arXiv subjects

Wenming Zou

Publications and source records attributed to Wenming Zou.

70 records · Page 4Linked to original sources

arXiv2016

On the equation $p \frac{Γ(\frac{n}{2}-\frac{s}{p-1})Γ(s+\frac{s}{p-1})}{Γ(\frac{s}{p-1})Γ(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{Γ(\frac{n+2s}{4})^2}{Γ(\frac{n-2s}{4})^2}$

The note is aimed at giving a complete characterization of the following equation: $$\displaystyle p\frac{Γ(\frac{n}{2}-\frac{s}{p-1})Γ(s+\frac{s}{p-1})}{Γ(\frac{s}{p-1})Γ(\frac{n-2s}{2}-\frac{s}{p-1})} =\frac{Γ(\frac{n+2s}{4})^2}{Γ(\frac{n-2s}{4})^2}.$$ The method is based on some key transformation and the properties of the Gamma function. Applications to fractional nonlinear Lane-Emden equations will be given.

math.AP↗

arXiv2016

The Nehari manifold for fractional systems involving critical nonlinearities

We study the combined effect of concave and convex nonlinearities on the number of positive solutions for a fractional system involving critical Sobolev exponents. With the help of the Nehari manifold, we prove that the system admits at least two positive solutions when the pair of parameters $(λ,μ)$ belongs to a suitable subset of $\R^2$.

math.AP↗

arXiv2015

On critical $p$-Laplacian systems

We consider the critical $p$-Laplacian system \begin{equation}\label{92} \begin{cases}-Δ_p u-\frac{λa}{p}|u|^{a-2}u|v|^b =μ_1|u|^{p^\ast-2}u+\frac{αγ}{p^\ast}|u|^{α-2}u|v|^β, &x\inΩ,\\ -Δ_p v-\frac{λb}{p}|u|^a|v|^{b-2}v =μ_2|v|^{p^\ast-2}v+\frac{βγ}{p^\ast}|u|^α|v|^{β-2}v, &x\inΩ,\\ u,v\ \text{in } D_0^{1,p}(Ω), \end{cases} \end{equation} where $Δ_p:=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian operator defined on $D^{1,p}(\mathbb{R}^N):=\{u\in L^{p^\ast}(\mathbb{R}^N):|\nabla u|\in L^p(\mathbb{R}^N)\}$, endowed with norm $\|u\|_{D^{1,p}}:=\big(\int_{\mathbb{R}^N}|\nabla u|^p\text{d}x\big)^{\frac{1}{p}}$, $N\ge3$, $1 1$ satisfy $a + b = p, α+ β= p^\ast:=\frac{Np}{N-p}$, the critical Sobolev exponent, $Ω$ is $\mathbb{R}^N$ or a bounded domain in $\mathbb{R}^N$, $D_0^{1,p}(Ω)$ is the closure of $C_0^\infty(Ω)$ in $D^{1,p}(\mathbb{R}^N)$. Under suitable assumptions, we establish the existence and nonexistence of a positive least energy solution. We also consider the existence and multiplicity of nontrivial nonnegative solutions.

math.AP↗

arXiv2015

On Elliptic Systems involving critical Hardy-Sobolev exponents

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.

math.AP↗

arXiv2015

On Elliptic Systems involving critical Hardy-Sobolev exponents (Part II)

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.

math.AP↗

arXiv2015

A perturbed nonlinear elliptic PDE with two Hardy-Sobolev critical exponents

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.

math.AP↗

arXiv2015

On a doubly critical Schrödinger system in $\bbr^4$ with steep potential wells

Study the following two-component elliptic system% \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0)u+u^3+βv^2u=0\quad&\text{in }\bbr^4,\\% &Δv-(λb(x)+b_0)v+v^3+βu^2v=0\quad&\text{in }\bbr^4,\\% &(u,v)\in\h\times\h,\endaligned\right.% \end{equation*} where $a_0,b_0\in\bbr$ are constants; $λ>0$ and $β\in\bbr$ are parameters and $a(x), b(x)\geq0$ are potential wells which are not necessarily to be radial symmetric. By using the variational method, we investigate the existence of ground state solutions and general ground state solutions (i.e., possibly semi-trivial) to this system. Indeed, to the best of our knowledge, even the existence of semi-trivial solutions is also unknown in the literature. We observe some concentration behaviors of ground state solutions and general ground state solutions. The phenomenon of phase separations is also excepted. It seems that this is the first result definitely describing the phenomenon of phase separation for critical system in the whole space $\bbr^4$. Note that both the cubic nonlinearities and the coupled terms of the system are all of critical growth with respect to the Sobolev critical exponent.

math.AP↗

arXiv2015

Existence of extremal functions for a family of Caffarelli-Kohn-Nirenberg inequalities

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 $V(x)$.

math.AP↗

arXiv2015

On coupled Schrödinger systems with double critical exponents and indefinite weights

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.

math.AP↗

arXiv2014

On a two-component Bose-Einstein condensate with steep potential wells

In this paper, we study the following two-component systems of nonlinear Schrödinger equations \begin{equation*} \left\{\aligned&Δu-(λa(x)+a_0(x))u+μ_1u^3+βv^2u=0\quad&\text{in }\bbr^3,\\ &Δv-(λb(x)+b_0(x))v+μ_2v^3+βu^2v=0\quad&\text{in }\bbr^3,\\ &u,v\in\h,\quad u,v>0\quad\text{in }\bbr^3,\endaligned\right. \end{equation*} where $λ,μ_1,μ_2>0$ and $β<0$ are parameters; $a(x), b(x)\geq0$ are steep potentials and $a_0(x),b_0(x)$ are sign-changing weight functions; $a(x)$, $b(x)$, $a_0(x)$ and $b_0(x)$ are not necessarily to be radial symmetric. By the variational method, we obtain a ground state solution and multi-bump solutions for such systems with $λ$ sufficiently large. The concentration behaviors of solutions as both $λ\to+\infty$ and $β\to-\infty$ are also considered. In particular, the phenomenon of phase separations is observed in the whole space $\bbr^3$. In the Hartree-Fock theory, this provides a theoretical enlightenment of phase separation in $\bbr^3$ for the 2-mixtures of Bose-Einstein condensates.

math.AP↗

arXiv2014

Infinitely many sign-changing and semi-nodal solutions for a nonlinear Schrodinger system

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics: {displaymath} {cases}-Δu_1 +\la_1 u_1 = μ_1 u_1^3+βu_1 u_2^2, \quad x\in Ω, -Δu_2 +\la_2 u_2 =μ_2 u_2^3+βu_1^2 u_2, \quad x\in \Om, u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.{cases}{displaymath} Here $\Om$ is a smooth bounded domain in $\R^N (N=2, 3)$ or $\Om=\RN$, $\la_1,\, \la_2$, $μ_1,\,μ_2$ are all positive constants and the coupling constant $\bb<0$. We show that this system has infinitely many sign-changing solutions. We also obtain infinitely many semi-nodal solutions in the following sense: one component changes sign and the other one is positive. The crucial idea of our proof, which has never been used for this system before, is turning to study a new problem with two constraints. Finally, when $\Om$ is a bounded domain, we show that this system has a least energy sign-changing solution, both two components of which have exactly two nodal domains, and we also study the asymptotic behavior of solutions as $β\to -\infty$ and phase separation is expected.

math.AP↗

arXiv2014

Standing waves for coupled nonlinear Schrodinger equations with decaying potentials

We study the following singularly perturbed problem for a coupled nonlinear Schrödinger system: {displaymath} {cases}-\e^2Δu +a(x) u = μ_1 u^3+βuv^2, \quad x\in \R^3, -\e^2Δv +b(x) v =μ_2 v^3+βvu^2, \quad x\in \R^3, u> 0, v> 0 \,\,\hbox{in $\R^3$}, u(x), v(x)\to 0 \,\,\hbox{as $|x|\to \iy$}.{cases}{displaymath} Here, $a, b$ are nonnegative continuous potentials, and $μ_1,μ_2>0$. We consider the case where the coupling constant $β>0$ is relatively large. Then for sufficiently small $\e>0$, we obtain positive solutions of this system which concentrate around local minima of the potentials as $\e\to 0$. The novelty is that the potentials $a$ and $b$ may vanish at someplace and decay to 0 at infinity.

math.AP↗

arXiv2014

Existence and symmetry of positive ground states for a doubly critical Schrodinger system

We study the following doubly critical Schrödinger system $$-Δu -\frac{\la_1}{|x|^2}u=u^{2^\ast-1}+ ν\al u^{\al-1}v^\bb, \quad x\in \RN, -Δv -\frac{\la_2}{|x|^2}v=v^{2^\ast-1} + ν\bb u^{\al}v^{\bb-1}, \quad x\in \RN, u, v\in D^{1, 2}(\RN),\quad u, v>0 in $\RN\setminus{0}$},$$ where $N\ge 3$, $\la_1, \la_2\in (0, \frac{(N-2)^2}{4})$, $2^\ast=\frac{2N}{N-2}$ and $\al>1, \bb>1$ satisfying $\al+\bb=2^\ast$. This problem is related to coupled nonlinear Schrödinger equations with critical exponent for Bose-Einstein condensate. For different ranges of $N$, $\al$, $\bb$ and $ν>0$, we obtain positive ground state solutions via some quite different methods, which are all radially symmetric. It turns out that the least energy level depends heavily on the relations among $\al, \bb$ and 2. Besides, for sufficiently small $ν>0$, positive solutions are also obtained via a variational perturbation approach. Note that the Palais-Smale condition can not hold for any positive energy level, which makes the study via variational methods rather complicated.

math.AP↗

arXiv2013

Positive Least Energy Solutions and Phase Separation for Coupled Schrodinger Equations with Critical Exponent: Higher Dimensional Case

We study the following nonlinear Schrödinger system which is related to Bose-Einstein condensate: {displaymath} {cases}-Δu +\la_1 u = μ_1 u^{2^\ast-1}+βu^{\frac{2^\ast}{2}-1}v^{\frac{2^\ast}{2}}, \quad x\in Ω, -Δv +\la_2 v =μ_2 v^{2^\ast-1}+βv^{\frac{2^\ast}{2}-1} u^{\frac{2^\ast}{2}}, \quad x\in \om, u\ge 0, v\ge 0 \,\,\hbox{in $\om$},\quad u=v=0 \,\,\hbox{on $\partial\om$}.{cases}{displaymath} Here $\om\subset \R^N$ is a smooth bounded domain, $2^\ast:=\frac{2N}{N-2}$ is the Sobolev critical exponent, $-\la_1(\om)<\la_1,\la_2<0$, $μ_1,μ_2>0$ and $β\neq 0$, where $λ_1(\om)$ is the first eigenvalue of $-Δ$ with the Dirichlet boundary condition. When $\bb=0$, this is just the well-known Brezis-Nirenberg problem. The special case N=4 was studied by the authors in (Arch. Ration. Mech. Anal. 205: 515-551, 2012). In this paper we consider {\it the higher dimensional case $N\ge 5$}. It is interesting that we can prove the existence of a positive least energy solution $(u_\bb, v_\bb)$ {\it for any $β\neq 0$} (which can not hold in the special case N=4). We also study the limit behavior of $(u_\bb, v_\bb)$ as $β\to -\infty$ and phase separation is expected. In particular, $u_\bb-v_\bb$ will converge to {\it sign-changing solutions} of the Brezis-Nirenberg problem, provided $N\ge 6$. In case $\la_1=\la_2$, the classification of the least energy solutions is also studied. It turns out that some quite different phenomena appear comparing to the special case N=4.

math.AP↗

arXiv2013

Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations

We study the following coupled Schrödinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-Δu_1 +\la_1 u_1 = μ_1 u_1^3+βu_1 u_2^2, \quad x\in Ω,\\ -Δu_2 +\la_2 u_2 =μ_2 u_2^3+βu_1^2 u_2, \quad x\in \Om,\\ u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.\end{cases}\end{displaymath} Here $\Om\subset\RN (N=2, 3)$ is a smooth bounded domain, $\la_1, \la_2$, $μ_1, μ_2$ are all positive constants. We show that, for each $k\in\mathbb{N}$ there exists $\bb_k>0$ such that this system has at least $k$ sign-changing solutions (i.e., both two components change sign) and $k$ semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$.

math.AP↗

arXiv2012

Standing Waves for nonlinear Schrodinger Equations involving critical growth

We consider the following singularly perturbed nonlinear elliptic problem: $$-\e^2Δu+V(x)u=f(u),\ u\in H^1(\mathbb{R^N}),$$ where $N\ge 3$ and the nonlinearity $f$ is of critical growth. In this paper, we construct a solution $u_\e$ of the above problem which concentrates at an isolated component of positive local minimum points of $V$ as $\e\to 0$ under certain conditions on $f$. Our result completes the study made in some very recent works in the sense that, in those papers only the subcritical growth was considered

math.AP↗