Ground state of indefinite coupled nonlinear Schrödinger systems

Abstract

In this paper, we study the ground state solutions of the following coupled nonlinear Schrödinger system (P) $-\Delta u_1-{\tau }_1{u}_1={\mu }_1{u}_1^3+\beta u_1{u}_2^2$, $-\Delta u_2-{\tau }_2{u}_2={\mu }_2{u}_2^3+\beta u_1^2{u}_2$ in $\Omega$, $u_1=u_2=0$ on $\partial \Omega$, where ${\mu }_1,{\mu }_2>0$, $\beta >0$ and $\Omega \subset {\mathbb{R}}^N(N\le 3)$ is a bounded domain with smooth boundary. We are concerned with the indefinite case, i.e., ${\tau }_1,{\tau }_2$ are greater than or equal to the principal eigenvalue of $-\Delta$ with the Dirichlet boundary datum. By delicate variational arguments, we obtain the existence of ground state solution to $(P)$, and also provide information on critical energy levels for coupling parameter $\beta$ in some ranges.