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Wenming Zou

Publications and source records attributed to Wenming Zou.

77 records · Page 5Linked to original sources
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↗