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

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 $N(t)=t^{σp}$, $σ\geq 1$, or a Young function of the form $N(t)=t^p\log(e+t)^σ$, $σ>1$. 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.