Positive solutions to nonhomogeneous quasilinear problems with singular and supercritical nonlinearities

In this article, we establish the existence of nonnegative solutions to the following quasilinear and singular elliptic problems with supercritical nonlinearity: \(\begin{aligned} \left\{ \begin{aligned} {} -\Delta _p z-\Delta _q z&{}= \lambda \frac{h(x)}{z^\gamma }+z^\theta , \ z>0&\quad \text{ in } \, \Omega , \\ z&{}= 0&\quad \text{ on } \partial \Omega , \end{aligned} \right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill -{\mathrm{\Delta }}_{p}z-{\mathrm{\Delta }}_{q}z& =\lambda \frac{h(x)}{z^{\gamma }}+z^{\theta },z>0\hfill & \hfill \text{in}\mathrm{\Omega },\\ \hfill z& =0\hfill & \hfill \text{on}\partial \mathrm{\Omega },\end{array}\right)\end{array}$$ where \(\Omega \) $\mathrm{\Omega }$ is an open, bounded subset of \(\mathbb {R}^N (N\ge 3)\) ${\mathbb{R}}^N(N\ge 3)$ with \(C^2\) $C^2$ boundary, h is a positive real-valued function, \(1<p<q<\infty \) $1<p<q<\infty$ and \(\lambda , \theta , \gamma \) $\lambda ,\theta ,\gamma$ are positive parameters. The main contribution of this paper is the treatment of singular and supercritical nonlinearities on the right-hand side in the presence of the nonhomogeneous \((p+q)\) $(p+q)$ -Laplace operator. To demonstrate the existence of a weak solution, we utilise the method of sub and supersolution.