Positive solutions to some systems of coupled nonlinear Schr\"odinger equations

TL;DR

This paper proves the existence of nontrivial bound state solutions, including positive solutions under certain conditions, for a system of coupled nonlinear Schrödinger equations in dimensions 1 to 3.

Contribution

It establishes the existence of nonnegative and positive bound state solutions for coupled nonlinear Schrödinger systems under specific parameter conditions.

Findings

Existence of nonnegative bound state solutions.

Conditions for positivity of solutions.

Results valid in dimensions 1 to 3.

Abstract

We study the existence of nontrivial bound state solutions to the following system of coupled nonlinear time-independent Schr\"odinger equations $$ - \Delta u_j+ \lambda_j u_j =\mu_j u_j^3+ \sum_{k=1;k\neq j}^N\beta_{jk} u_ju_k^2,\quad u_j\in W^{1,2}(\mathbb{R}^n);\: j=1,\ldots,N $$ where $n=1,\,2,\, 3; \,\lambda_j,\, \mu_j>0$ for $j=1,\ldots,N$, the coupling parameters $\beta_{jk}=\beta_{kj}\in \mathbb{R}$ for $j,k=1,\ldots,N$, $j\neq k$. Precisely, we prove the existence of nonnegative bound state solutions for suitable conditions on the coupling factors. Additionally, with more restrictive conditions on the coupled parameters, we show that the bound states founded are positive.