Infinitely many solutions for a generalized p(·)-Laplace equation involving Leray–Lions type operators

Abstract

We study the existence of infinitely many solutions for a generalized p(·)-Laplace equation involving Leray–Lions operators. Firstly, under a p(·)-sublinear condition for nonlinear term, we obtain a sequence of solutions approaching 0 by showing a new a priori bound for solutions. Secondly, for a p(·)-superlinear condition, we produce a sequence of solutions whose Sobolev norms diverge to infinity when the nonlinear term satisfies a couple of generalized Ambrosetti–Rabinowitz type conditions in which each associated energy functional holds the Palais–Smale condition. Lastly, we deal with a case without the Ambrosetti–Rabinowitz type condition in which an associated energy functional holds the Cerami condition and establish a sequence of solutions whose Sobolev norms diverge to infinity.