Ground state of indefinite coupled nonlinear Schr\"odinger systems

TL;DR

This paper investigates the existence and properties of ground state solutions for a coupled nonlinear Schrödinger system with indefinite parameters, using variational methods to analyze critical energy levels and solution existence.

Contribution

The study introduces new variational techniques to establish ground state solutions for indefinite coupled nonlinear Schrödinger systems, including analysis of energy levels relative to the coupling parameter.

Findings

Existence of ground state solutions under certain conditions.

Identification of critical energy levels for the coupling parameter.

Application of variational methods to indefinite systems.

Abstract

In this paper, we study the ground state solutions of the following coupled nonlinear Schr\"odinger system (P) $-\Delta u_1-\tau_1 u_1 =\mu_1u_1^3+\beta u_1u_2^2$, $ -\Delta u_2-\tau_2 u_2 =\mu_2u_2^3+\beta u_1^2u_2$ in $\Omega$, $u_1=u_2=0$ on $\partial\Omega$, where $\mu_1, \mu_2>0$, $\beta>0$ and $\Omega\subset \mathbb{R}^N (N\le3)$ 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.