Necessary and sufficient condition for existence at resonance for eigenvalues of multiplicity two

TL;DR

This paper provides a precise criterion for the existence of solutions at resonance for certain semilinear Dirichlet problems with double eigenvalue multiplicity, and applies it to analyze solution unboundedness in related heat equations.

Contribution

It introduces a necessary and sufficient condition for solutions at resonance when the linear operator has eigenvalues of multiplicity two, advancing the understanding of resonance phenomena.

Findings

Established a criterion for solutions at resonance with double eigenvalues

Applied the criterion to determine unboundedness of solutions in semilinear heat equations

Enhanced the theoretical framework for resonance in nonlinear PDEs

Abstract

We establish necessary and sufficient condition for existence of solutions for a class of semilinear Dirichlet problems with the linear part at resonance at eigenvalues of multiplicity two. The result is applied to give a condition for unboundness of all solutions of the corresponding semilinear heat equation.