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

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.