The non-homogeneous Dirichlet problem for the p(x)-Laplacian with unbounded p(x) on a smooth domain

Abstract

This paper examines the solvability of the Dirichlet problem for the variable exponent p-Laplacian in the case of unbounded $p\left(x\right)$. For a bounded domain $\Omega \subset R^n$ with a smooth boundary and p satisfying $p_{-}=\underset{\Omega }{\mathrm{essinf}}p>n$ and φ in the Sobolev space $W^{1,p\left(x\right)}\left(\Omega \right)$, we investigate the problem${\Delta }_p\left(u\right)=0\text{in}\Omega ,u{|}_{\partial \Omega }=\phi .$ We introduce the space $V_0^{1,p}\left(\Omega \right)$, which is the natural solution space for the minimization of the Dirichlet integral given the unbounded nature of $p\left(x\right)$. Our main results establish the existence and uniqueness of solutions within this space. Since $V_0^{1,p}\left(\Omega \right)$ is not defined via a TVS topology, the paper includes the description of the necessary modular topological framework and discusses Clarkson-type inequalities for unbounded variable exponents, which are interesting in their own right.