Positive solutions to a Dirichlet problem with non-Lipschitz nonlinearities

Abstract

Let \(Omega\) be a bounded smooth domain in \(R^N\). We study the existence of positive solutions to the Dirichlet problem $$\displaylines{ -\Delta u=(1-u)u^{s-1}-\lambda u^{r-1}, \quad\text{in } \Omega,\cr u=0, \quad \text{on } \partial\Omega,}$$ where \(1<;r<\leq 2\),  and \(\lambda>0\). In particular, we answer some questions posed in the recent paper [3] where this problem was considered. For more information see https://ejde.math.txstate.edu/Volumes/2021/30/abstr.html