Limit as $p(x)\rightarrow \infty$ of $p(x)$-Harmonic functions for unbounded $p(x)$

TL;DR

This paper studies the convergence of solutions to variable exponent p-Laplacian equations as the exponents grow unbounded, showing they tend to a viscosity solution of a related differential operator.

Contribution

It demonstrates that solutions of p(x)-Laplacian problems with unbounded, variable exponents converge to a viscosity solution, extending previous results to unbounded exponent sequences.

Findings

Solutions converge to a viscosity solution of a differential operator.

The convergence holds for unbounded, variable exponents.

The sequence of exponents can be unbounded in the domain.

Abstract

It is shown that if $p_n$ is a sequence of continuous, unbounded exponents on a bounded, smooth domain $\Omega\subset {\mathbb R}^n$ with $1<\inf\limits_{x\in \Omega}p_n(x)$ and $p_n\rightarrow \infty$ uniformly, then the sequence $(u_n)$ of solutions of the $p_n(\cdot)$-Laplacian converges to the viscosity solution of a suitable differential operator. The novelty here is that each term of the sequence of exponents $(p_n)$ is allowed to be unbounded in $\Omega$.