Existence and regularity results for a class of non-uniformly elliptic Robin problems

TL;DR

This paper investigates the existence and regularity of solutions to a class of Robin boundary value problems involving non-uniformly elliptic operators characterized by a variable coefficient function that may vanish at infinity, within bounded Lipschitz domains.

Contribution

It establishes existence and summability results for solutions to non-uniformly elliptic Robin problems with variable coefficients that can degenerate at infinity.

Findings

Solutions exist under certain conditions on the data.

Solutions exhibit specific regularity and summability properties.

The operator's non-uniform ellipticity is addressed in the analysis.

Abstract

In this paper, we study the existence and the summability of solutions to a Robin boundary value problem whose prototype is the following: $$ \begin{cases} -\text{div}(b(|u|)\nabla u)=f &\text{in }\Omega,\\[.2cm] \displaystyle\frac{\partial u}{\partial \nu}+\beta u=0 &\text{on }\partial\Omega \end{cases} $$ where $\Omega$ is a bounded Lipschitz domain in $\mathbb R^N$, $N>2$, $\beta>0$, $b(s)$ is a positive function which may vanish at infinity and $f$ belongs to a suitable Lebesgue space. The presence of such a function $b$ in the principal part of the operator prevents it from being uniformly elliptic when $u$ is large.