in}\quad \mathbb{R}^2, \end{array}\right.\endaligned \end{equation} where $V\in C(\mathbb{R}^2)$ has local minimum points, and $f,g\in C^1(\mathbb{R})$ are assumed to be either superlinear or asymptotically linear at infinity and of subcritical exponential growth in the sense of Trudinger-Moser inequality. Under only a local condition on $V$, we obtain a family of semiclassical states concentrating around local minimum points of $V$. In addition, in the case that $f$ and $g$ are superlinear at infinity, the decay and positivity of semiclassical states are also given. The proof is based on a reduction method, variational methods and penalization techniques.
In present paper, we prove the existence of solutions $(λ, u)\in \R\times H^1(\R^N)$ to the following Schrödinger equation $$ \begin{cases} -Δu(x)+V(x)u(x)+λu(x)=g(u(x))\quad &\hbox{in}~\R^N\\ 0\leq u(x)\in H^1(\R^N), N\geq 3 \end{cases} $$ satisfying the normalization constraint $\displaystyle \int_{\R^N}u^2 dx=a$. We treat the so-called mass super-critical case here. Under an explicit smallness assumption on $V$ and some Ambrosetti-Rabinowitz type conditions on $g$, we can prove the existence of ground state normalized solutions for prescribed mass $a>0$. Furthermore, we emphasize that the mountain pass characterization of a minimizing solution of the problem $$\inf\left\{\int \left[\frac{1}{2}|\nabla u|^2+\frac{1}{2}V(x)u^2-G(u)\right]dx : \|u\|_{L^2(\R^N)}^{2}=a, P[u]=0\right\},$$ where $G(s)=\int_0^s g(τ)dτ$ and $$P[u]=\int\left[|\nabla u|^2-\frac{1}{2}\langle \nabla V(x), x\rangle u^2 -N\left(\frac{1}{2}g(u)u-G(u)\right)\right]dx.$$
In present paper, we study the normalized solutions $(λ_c, u_c)\in \R\times H^1(\R^N)$ to the following Kirchhoff problem $$ -\left(a+b\int_{\R^N}|\nabla u|^2dx\right)Δu+λu=g(u)~\hbox{in}~\R^N,\;1\leq N\leq 3 $$ satisfying the normalization constraint $ \displaystyle\int_{\R^N}u^2=c, $ which appears in free vibrations of elastic strings. The parameters $a,b>0$ are prescribed as is the mass $c>0$. The nonlinearities $g(s)$ considered here are very general and of mass super-critical. Under some suitable assumptions, we can prove the existence of ground state normalized solutions for any given $c>0$. After a detailed analysis via the blow up method, we also make clear the asymptotic behavior of these solutions as $c\rightarrow 0^+$ as well as $c\rightarrow+\infty$.
In the present paper, we prove the existence of solutions $(λ, u)\in \R\times H^1(\R^N)$ to the following elliptic equations with potential $\displaystyle -Δu+(V(x)+λ)u=g(u)\;\hbox{in}\;\R^N, $ satisfying the normalization constraint $\displaystyle \int_{\R^N}u^2=a>0,$ which is deduced by searching for solitary wave solution to the time-dependent nonlinear Schrödinger equations. Besides the importance in the applications, not negligible reasons of our interest for such problems with potential $V(x)$ are their stimulating and challenging mathematical difficulties. We develop an interesting way basing on iteration and give a new proof of the so-called "sub-additive inequality", which can simply the standard process in the traditional sense. Under some very relax assumption on the potential $V(x)$ and some other suitable assumptions on $g$, we can obtain the existence of ground state solution for prescribed $a>0$.
In this paper, we prove the existence of positive solutions $(λ_1,λ_2, u,v)\in \R^2\times H^1(\R^N, \R^2)$ to the following coupled Schrödinger system $$\begin{cases} -Δu + λ_1 u= μ_1|u|^{p-2}u+βv \quad &\hbox{in}\;\RN, \\ -Δv + λ_2 v= μ_2|v|^{q-2}v+βu \quad &\hbox{in}\;\RN, \end{cases}$$ satisfying the normalization constraints $\displaystyle\int_{\RN}u^2 =a, ~ \int_{\RN}v^2 =b$. The parameters $μ_1,μ_2,β>0$ are prescribed and the masses $a,b>0$. Here $2+\frac{4}{N} 0$ and $β>0$, provided $2\leqslant N\leqslant 4$. For the Sobolev critical case with $N=3,4$, it can be viewed as a counterpart of the Brezis-Nirenberg critical semilinear elliptic problem for the system case in the context of normalized solutions. Under some suitable assumptions, we obtain the existence or non-existence of positive normalized ground state solution.
In the present paper, we prove the existence of solutions $(λ_1,λ_2,u,v)\in\mathbb{R}^2\times H^1(\mathbb{R}^3,\mathbb{R}^2)$ to systems of coupled Schrödinger equations $$ \begin{cases} -Δu+λ_1u=μ_1 u^3+βuv^2\quad &\hbox{in}\;\mathbb{R}^3\\ -Δv+λ_2v=μ_2 v^3+βu^2v\quad&\hbox{in}\;\mathbb{R}^3\\ u,v>0&\hbox{in}\;\mathbb{R}^3 \end{cases} $$ satisfying the normalization constraint $ \displaystyle\int_{\mathbb{R}^3}u^2=a^2\quad\hbox{and}\;\int_{\mathbb{R}^3}v^2=b^2, $ which appear in binary mixtures of Bose-Einstein condensates or in nonlinear optics. The parameters $μ_1,μ_2,β>0$ are prescribed as are the masses $a,b>0$. The system has been considered mostly in the fixed frequency case. And when the masses are prescribed, the standard approach to this problem is variational with $λ_1,λ_2$ appearing as Lagrange multipliers. Here we present a new approach based on bifurcation theory and the continuation method. We obtain the existence of normalized solutions for any given $a,b>0$ for $β$ in a large range. We also give a result about the nonexistence of positive solutions. From which one can see that our existence theorem is almost the best. Especially, if $μ_1=μ_2$ we prove that normalized solutions exist for all $β>0$ and all $a,b>0$.
In this paper, we will study the following PDE in $\mathbb{R}^N$ involving multiple Hardy-Sobolev critical exponents: $$ \begin{cases} Δu+\sum_{i=1}^{l}λ_i \frac{u^{2^*(s_i)-1}}{|x|^{s_i}}+u^{2^*-1}=0\;\hbox{in}\;\mathbb{R}^N, u\in D_{0}^{1,2}(\mathbb{R}^N), \end{cases} $$ where $0 0$ for $1\leq i\leq k$; $λ_i<0$ for $k+1\leq i\leq l$. We develop an interesting way to study this class of equations involving mixed sign parameters. We prove the existence and non-existence of the positive ground state solution. The regularity of the least-energy solution are also investigated.
Let $Ω\subset \mathbb{R}^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. The sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{α,β,λ,μ}(Ω) \Big(\int_Ω\big(λ\frac{|u|^{2^*(s)}}{|x|^s}+μ\frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)κ\frac{|u|^α|v|^β}{|x|^s}\big)dx\Big)^{\frac{2}{2^*(s)}}$$ $$\leq \int_Ω\big(|\nabla u|^2+|\nabla v|^2\big)dx$$ for $(u,v)\in {\mathscr{D}}:=D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω)$ will be explored. Further results about the sharp constant $S_{α,β,λ,μ}(Ω)$ with its extremal functions when $Ω$ is a general open domain will be involved. For this goal, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -Δu-λ\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=κα\frac{1}{|x|^{s_2}}|u|^{α-2}u|v|^β\quad &\hbox{in}\;Ω, -Δv-μ\frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=κβ\frac{1}{|x|^{s_2}}|u|^α|v|^{β-2}v\quad &\hbox{in}\;Ω, (u,v)\in \mathscr{D}:=D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω), \end{cases}$$ where $s_1,s_2\in (0,2), α>1,β>1, λ>0,μ>0,κ\neq 0, α+β\leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $α+β=2^*(s_2)$) when $Ω$ is a cone (in particular, $Ω=\mathbb{R}_+^N$ or $Ω=\mathbb{R}^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.}
Let $Ω\subset \R^N$ ($N\geq 3$) be an open domain which is not necessarily bounded. By using variational methods, we consider the following elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -Δu-λ\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=κα\frac{1}{|x|^{s_2}}|u|^{α-2}u|v|^β\quad &\hbox{in}\;Ω,\\ -Δv-μ\frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=κβ\frac{1}{|x|^{s_2}}|u|^α|v|^{β-2}v\quad &\hbox{in}\;Ω,\\ (u,v)\in \mathscr{D}:=D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω), \end{cases}$$ where $s_1,s_2\in (0,2), α>1,β>1, λ>0,μ>0,κ\neq 0, α+β\leq 2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. We mainly study the critical case (i.e., $α+β=2^*(s_2)$) when $Ω$ is a cone (in particular, $Ω=\R_+^N$ or $Ω=\R^N$). We will establish a sequence of fundamental results including regularity, symmetry, existence and multiplicity, uniqueness and nonexistence, {\it etc.} In particular, the sharp constant and extremal functions to the following kind of double-variable inequalities $$ S_{α,β,λ,μ}(Ω) \Big(\int_Ω\big(λ\frac{|u|^{2^*(s)}}{|x|^s}+μ\frac{|v|^{2^*(s)}}{|x|^s}+2^*(s)κ\frac{|u|^α|v|^β}{|x|^s}\big)dx\Big)^{\frac{2}{2^*(s)}}$$ $$\leq \int_Ω\big(|\nabla u|^2+|\nabla v|^2\big)dx$$ for $(u,v)\in {\mathscr{D}} $ will be explored. Further results about the sharp constant $S_{α,β,λ,μ}(Ω)$ with its extremal functions when $Ω$ is a general open domain will be involved.
This paper is the second part of a work devoted to the study of elliptic systems involving multiple Hardy-Sobolev critical exponents: $$\begin{cases} -Δu-λ\frac{|u|^{2^*(s_1)-2}u}{|x|^{s_1}}=κα\frac{1}{|x|^{s_2}}|u|^{α-2}u|v|^β\quad &\hbox{in}\;Ω,\\ -Δv-μ\frac{|v|^{2^*(s_1)-2}v}{|x|^{s_1}}=κβ\frac{1}{|x|^{s_2}}|u|^α|v|^{β-2}v\quad &\hbox{in}\;Ω,\\ κ>0,(u,v)\in \mathscr{D}:=D_{0}^{1,2}(Ω)\times D_{0}^{1,2}(Ω), \end{cases}$$ where $s_1\neq s_2\in (0,2), α>1,β>1, λ>0,μ>0,κ>0, α+β=2^*(s_2)$. Here, $2^*(s):=\frac{2(N-s)}{N-2}$ is the critical Hardy-Sobolev exponent. When $Ω$ is a cone (especially $Ω=\R_+^N$ or $Ω=\R^N$), we study the existence of positive ground state solution.
Let $Ω$ be a $C^1$ open bounded domain in $\R^N$ ($N\geq 3$) with $0\in \partial Ω$. Suppose that $\partialΩ$ is $C^2$ at $0$ and the mean curvature of $\partialΩ$ at $0$ is negative. Consider the following perturbed PDE involving two Hardy-Sobolev critical exponents: $$ \begin{cases} &Δu+λ_1 \frac{u^{2^*(s_1)-1}}{|x|^{s_1}}+λ_2\frac{u^{2^*(s_2)-1}}{|x|^{s_2}}+λ_3\frac{u^p}{|x|^{s_3}}=0\;\quad \hbox{in}\;Ω,\\ &u(x)>0\;\hbox{in}\;Ω,\;\, u(x)=0\;\hbox{on}\;\partialΩ, \end{cases} $$ where $0 0, 1< p\leq 2^*(s_3)-1$. The existence of ground state solution is studied under different assumptions via the concentration compactness principle and the Nehari manifold method. We also apply a perturbation method to study the existence of positive solution.
Consider the following inequalities due to Caffarelli, Kohn and Nirenberg {\it (Compositio Mathematica,1984):} $$\Big(\int_Ω\frac{|u|^r}{|x|^s}dx\Big)^{\frac{1}{r}}\leq C(p,q,r,μ,σ,s)\Big(\int_Ω\frac{|\nabla u|^p}{|x|^μ}dx\Big)^{\frac{a}{p}}\Big(\int_Ω\frac{|u|^q}{|x|^σ}dx\Big)^{\frac{1-a}{q}},$$ where $Ω\subset \R^N (N\geq 2)$ is an open set; $p, q, r, μ, σ, s, a$ are some parameters satisfying some balanced conditions. When $Ω$ is a cone in $\R^N$ (for example, $Ω=\R^N)$, we prove the sharp constant $C(p,q,r,μ,σ,s)$ can be achieved for a very large parameter space. Besides, we find some sufficient conditions which guarantee that the following Sobolev spaces $$W_μ^{1,p}(Ω),\; W_μ^{1,p}(Ω)\cap L^p(Ω), \; H^{1,p}(\R^N) $$ are compactly embedded into $L^r(\R^N, \frac{dx}{|x|^s})$ for some new ranges of parameters, where $\displaystyle W_μ^{1,p}(Ω)$ is the completion of $C_0^\infty(Ω)$ with respect to the norm $\displaystyle \Big(\int_Ω\frac{ |\nabla u|^p}{|x|^μ}dx\Big)^{\frac{1}{p}}. $ As applications, we also study the equation $$\displaystyle -div\Big(\frac{|\nabla u|^{p-2}\nabla u}{|x|^μ}\Big)=λV(x)|u|^{q-2}u, \;\;\; u\in W_μ^{1,p}(Ω)$$ under some proper conditions on $V(x)$.
By using variational methods, we study the existence of mountain pass solution to the following doubly critical Schrödinger system: $$ \begin{cases} -Δu-μ_1\frac{u}{|x|^2}-|u|^{2^{*}-2}u &=h(x)α|u|^{α-2}|v|^βu\quad \rm{in}\; \R^N, -Δv-μ_2\frac{v}{|x|^2}-|v|^{2^{*}-2}v &= h(x)β|u|^α|v|^{β-2}v\quad \rm{in}\; \R^N, \end{cases} $$ where $α\geq 2, β\geq 2, α+β\leq 2^*$;\; $ μ_1, μ_2\in [0, \frac{(N-2)^2}{4})$. The weight function $h(x)$ is allowed to be sign-changing so that the nonlinearities include a large class of indefinite weights. We show that the $PS$ condition is satisfied at higher energy level when $α+β=2^*$ and obtain the existence of mountain pass solution. Besides, a nonexistence result of the ground state is given.