Existence results for a class of degenerate elliptic equations

Abstract

In the present paper we prove existence results for a class of nonlinear elliptic equations whose prototype is: $$ - {\rm div} \left( \left\vert { \nabla u} \right\vert ^{{ p-2}}{ \nabla u\varphi (x)}\right) { +b(x)}\left\vert { \nabla u}\right\vert ^{ { \sigma }}{ \varphi (x)=g\varphi ,} $$ where $ \Omega $ is an open set, $ u=0 $ on $ \partial \Omega , $ where the function $ \varphi (x)=(2\pi )^{-\frac{n}{2}}$ $\exp \left( -\left\vert x\right\vert ^{2}/2\right) $ is the density of Gauss measure and $ g \! \in \! L^{r}(\log L)^{-\frac{1}{2}}(\varphi ,\Omega )$ for $ 1 < r < p^{\prime }.$