Symmetry and asymptotic behavior of ground state solutions for schrödinger systems with linear interaction
Zhitao Zhang Haijun Luo
Communications on Pure & Applied Analysis 2018, 17(3): 787-806 doi: 10.3934/cpaa.2018040
We study symmetry and asymptotic behavior of ground state solutions for the doubly coupled nonlinear Schrödinger elliptic system
$\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$
$ N≤3, Ω\subseteq\mathbb{R}^N$
is a smooth domain. First we establish the symmetry of ground state solutions, that is, when
$ Ω$
is radially symmetric, we show that ground state solution is foliated Schwarz symmetric with respect to the same point. Moreover, ground state solutions must be radially symmetric under the condition that
$ Ω$
is a ball or the whole space
$ \mathbb{R}^N$
. Next we investigate the asymptotic behavior of positive ground state solution as
$ κ\to 0^-$
, which shows that the limiting profile is exactly a minimizer for
$ c_0$
(the minimized energy constrained on Nehari manifold corresponds to the limit systems). Finally for a system with three equations, we prove the existence of ground state solutions whose all components are nonzero.
keywords: Nonlinear elliptic system ground state solution foliated Schwarz symmetric asymptotic limits
Critical point, anti-maximum principle and semipositone p-laplacian problems
E. N. Dancer Zhitao Zhang
Conference Publications 2005, 2005(Special): 209-215 doi: 10.3934/proc.2005.2005.209
In this paper, we use Nehari manifold to extend the anti-maximum principle of Laplacian operator to an existence theorem for p-Laplacian ($p\not=2$), then consider the existence of nonnegative solutions to semipositone quasilinear elliptic problems $-\Delta_p u=\lambda f(u), x\in \Om; u>0, x\in \Om; u=0, x\in \Po$.
keywords: Critical point Quasilinear Elliptic Equation Anti-Maximum principle. Supported by Humboldt Foundation and National Natural Science Foundation of China.
Multiple solutions theorems for semilinear elliptic boundary value problems with resonance at infinity
Shujie Li Zhitao Zhang
Discrete & Continuous Dynamical Systems - A 1999, 5(3): 489-493 doi: 10.3934/dcds.1999.5.489
In this paper, we use Lyapunov-Schmidt method and Morse theory to study semilinear elliptic boundary value problems with resonance at infinity, and get new multiple solutions theorems.
keywords: Dirichlet problems resonance. multiple solutions
A perturbation result for system of Schrödinger equations of Bose-Einstein condensates in $\mathbb{R}^3$
Kui Li Zhitao Zhang
Discrete & Continuous Dynamical Systems - A 2016, 36(2): 851-860 doi: 10.3934/dcds.2016.36.851
We are concerned with the existence of positive solutions for a coupled Schrödinger system \begin{equation*} \left\{ \begin{aligned} &-\Delta{u}_1+{\lambda}_1 {u}_1={\mu}_1 {u}_1^3+\varepsilon \beta(x) {u}_1 {u}_2^2 & ~~in &~~~~ \mathbb{R}^3,\\ &-\Delta{u}_2+{\lambda}_2 {u}_2={\mu}_2 {u}_2^3+\varepsilon \beta(x) {u}_1^2 {u}_2 & ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1>0, ~~{u}_2>0& ~~in & ~~~~\mathbb{R}^3,\\ &{u}_1\in H^1(\mathbb{R}^3),~~{u}_2\in H^1(\mathbb{R}^3), \end{aligned} \right. \end{equation*} where ${\lambda}_1,{\lambda}_2,{\mu}_1,{\mu}_2$ are positive constants. We use perturbation methods to prove that if $\beta \in L^r(\mathbb{R}^3)(r\geq 3)$ doesn't change sign, as corresponding $\varepsilon$ is sufficiently small the system has a positive solution of which both components are positive. Our results is also true for domain $\mathbb{R}^{2}$ and for domain $ \mathbb {R}^{N}, N \geq 4 $ when the similar system is subcritical.
keywords: perturbation. Positive solutions coupled Schrödinger problem
On some nonlocal eigenvalue problems
Ravi P. Agarwal Kanishka Perera Zhitao Zhang
Discrete & Continuous Dynamical Systems - S 2012, 5(4): 707-714 doi: 10.3934/dcdss.2012.5.707
We study a class of nonlocal eigenvalue problems related to certain boundary value problems that arise in many application areas. We construct a nondecreasing and unbounded sequence of eigenvalues that yields nontrivial critical groups for the associated variational functional using a nonstandard minimax scheme that involves the $\mathbb{Z}_2$-cohomological index. As an application we prove a multiplicity result for a class of nonlocal boundary value problems using Morse theory.
keywords: multiplicity $\mathbb{Z}_2$-cohomological index Nonlocal eigenvalue problems minimax eigenvalues Morse theory. nontrivial critical groups nonlocal boundary value problems
Global bifurcations and a priori bounds of positive solutions for coupled nonlinear Schrödinger Systems
Guowei Dai Rushun Tian Zhitao Zhang
Discrete & Continuous Dynamical Systems - S 2018, 0(0): 1905-1927 doi: 10.3934/dcdss.2019125
In this paper, we consider the following coupled elliptic system
$ \begin{equation} \left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber \end{equation} $
Under symmetric assumptions
$ \lambda_1 = \lambda_2, \mu_1 = \mu_2 $
, we determine the number of
$ \gamma $
-bifurcations for each
$ \beta\in(-1, +\infty) $
, and study the behavior of global
$ \gamma $
-bifurcation branches in
$ [-1, 0]\times H_r^1\left( \mathbb{R} ^N\right)\times H_r^1\left( \mathbb{R} ^N\right) $
. Moreover, several results for
$ \gamma = 0 $
, such as priori bounds, are of independent interests, which are improvements of corresponding theorems in [6] and [35].
keywords: Bifurcation Schrödinger systems positive solutions

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