Configuration spaces and multiple positive solutions to a singularly perturbed elliptic system

TL;DR

This paper investigates how the number of positive solutions to a singularly perturbed elliptic system increases with the number of equations, using variational methods, asymptotic analysis, and topological estimates.

Contribution

It introduces new estimates on configuration space categories and demonstrates the growth of solutions in the competitive regime.

Findings

Number of solutions increases with equations in the competitive regime

Develops new topological estimates for configuration spaces

Combines variational and asymptotic methods effectively

Abstract

We consider a weakly coupled singularly perturbed variational elliptic system in a bounded smooth domain with Dirichlet boundary conditions. We show that, in the competitive regime, the number of fully nontrivial solutions with nonnegative components increases with the number of equations. Our proofs use a combination of four key elements: a convenient variational approach, the asymptotic behavior of solutions (concentration), the Lusternik-Schnirelman theory, and new estimates on the category of suitable configuration spaces.