Positive and sign-changing solutions for the nonlinear Schr\"{o}dinger systems with synchronization and separation

TL;DR

This paper constructs new complex solutions for a nonlinear Schrödinger system with radial potentials, exploring effects of coupling on solution structure, including positive, sign-changing, and segregated solutions with intricate concentration patterns.

Contribution

It introduces a new family of solutions with complex concentration structures and analyzes the impact of nonlinear coupling, including the existence of infinitely many sign-changing solutions.

Findings

Constructed solutions with concentration on top and bottom circles of a cylinder.

Established existence of unbounded sequences of positive solutions in both repulsive and attractive cases.

Proved the existence of infinitely many sign-changing solutions with arbitrarily large energy.

Abstract

In this paper, we consider the following nonlinear Schr\"odinger system: -$\Delta$ u+P(x)u=$\mu_1$ $u^3$+$\beta$ u$v^2$, x $\in$ $R^3$,\\ -$\Delta$ v+Q(x)v=$\mu_2$ $v^3$+$\beta$ $u^2$v, x $\in$ $R^3$, where $P(x),Q(x)$ are positive radial potentials,~$\mu_1,\,\mu_2>0$, $\beta$ $\in$ $R$ is a coupling constant. We constructed a new type of solutions which are different from the ones obtained in \cite{PW}. This new family of solutions to system have a more complex concentration structure and are centered at the points lying on the top and the bottom circles of a cylinder with height $h$. Moreover, we examine the effect of nonlinear coupling on the solution structure. In the repulsive case, we construct an unbounded sequence of non-radial positive vector solutions of segregated type. In the attractive case, we construct an unbounded sequence of non-radial positive vector solutions of synchronized type. Moreover, we prove that there exist infinitely many sign-changing solutions whose energy can be arbitrarily large.