# American Institute of Mathematical Sciences

## Journals

CPAA
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
 $\left\{ {\begin{array}{*{20}{l}}{ - \Delta u + {\lambda _1}u + \kappa v = {\mu _1}{u^3} + \beta u{v^2}}&{\quad {\rm{ in}}\;\;\Omega ,}\\{ - \Delta v + {\lambda _2}v + \kappa u = {\mu _2}{v^3} + \beta {u^2}v}&{\quad {\rm{ in}}\;\;\Omega ,}\\{u = v = 0\;on\;\;\partial \Omega \;({\rm{or}}\;u,v \in {H^1}({\mathbb{R}^N})\;{\rm{as}}\;\Omega = {\mathbb{R}^N}),}&{}\end{array}} \right.$
where
 $N≤3, Ω\subseteq\mathbb{R}^N$
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.
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PROC
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:
DCDS
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:
DCDS
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:
DCDS-S
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:
DCDS-S
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
 $$$\left\{ \begin{array}{ll} -\Delta u+\lambda_1 u = \mu_1 u^3+\beta uv^2-\gamma v &\text{in } \mathbb{R}^N, \\ -\Delta v+\lambda_2 v = \mu_2 v^3+\beta vu^2-\gamma u &\text{in } \mathbb{R}^N, \\ u(x), v(x)\rightarrow 0 \text{ as } \vert x\vert\rightarrow+\infty. \end{array} \right.\nonumber$$$
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].
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