Electron. J. Diff. Eqns., Vol. 2008(2008), No. 119, pp. 1-10.

Existence of weak solutions for a nonuniformly elliptic nonlinear system in $R^N$

Nguyen Thanh Chung

We study the nonuniformly elliptic, nonlinear system
 - \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.

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Nguyen Thanh Chung
Department of Mathematics and Informatics
Quang Binh University, 312 Ly Thuong Kiet, Dong Hoi, Vietnam
email: ntchung82@yahoo.com

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