Topological structure of the set of solution for singular elliptic equations with a convective term

Abstract

In this paper we establish existence and nonexistence of solution to the quasilinear singular elliptic equation $-{\Delta }_{p}u=\lambda \beta \left(u\right)|\nabla u{|}^p+f\text{in}\Omega$, under Dirichlet boundary conditions, where $\Omega \subset R^N$ is a bounded domain with smooth boundary ∂Ω, $\lambda >0$ is a real parameter, $\beta :(0,\infty )\to (0,\infty )$ is a $C^1$ function, possibly singular at zero, in the sense that $\beta \left(s\right)\stackrel{s\to 0}{\to }\infty$, and $f:\bar{\Omega }\to [0,\infty )$ is continuous. No monotonicity condition whatsoever is imposed upon β.