Abstract view
Ground State Solutions of NehariPankov Type for a Superlinear Hamiltonian Elliptic System on ${\mathbb{R}}^{N}$


Published:20150521
Printed: Sep 2015
Xianhua Tang,
School of Mathematics and Statistics, Central South University, Changsha, Hunan 410083, P.R. China
Abstract
This paper is concerned with the following
elliptic system of Hamiltonian type
\[
\left\{
\begin{array}{ll}
\triangle u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
\triangle v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},
\\
u, v\in H^{1}({\mathbb{R}}^{N}),
\end{array}
\right.
\]
where the potential $V$ is periodic and $0$ lies in a gap of
the spectrum of $\Delta+V$, $W(x, s, t)$ is
periodic in $x$ and superlinear in $s$ and $t$ at infinity.
We develop a direct approach to find ground
state solutions of NehariPankov type for the above system.
Especially, our method is applicable for the
case when
\[
W(x, u, v)=\sum_{i=1}^{k}\int_{0}^{\alpha_iu+\beta_iv}g_i(x,
t)t\mathrm{d}t
+\sum_{j=1}^{l}\int_{0}^{\sqrt{u^2+2b_juv+a_jv^2}}h_j(x,
t)t\mathrm{d}t,
\]
where $\alpha_i, \beta_i, a_j, b_j\in \mathbb{R}$ with $\alpha_i^2+\beta_i^2\ne
0$ and $a_j\gt b_j^2$, $g_i(x, t)$
and $h_j(x, t)$ are nondecreasing in $t\in \mathbb{R}^{+}$ for every
$x\in \mathbb{R}^N$ and $g_i(x, 0)=h_j(x, 0)=0$.