Existence of Solutions for a Modified Nonlinear Schrödinger System

Abstract

We are concerned with the following modified nonlinear Schrödinger system: $-\mathrm{\Delta }u$$+u-(\mathrm{1}/\mathrm{2})u\mathrm{\Delta }(u^{\mathrm{2}})=(\mathrm{2}\alpha /(\alpha +\beta ))|u{|}^{\alpha -\mathrm{2}}|v{|}^{\beta }u$, $x\in \mathrm{\Omega }$, $-\mathrm{\Delta }v+v-(\mathrm{1}/\mathrm{2})v\mathrm{\Delta }(v^{\mathrm{2}})=(\mathrm{2}\beta /(\alpha +\beta ))|u{|}^{\alpha }|v{|}^{\beta -\mathrm{2}}v$, $x\in \mathrm{\Omega }$, $u=\mathrm{0}$, $v=\mathrm{0}$, $x\in \partial \mathrm{\Omega }$, where $\alpha >\mathrm{2}$, $\beta >\mathrm{2}$, , ${\mathrm{2}}^{\mathrm{*}}=\mathrm{2}N/(N-\mathrm{2})$ is the critical Sobolev exponent, and ${\mathrm{\Omega }\subset ℝ}^N$ $(N\ge \mathrm{3})$ is a bounded smooth domain. By using the perturbation method, we establish the existence of both positive and negative solutions for this system.