Existence of positive solution for a class of elliptic problems with indefinite nonlinearities with critical and supercritical growth

Abstract

We are interested in problems as follows \begin{equation} \tag{$P_{\mu,\beta}$} \left\{ \begin{array}[c]{ll} - \Delta u = \lambda_1 u + \mu g(x,u) + W(x)f(u) + u^{\beta-1}\, \, \, \mbox{in} \, \, \Omega, & \\ u= 0, \, \, \, \mbox{on}\, \, \partial \Omega, &\\ u(x)\geq 0\, \, \, \mbox{in}\, \, \Omega, \end{array} \right. \end{equation} where $\beta\geq 2^*$, $\Omega \subset \mathbb{R}^{N}$ is a bounded domain with regular boundary $\partial \Omega$, $N\geq 3$, $\mu > 0$ is a parameter, $\lambda_1$ is the first eigenvalue of the operator $-\Delta$, $W$ is a weight function that changes signal and under suitable conditions on functions $f$ and $g$. We apply variational and sub-supersolutions methods to obtain a non-negative and nontrivial solution for problem $(P_{\mu,s})$. Our results are related to the critical and supercritical cases of the work [13].