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

DOI: http://dx.doi.org/10.12775/TMNA.2009.029

#### 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

[< i> Infinitely many solutions of a symmetric Dirichlet problem< /i> , Nonlinear

Anal. < b> 20< /b> (1993), 1205–1216]

and Bartsch-Willem [< i> On an elliptic equation with concave and convex nonlinearities< /i> ,

Proc. Amer. Math. Soc. < b> 123< /b> (1995), 3555–3561], respectively), we will study the multiplicity

of solutions for operator equation

$$

J_{\varphi}u=G^{\prime}(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.

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

[< i> Infinitely many solutions of a symmetric Dirichlet problem< /i> , Nonlinear

Anal. < b> 20< /b> (1993), 1205–1216]

and Bartsch-Willem [< i> On an elliptic equation with concave and convex nonlinearities< /i> ,

Proc. Amer. Math. Soc. < b> 123< /b> (1995), 3555–3561], respectively), we will study the multiplicity

of solutions for operator equation

$$

J_{\varphi}u=G^{\prime}(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.

#### Keywords

Critical points; fountain theorem; dual fountain theorem; duality mappings; Orlicz-Sobolev spaces

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