An existence result for a class of quasilinear elliptic eigenvalue problems in unbounded domains

TL;DR

This paper proves the existence of nontrivial solutions for a class of nonlinear eigenvalue problems involving weighted p-Laplacian operators in unbounded domains with Robin boundary conditions, extending previous results.

Contribution

It introduces a new existence result for eigenvalue problems with weighted p-Laplacian in unbounded domains, using Morse theory and cohomological methods.

Findings

Existence of nontrivial weak solutions for all eigenvalues

Results valid even in the semilinear case p=2

Complements recent findings in related eigenvalue problems

Abstract

We consider a nonlinear eigenvalue problem under Robin boundary conditions in a domain with (possibly noncompact) smooth boundary. The problem involves a weighted p-Laplacian operator and subcritical nonlinearities satisfying Ambrosetti-Rabinowitz type conditions. Using Morse theory and a cohomological local splitting as in Degiovanni et al. [5], we prove the existence of a nontrivial weak solution for all (real) values of the eigenvalue parameter. Our result is new even in the semilinear case p = 2 and complements some recent results obtained in Autuori et al. [1].