Ground State Solutions of Nehari–Pankov Type for a Superlinear Hamiltonian Elliptic System on RN

Tang, Xianhua

Canadian mathematical bulletin, Tome 58 (2015) no. 3, pp. 651-663

Résumé

This paper is concerned with the following elliptic system of Hamiltonian type $$\left\{ \begin{align}& -\Delta u+V\left( x \right)u={{W}_{v}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\& -\Delta v+V\left( x \right)v={{W}_{u}}\left( x,u,v \right),\,x\in {{\mathbb{R}}^{N}}, \\& u,v\in {{H}^{1}}\left( {{\mathbb{R}}^{N}} \right), \\ \end{align} \right.$$ where the potential $V$ is periodic and $0$ lies in a gap of the spectrum of $-\Delta +V,W\left( x,u,v \right)$ is periodic in $x$ and superlinear in $u$ and $v$ at infinity. We develop a direct approach to finding ground state solutions of Nehari–Pankov type for the above system. Our method is especially applicable to the case when $$W\left( x,u,v \right)=\sum\limits_{i=1}^{k}{\,\int _{0}^{\left| \alpha iu+\beta iv \right|}\,{{g}_{i}}\left( x,\,t \right)t\text{d}t+\sum\limits_{j=1}^{l}{\int_{0}^{\sqrt{{{u}^{2}}+2{{b}_{juv+aj{{v}^{2}}}}}}{{{h}_{j}}\left( x,t \right)tdt,}}}$$ where ${{\alpha }_{i}},{{\beta }_{i}},{{a}_{j}},{{b}_{j}}\in \mathbb{R}$ with $\alpha _{i}^{2}+\beta _{i}^{2}\ne 0$ , and ${{a}_{j}}>b_{j}^{2},{{g}_{i}}\left( x,t \right)$ and ${{h}_{j}}\left( x,t \right)$ are nondecreasing in $t\in {{\mathbb{R}}^{+}}$ for every $x\in {{\mathbb{R}}^{N}}$ and ${{g}_{i}}\left( x,0 \right)={{h}_{j}}\left( x,0 \right)=0$ .

Détail

DOI : 10.4153/CMB-2015-019-2

Mots-clés : 35J50, 35J55, Hamiltonian elliptic system, superlinear, ground state solutions of Nehari–Pankov type, strongly indefinite functionals