On Caffarelli–Kohn–Nirenberg Type Problems with a Sign-Changing Term

Abstract

In this work we show existence and multiplicity of positive solutions using the sub-supersolution method in Caffarelli–Kohn–Nirenberg type problems with a sign-changing term. More precisely, using the sub-supersolution method, we study the following class of singular problem:

$$\begin{aligned} {\left\{ \begin{array}{ll} -\hbox {div}\left( |x|^{-ap}|\nabla u|^{p-2}\nabla u\right) = |x|^{-(a+1)p+c} h(x)u^{-\gamma }+ |x|^{-(a+1)p+c} f(x,u) \hbox { in } \Omega \hbox {,}\\ u>0\hbox { in }\Omega \hbox {,}\\ u=0\hbox { on } \partial \Omega \hbox {,} \end{array}\right. } \end{aligned}$$

where $\Omega$ is a bounded smooth domain in ${\mathbb {R}}^{N}$ with $N\ge 3$, $1< p<N$, $0\le a< \frac{N-p}{p}$, $c>0$, and $\gamma >0$. The hypotheses on the functions h and f allow to use sub-supersolutions and Mountain Pass Theorem.