Existence of solutions for singular quasilinear elliptic problems with dependence of the gradient

Abstract

In this paper we establish existence of solutions to the following boundary value problem involving a $p$-gradient term $$\displaystyle -\Delta_{p} u + g(u)|\nabla u|^p = \lambda u^\sigma+ \Psi(x), \quad u>0 \quad\mbox{in} ~\Omega, \quad u = 0 \quad\mbox{on} ~ \partial\Omega,$$ where $\Delta_{p}:= \operatorname{div}(|\nabla u|^{p-2}\nabla u)$ is $p$-Laplacian operator, $\Omega\subset \mathbb{R}^N$ $\left (N\geq 3\right )$ is a bounded domain with smooth boundary, $1<p<N$, $0<\sigma<p^*-1$ with $p^*:= pN/\left ( N-p\right )$, $\Psi$ is a measurable function and $g(s)\geq 0$ is a continuous function on the interval $(0,+\infty)$ which may have a singularity at the origin, i.e.~$g(s)\to +\infty$ as $s\to 0$. Using the topological degree theory, under certain assumptions on $\Psi$, we prove the existence of a continuum of positive solutions.

Keywords

• $p$-gradient term
• singular equations
• elliptic equations