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 \[ \|φ\|_{L^N(v,Ω)}\leq S_N\|\sqrt{Q} φ\|_{L^p(Ω)}, \] where $N$ is a power function of the form $N(t)=t^{\sigma p}$, $\sigma \ge 1$, 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.

Substantially revised based on reviewer comments. Now includes boundedness of solutions to equations with a divergence term on the righthand side