A note on the equation $-\Delta u+e^{u}-1=0$

TL;DR

This paper investigates boundary measure conditions for solving a nonlinear elliptic PDE involving the Laplacian and exponential nonlinearity, using capacity theory to characterize solvability in bounded domains.

Contribution

It provides new capacity-based criteria on boundary measures for the existence of solutions to the nonlinear PDE in bounded domains.

Findings

Capacity conditions characterize solvability.

Boundary measure conditions are explicitly formulated.

Results extend understanding of nonlinear boundary value problems.

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 nonlinear capacities.