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

Abstract

In this paper, we prove the existence of solutions to quasilinear elliptic equations on a bounded domain of ${\mathbb{R}}^N$ 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.