Infinitely many solutions for an anisotropic differential inclusion on unbounded domains

Abstract

The problem deals with the anisotropic $p(x)$-Laplacian operator where $p_i$ are Lipschitz continuous functions $2\leq p_i(x)<N$ for all $x\in \mathbb{R}^N$ and $i\in\{1,\dots,N\}$. Assume $p^o_N(x)=\max_{1\leq i\leq N} p_i(x)$, $a\in L^1_+(\mathbb{R}^N)\cap L^{\frac{N}{p^o_N(x)-1}}(\mathbb{R}^N)$, $F(x,t)$ is locally Lipschitz in the $t$-variable integrand and $\partial F(x,t)$ is the subdifferential with respect to the $t$-variable in the sense of Clarke. By establishing the existence of infinitely many solutions, we achieve a first result within the anisotropic framework.

Keywords

• anisotropic $p(x)$-laplacian
• differential inclusion problem
• locally lipschitz function
• infinitely many solutions
• variational method