Resonant semilinear Robin problems with a general potential

TL;DR

This paper studies a complex Robin boundary value problem involving an indefinite potential and resonance phenomena, proving the existence of multiple solutions using advanced reduction techniques.

Contribution

It introduces a novel application of the reduction method to establish multiple solutions for a resonant semilinear Robin problem with a general potential.

Findings

Existence of at least two nontrivial smooth solutions

Applicable to problems with indefinite, unbounded potentials

Handles resonance at multiple eigenvalues

Abstract

We consider a semilinear Robin problem driven by the Laplacian plus an indefinite and unbounded potential. The reaction term is a Carath\'eodory function which is resonant with respect to any nonprincipal eigenvalue both at $\pm \infty$ and 0. Using a variant of the reduction method, we show that the problem has at least two nontrivial smooth solutions.