Existence and multiplicity of solutions for double-phase Robin problems

TL;DR

This paper investigates the existence and multiplicity of solutions for a double phase Robin problem with nonlinearities, employing Morse theory and homological linking to handle superlinear and resonant cases.

Contribution

It introduces a Morse theoretic approach using homological local linking to establish existence and multiplicity results without the Ambrosetti-Rabinowitz condition.

Findings

Proves existence of solutions in superlinear case

Establishes multiplicity of solutions in resonant case

Develops Morse theoretic framework for double phase problems

Abstract

We consider a double phase Robin problem with a Carath\'{e}odory nonlinearity. When the reaction is superlinear but without satisfying the Ambrosetti-Rabinowitz condition, we prove an existence theorem. When the reaction is resonant, we prove a multiplicity theorem. Our approach is Morse theoretic, using the notion of homological local linking.