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

TL;DR

This paper establishes new existence criteria for weak solutions to supercritical elliptic equations with boundary measure data, using capacity conditions and addressing equations with mixed nonlinear source terms.

Contribution

It introduces novel criteria based on Bessel capacities for the existence of solutions in supercritical cases and extends results to equations with mixed nonlinear source terms.

Findings

Existence criteria expressed via Bessel capacities.

Sufficient conditions for equations with mixed source terms.

Applicable to bounded smooth domains and half-spaces.

Abstract

We give new criteria for the existence of weak solutions to an equation with a super linear source term \begin{align*}-\Delta u = u^q ~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega\end{align*}where $\Omega$ is a either a bounded smooth domain or $\mathbb{R}\_+^{N}$, $q\textgreater{}1$ and $\sigma\in \mathfrak{M}^+(\partial\Omega)$ is a nonnegative Radon measure on $\partial\Omega$. One of the criteria we obtain 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 terms. \begin{align*} -\Delta u = |u|^{q\_1-1}u|\nabla u|^{q\_2} ~~\text{in}~\Omega,~~u=\sigma~~\text{on }~\partial\Omega \end{align*} where $q\_1,q\_2\geq 0, q\_1+q\_2\textgreater{}1, q\_2\textless{}2$, $\sigma\in \mathfrak{M}(\partial\Omega)$ is a Radon measure on $\partial\Omega$.