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 $$-v^{-1}\mathrm{D}\mathrm{i}\mathrm{v}(\mathrm{\mid }\sqrt{Q}\mathrm{\nabla }u{\mathrm{\mid }}^{p-2}Q\mathrm{\nabla }u)=f\mathrm{\mid }f{\mathrm{\mid }}^{p-2}-v^{-1}\mathrm{D}\mathrm{i}\mathrm{v}(v\mathrm{\mid }g{\mathrm{\mid }}^{p-2}g\mathbf{t}),1<p<\mathrm{\infty },$$ assuming that there is a Sobolev inequality of the form $$\mathrm{\parallel }\phi {\mathrm{\parallel }}_{L^N(v,\mathrm{\Omega })}\le S_N\mathrm{\parallel }\sqrt{Q}\phi {\mathrm{\parallel }}_{L^p(\mathrm{\Omega })},$$ 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.