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

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: \(\begin{aligned} {\left\{ \begin{array}{ll} -\hbox {div}\left( |x|^{-ap}|\nabla u|^{p-2}\nabla u\right) = |x|^{-(a+1)p+c} h(x)u^{-\gamma }+ |x|^{-(a+1)p+c} f(x,u) \hbox { in } \Omega \hbox {,}\\ u>0\hbox { in }\Omega \hbox {,}\\ u=0\hbox { on } \partial \Omega \hbox {,} \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}-\text{div}\left({|x|}^{-ap},{|\mathrm{\nabla }u|}^{p-2},\mathrm{\nabla },u\right)={|x|}^{-(a+1)p+c}h(x)u^{-\gamma }+{|x|}^{-(a+1)p+c}f(x,u)\text{in}\mathrm{\Omega }\text{,}\hfill \\ u>0\text{in}\mathrm{\Omega }\text{,}\hfill \\ u=0\text{on}\partial \mathrm{\Omega }\text{,}\hfill \end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is a bounded smooth domain in \({\mathbb {R}}^{N}\) ${\mathbb{R}}^N$ with \(N\ge 3\) $N\ge 3$ , \(1< p<N\) $1<p<N$ , \(0\le a< \frac{N-p}{p}\) $0\le a<\frac{N-p}{p}$ , \(c>0\) $c>0$ , and \(\gamma >0\) $\gamma >0$ . The hypotheses on the functions h and f allow to use sub-supersolutions and Mountain Pass Theorem.