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

This paper examines the solvability of the Dirichlet problem for the variable exponent p-Laplacian in the case of unbounded p(x). For a bounded domain Ω⊂Rⁿ with a smooth boundary and p satisfying p₋=essinfΩp>n and φ in the Sobolev space W^1,p(x)(Ω), we investigate the problem Δ_p(u)=0inΩ,u|_∂Ω=φ. We introduce the space V₀^1,p(Ω), which is the natural solution space for the minimization of the Dirichlet integral given the unbounded nature of p(x). Our main results establish the existence and uniqueness of solutions within this space. Since V₀^1,p(Ω) 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.