Affiliation:
1. School of Mathematics and Finance, Xiangnan University , Chenzhou , , Hunan , P.R.China
2. School of Mathematics and Statistics, Hunan University of Technology and Business , Changsha , Hunan , P. R. China
3. Key Laboratory of Hunan Province for Statistical Learning and Intelligent Computation, Hunan University of Technology and Business , Changsha , Hunan , P. R. China
4. Department of Mathematics, University of Craiova , Street A.I. Cuza no. 13 , Craiova , Romania
Abstract
Abstract
This paper is concerned with the following Hamiltonian elliptic system
−
Δ
u
+
V
(
x
)
u
=
W
v
(
x
,
u
,
v
)
,
x
∈
R
N
,
−
Δ
v
+
V
(
x
)
v
=
W
u
(
x
,
u
,
v
)
,
x
∈
R
N
,
$$
\left\{
\begin{array}{ll}
-\Delta u+V(x)u=W_{v}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\
-\Delta v+V(x)v=W_{u}(x, u, v), \ \ \ \ x\in {\mathbb{R}}^{N},\\
\end{array}
\right.
$$
where z = (u, v) : ℝ
N
→ ℝ2, V(x) and W(x, z) are 1-periodic in x. By making use of variational approach for strongly indefinite problems, we obtain a new existence result of nontrivial solution under new conditions that the nonlinearity
W
(
x
,
z
)
:=
1
2
V
∞
(
x
)
|
A
z
|
2
+
F
(
x
,
z
)
$ W(x,z):=\frac{1}{2}V_{\infty}(x)|Az|^2+F(x, z) $
is general asymptotically quadratic, where V
∞(x) ∈ (ℝ
N
, ℝ) is 1-periodic in x and infℝ
N
V
∞(x) > minℝ
N
V(x), and A is a symmetric non-negative definite matrix.
Cited by
5 articles.
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