DCDS
On a diffusion system with bounded potential
Yanheng Ding Fukun Zhao
Discrete & Continuous Dynamical Systems - A 2009, 23(3): 1073-1086 doi: 10.3934/dcds.2009.23.1073
This paper is concerned with the following non-periodic diffusion system

$\partial_tu-\Delta_x u+b(t,x)\cdot\nabla_x u+V(x)u=H_v(t,x,u,v)$in $\mathbb{R}\times\mathbb{R}^N,$ $-\partial_tv-\Delta_x v-b(t,x)\cdot\nabla_x v+V(x)v=H_u(t,x,u,v)$in$\mathbb{R}\times\mathbb{R}^N,$
$u(t,x)\to 0$and$v(t,x)\to0$as$|t|+|x|\to\infty.$

Assuming the potential $V$ is bounded and has a positive bound from below, existence and multiplicity of solutions are obtained for the system with asymptotically quadratic nonlinearities via variational approach.

keywords: strongly indefinite functionals Diffusion system variational methods
DCDS
Multiple solutions for superlinear elliptic systems of Hamiltonian type
Rumei Zhang Jin Chen Fukun Zhao
Discrete & Continuous Dynamical Systems - A 2011, 30(4): 1249-1262 doi: 10.3934/dcds.2011.30.1249
This paper is concerned with the following periodic Hamiltonian elliptic system

$\-\Delta \varphi+V(x)\varphi=G_\psi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\-\Delta \psi+V(x)\psi=G_\varphi(x,\varphi,\psi)$ in $\mathbb{R}^N,$
$\varphi(x)\to 0$ and $\psi(x)\to0$ as $|x|\to\infty.$

Assuming the potential $V$ is periodic and $0$ lies in a gap of $\sigma(-\Delta+V)$, $G(x,\eta)$ is periodic in $x$ and superquadratic in $\eta=(\varphi,\psi)$, existence and multiplicity of solutions are obtained via variational approach.
keywords: strongly indefinite functionals. Hamiltonian elliptic system variational methods
CPAA
A note on a superlinear and periodic elliptic system in the whole space
Shuying He Rumei Zhang Fukun Zhao
Communications on Pure & Applied Analysis 2011, 10(4): 1149-1163 doi: 10.3934/cpaa.2011.10.1149
This paper is concerned with the following periodic Hamiltonian elliptic system

$ -\Delta u+V(x)u=g(x,v)$ in $R^N,$

$ -\Delta v+V(x)v=f(x,u)$ in $R^N,$

$ u(x)\to 0$ and $v(x)\to 0$ as $|x|\to\infty,$

where the potential $V$ is periodic and has a positive bound from below, $f(x,t)$ and $g(x,t)$ are periodic in $x$ and superlinear but subcritical in $t$ at infinity. By using generalized Nehari manifold method, existence of a positive ground state solution as well as multiple solutions for odd $f$ and $g$ are obtained.

keywords: variational method strongly indefinite functionals. Hamiltonian elliptic system

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