Existence of multiple positive solutions for Brezis-Nirenberg-type problems involving singular nonlinearities
Summary: This article is devoted to the existence and multiplicity to the following Brezis-Nirenberg-type problems involving singular nonlinearities: \[ \begin{cases} -\Delta u = |u|^{p - 1}u + \lambda(|u|^{-1-\beta}/|x|^\alpha)u & \text{in } \Omega, \\ u = 0 & \text{on } \partial\Omega, \end{cases} \] where \(\Omega\) is a smooth bounded domain in \(\mathbb{R}^N\) (\(N \geq 3\)), \(0\in\Omega\), \(\lambda>0\), \(p= 2^\ast-1\) with \(2^\ast:=2N/(N - 2)\) is the critical Sobolev exponent, \(0\leq\alpha<N(p + \beta)/(p + 1)\), and \(0<\beta<1\). By using the Nehari manifold and maximum principle theorem, the existence of at least two distinct positive solutions is obtained.
scientific article; zbMATH DE number 7433068
scientific article