# An elliptic semilinear equation with source term and boundary measure data: the supercritical case

created by nguyen on 22 Jul 2018

[BibTeX]

Published Paper

Inserted: 22 jul 2018
Last Updated: 22 jul 2018

Journal: Journal of Functional Analysis
Year: 2015
Doi: https://www.sciencedirect.com/science/article/pii/S002212361500261X

Abstract:

We give new criteria for the existence of weak solutions to equation with source term $-\Delta u = u^q$ in $\Omega$, $u=\sigma$ on $\partial\Omega$ where $q>1$, $\Omega$ is a either a bounded smooth domain or $\mathbb{R}_+^{N}$ and $\sigma\in \mathbf{M}^+(\partial\Omega)$ is a nonnegative Radon measure on $\partial\Omega$. In particular, one of the criteria is expressed in terms of some Bessel capacities on $\partial\Omega$. We also give a sufficient condition for the existence of weak solutions to equation with source mixed term. $-\Delta u = u ^{q_1-1}u \nabla u ^{q_2}$ in $\Omega$, $u=\sigma$ on $\partial\Omega$ where $q_1,q_2\geq 0, q_1+q_2>1, q_2<2$, $\sigma\in \mathbf{M}(\partial\Omega)$ is a Radon measure on $\partial\Omega$.