Infinitely many solutions for operator equations involving duality mappings on Orlicz-Sobolev spaces

Abstract

Let $X$ be a real reflexive and separable Banach space having the Kadeč-Klee property, compactly imbedded in the real Banach space $V$ and let $G\colon V\rightarrow\mathbb{R}$ be a differentiable functional.

By using ``fountain theorem'' and ``dual fountain theorem'' (Bartsch [Infinitely many solutions of a symmetric Dirichlet problem, Nonlinear Anal. 20 (1993), 1205–1216] and Bartsch-Willem [On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), 3555–3561], respectively), we will study the multiplicity of solutions for operator equation $$ J_{\varphi}u=G'(u), $$ where $J_{\varphi}$ is the duality mapping on $X$, corresponding to the gauge function $\varphi$.

Equations having the above form with $J_{\varphi}$ a duality mapping on Orlicz-Sobolev spaces are considered as applications. As particular cases of the latter results, some multiplicity results concerning duality mappings on Sobolev spaces are derived.