On a Robin problem with $p$-Laplacian and reaction bounded only from above

TL;DR

This paper proves the existence of three distinct solutions for a Robin boundary value problem involving the p-Laplacian without growth restrictions, using variational and Morse theory techniques.

Contribution

It establishes the existence of multiple solutions for a p-Laplacian Robin problem under minimal growth conditions, including the case p=2.

Findings

Existence of three solutions: negative, positive, and nodal.

Additional nodal solution for p=2 case via Morse theory.

No sub-critical growth condition required.

Abstract

The existence of three smooth solutions, one negative, one positive, and one nodal, to a homogeneous Robin problem with $p$-Laplacian and Carath\'eodory reaction is established. No sub-critical growth condition is taken on. Proofs exploit variational as well as truncation techniques. The case $p=2$ is separately examined, obtaining a further nodal solution via Morse's theory.