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

Mónica Clapp, Alberto Saldaña, Andrzej Szulkin

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.

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

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.
Paper Structure (5 sections, 13 theorems, 68 equations)

This paper contains 5 sections, 13 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. The variational problem
  3. The sign of low energy solutions
  4. The effect of the configuration space
  5. The category of configuration spaces

Key Result

Theorem 1.1

For $\varepsilon$ small enough, the system eq:system has at least $\ell$ nonnegative solutions.

Theorems & Definitions (26)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • ...and 16 more