We consider the boundedness and exponential integrability of solutions to the Dirichlet problem for the degenerate quasilinear elliptic equation \(\begin{aligned} -v^{-1}{{\,\textrm{div}\,}}(|\sqrt{Q}{{\,\mathrm{\nabla }\,}}u|^{p-2}Q{{\,\mathrm{\nabla }\,}}u)=f|f|^{p-2} -v^{-1}{{\,\textrm{div}\,}}(v|g|^{p-2}g\textbf{t}), \quad 1<p<\infty , \end{aligned}\) assuming that there is a Sobolev inequality of the form \(\begin{aligned} \Vert \varphi \Vert _{L^N(v,\Omega )}\le S_N\Vert \sqrt{Q} \nabla \varphi \Vert _{L^p(\Omega )}, \end{aligned}\) 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 [13, 43].
