Mountain pass solutions to equations with subcritical Musielak-Orlicz-Sobolev growth

TL;DR

This paper proves the existence of solutions for quasilinear elliptic equations with subcritical Musielak-Orlicz-Sobolev growth using Mountain Pass Theorem, introducing a novel approach even for the Orlicz case.

Contribution

It introduces a new variational approach to establish solutions for equations with Musielak-Orlicz-Sobolev growth, demonstrating the sharpness of key assumptions.

Findings

Existence of solutions under subcritical Musielak-Orlicz-Sobolev growth

Novel application of Mountain Pass Theorem in this context

Sharpness of the main assumptions

Abstract

In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of $\R^N$ under subcritical Musielak-Orlicz-Sobolev growth. Our proofs rely essentially on Mountain Pass Theorem with corresponding variational techniques. Furthermore, we establish the sharpness of our central assumptions. As far as we know, our approach is new, even for the Orlicz case.