Bounded solutions of degenerate elliptic equations with an Orlicz-gain Sobolev inequality

TL;DR

This paper investigates the boundedness and exponential integrability of solutions to a class of degenerate elliptic equations under specific Sobolev inequalities involving Orlicz spaces, extending previous results in the field.

Contribution

The authors establish new regularity results for solutions of degenerate elliptic equations assuming Orlicz-Sobolev inequalities, generalizing prior work.

Findings

Solutions are bounded under certain Sobolev and regularity conditions.

Solutions exhibit exponential integrability given specific assumptions.

The results extend classical regularity theory to Orlicz space settings.

Abstract

We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate elliptic equation \[ -v^{-1}\mathrm{Div}(|\sqrt{Q}\nabla u|^{p-2}Q\nabla u)=f|f|^{p-2}- v^{-1}\mathrm{Div}(v|g|^{p-2}g \mathbf{t}), \quad 1<p<\infty, \] assuming that there is a Sobolev inequality of the form \[ \|\varphi\|_{L^N(v,\Omega)}\leq S_N\|\sqrt{Q} \varphi\|_{L^p(\Omega)}, \] where $N$ is a power function of the form $N(t)=t^{\sigma p}$, $\sigma\geq 1$, or a Young function of the form $N(t)=t^p\log(e+t)^\sigma$, $\sigma>1$. In our results we study the interplay between the Sobolev inequality and the regularity assumptions needed on $f$ and $g$ to prove that the solution is bounded or is exponentially integrable. Our results generalize those previously proved in previous work by the authors.