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

TL;DR

This paper investigates the behavior of solutions to p(x)-Laplacian problems with unbounded exponents in a subdomain, establishing existence, uniqueness, and characterizations of the limit solutions as the exponent tends to infinity.

Contribution

It introduces a rigorous framework for defining solutions when the exponent becomes unbounded, including variational and viscosity characterizations, extending the theory of p(x)-harmonic functions.

Findings

Limit solutions exist and are unique under certain conditions.

The limit solutions are characterized as infinity-harmonic in the subdomain.

The solutions satisfy a specific boundary value problem in the viscosity sense.

Abstract

We study the Dirichlet problem $-\div(|\nabla u|^{p(x)-2} \nabla u) =0 $ in $\Omega$, with $u=f$ on $\partial \Omega$ and $p(x) = \infty$ in $D$, a subdomain of the reference domain $\Omega$. 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, $\infty$-harmonic within $D$. Moreover, we examine this limit in the viscosity sense and find the boundary value problem it satisfies in the whole of $\Omega$.