Existence results for nonlinear degenerate elliptic problems involving singular and supercritical nonlinearities

In this article, we establish the existence of nonnegative solutions to a class of nonlinear degenerate elliptic equations subject to zero Dirichlet boundary conditions on \(\Omega \) $\mathrm{\Omega }$ , an open, bounded subset of \({\mathbb {R}}^N (N\ge 3)\) ${\mathbb{R}}^N(N\ge 3)$ . The problem is modeled by: \(\begin{aligned} {\left\{ \begin{array}{ll} -\text {div}\left( \dfrac{a(x)\nabla u}{(1+u)^{\theta }}\right) = \dfrac{\lambda }{u^\gamma }+ u^p &{}\text { in } \Omega ,\\ u\ge 0 &{}\text { in } \Omega ,\\ u=0 &{}\text { on } \partial \Omega , \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\text{div}\left({\displaystyle \frac{a(x)\mathrm{\nabla }u}{{(1+u)}^{\theta }}}\right)={\displaystyle \frac{\lambda }{u^{\gamma }}}+u^p\hfill & \text{in}\mathrm{\Omega },\hfill \\ u\ge 0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$ where \(\lambda , \theta , \gamma \) $\lambda ,\theta ,\gamma$ and p are positive parameters and a(x) is a real-valued function defined on \(\Omega \) $\mathrm{\Omega }$ such that for \(\alpha , \beta \in {\mathbb {R}}, \) $\alpha ,\beta \in \mathbb{R},$ it satisfies \(0<\alpha \le a(x)\le \beta \) $0<\alpha \le a(x)\le \beta$ . The main contributions of this paper are the treatment of singular and supercritical nonlinearities on the right-hand side, as well as the lack of coercivity in the principle part.