$p(x)$-Harmonic functions with unbounded exponent in a subdomain

Abstract

We study the Dirichlet problem $-\mathrm{div}(|\nabla u{|}^{p(x)-2}\nabla u)=0$ in Ω, with $u=f$ on ∂Ω and $p(x)=\infty$ in D, a subdomain of the reference domain Ω. The main issue is to give a proper sense to what a solution is. To this end, we consider the limit as $n\to \infty$ of the solutions $u_n$ to the corresponding problem when $p_n(x)=p(x)\wedge n$, in particular, with $p_n=n$ in D. Under suitable assumptions on the data, we find that such a limit exists and that it can be characterized as the unique solution of a variational minimization problem which is, in addition, ∞-harmonic within D. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of Ω.