Electronic Journal of Differential Equations,
Vol. 2008(2008), No. 119, pp. 1-10.
Title: Existence of weak solutions for a nonuniformly elliptic nonlinear
system in $R^N$
Author: Nguyen Thanh Chung (Quang Binh Univ., Dong Hoi, Vietnam)
Abstract:
We study the nonuniformly elliptic, nonlinear system
$$\displaylines{
- \hbox{div}(h_1(x)\nabla u)+ a(x)u = f(x,u,v) \quad
\text{in } \mathbb{R}^N,\cr
- \hbox{div}(h_2(x)\nabla v)+ b(x)v = g(x,u,v) \quad
\text{in } \mathbb{R}^N.
}$$
Under growth and regularity conditions on the nonlinearities
$f$ and $g$, we obtain weak solutions in a subspace
of the Sobolev space $H^1(\mathbb{R}^N, \mathbb{R}^2)$ by applying
a variant of the Mountain Pass Theorem.
Submitted March 27, 2008. Published August 25, 2008.
Math Subject Classifications: 35J65, 35J20.
Key Words: Nonuniformly elliptic; nonlinear systems; mountain pass theorem;
weakly continuously differentiable functional.