Existence of Nontrivial Solutions for Perturbed Elliptic System in ℝN

Abstract

We consider the perturbed nonlinear elliptic system $-{\epsilon }^2\mathrm{\Delta }u+V(x)u=K(x)|u{|}^{2^{\mathrm{*}}-2}u+H_u(u,v),\mathrm{ }\mathrm{ }x\in {ℝ}^N$, $-{\epsilon }^2\mathrm{\Delta }v+V(x)v=K(x)|v{|}^{2^{\mathrm{*}}-2}v+H_v(u,v),\mathrm{ }\mathrm{ }x\in {ℝ}^N$, where $N\ge 3,2^{*}=2N/(N-2)$ is the Sobolev critical exponent. Under proper conditions on $V$, $H$, and $K$, the existence result and multiplicity of the system are obtained by using variational method provided $\epsilon$ is small enough.