Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth

Abstract

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is $\{-\mathrm{d}\mathrm{i}\mathrm{v}[(\mathrm{1}+{\left|u\right|}^q)\nabla u]+u=\gamma \left({\left|\nabla u\right|}^{\mathrm{2}}/{\left(\mathrm{1}+\left|u\right|\right)}^{\mathrm{1}-q}\right)+f\hspace{0.17em}\hspace{0.17em}\mathrm{i}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\mathrm{\Omega },\hspace{0.17em}\hspace{0.17em}u=\mathrm{0}\hspace{0.17em}\hspace{0.17em}\mathrm{o}\mathrm{n}\hspace{0.17em}\hspace{0.17em}\partial \mathrm{\Omega },\}$, where , $\gamma >\mathrm{0}$ and $f$ belongs to some appropriate Lebesgue space. We give assumptions on $f$ with respect to $q$ and $\gamma$ to show the existence and regularity results for the solutions of previous equation.