$p(x)$-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 $\Omega$, 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 $\Omega$.