Singular elliptic problems with convection term in anisotropic media

Abstract

We are concerned with singular elliptic problems of the form $-Δu\pm p(d(x))g(u)=\la f(x,u)+μ|\nabla u|^a$ in $\Omega ,$ where $\Omega$ is a smooth bounded domain in $\RR^N$, $d(x)=\mathrm{dist}(x,\partial \Omega ),$ $\la>0,$ $μ\in\RR$, $0<a\le 2$, and $f,k$ are nonnegative and nondecreasing functions. We assume that $p(d(x))$ is a positive weight with possible singular behavior on the boundary of $\Omega$ and that the nonlinearity $g$ is unbounded around the origin. Taking into account the competition between the anisotropic potential $p(d(x))$, the convection term $|\nabla u{|}^a$, and the singular nonlinearity $g$, we establish various existence and nonexistence results.