On the equation $-\Delta u+e^{u}-1=0$ with measures as boundary data

TL;DR

This paper investigates conditions on boundary measures for solving a nonlinear PDE involving the Laplacian and exponential nonlinearity, using Orlicz capacities to characterize solvability in bounded domains.

Contribution

It introduces measure conditions expressed via Orlicz capacities that determine solvability of the PDE with measure boundary data.

Findings

Conditions on boundary measures are characterized by Orlicz capacities.

The results provide a criterion for existence of solutions based on boundary measure properties.

The approach links nonlinear PDE solvability to capacity theory in potential analysis.

Abstract

If $\Omega$ is a bounded domain in $\mathbb R^N$, we study conditions on a Radon measure $\mu$ on $\partial\Omega$ for solving the equation $-\Delta u+e^{u}-1=0$ in $\Omega$ with $u=\mu$ on $\partial\Omega$. The conditions are expressed in terms of Orlicz capacities.