On a Schr\"odinger system with shrinking regions of attraction

TL;DR

This paper investigates a coupled elliptic system modeling competing species attracted to shrinking regions, establishing existence, concentration behavior, and the limiting profile of solutions as attraction regions diminish.

Contribution

It proves the existence of infinitely many solutions and a least energy solution, and characterizes their concentration and decoupling behavior in the shrinking regions limit.

Findings

Existence of infinitely many solutions and a least energy solution.

Concentration of solutions as attraction regions shrink.

Decoupling of components when attracted to different regions.

Abstract

In this paper we consider a competitive weakly coupled elliptic system in which each species is attracted to a small region and repelled from its complement. In this setting, we establish the existence of infinitely many solutions and of a nonnegative least energy solution. We show that, as the regions of attraction shrink, least energy solutions of the system concentrate. We study this behavior and characterize their limit profile. In particular, we show that if each component of a least energy solution is attracted to a different region, then the components decouple in the limit, whereas if all the components are attracted to the same region, they remain coupled.