Multiple sign-changing and semi-nodal solutions for coupled Schrodinger equations

TL;DR

This paper proves the existence of multiple sign-changing and semi-nodal solutions for coupled Schrödinger equations with specific boundary conditions, expanding understanding of solution multiplicity in nonlinear PDE systems.

Contribution

It establishes the existence of infinitely many sign-changing and semi-nodal solutions for coupled Schrödinger equations with positive parameters, for each fixed coupling parameter within a certain range.

Findings

Existence of at least k sign-changing solutions for each natural number k.

Existence of at least k semi-nodal solutions for each natural number k.

Solutions exist for coupling parameters below a certain positive threshold.

Abstract

We study the following coupled Schr\"{o}dinger equations which have appeared as several models from mathematical physics: \begin{displaymath} \begin{cases}-\Delta u_1 +\la_1 u_1 = \mu_1 u_1^3+\beta u_1 u_2^2, \quad x\in \Omega,\\ -\Delta u_2 +\la_2 u_2 =\mu_2 u_2^3+\beta u_1^2 u_2, \quad x\in \Om,\\ u_1=u_2=0 \,\,\,\hbox{on \,$\partial\Om$}.\end{cases}\end{displaymath} Here $\Om\subset\RN (N=2, 3)$ is a smooth bounded domain, $\la_1, \la_2$, $\mu_1, \mu_2$ are all positive constants. We show that, for each $k\in\mathbb{N}$ there exists $\bb_k>0$ such that this system has at least $k$ sign-changing solutions (i.e., both two components change sign) and $k$ semi-nodal solutions (i.e., one component changes sign and the other one is positive) for each fixed $\bb\in (0, \bb_k)$.