The existence of three positive solutions of a singular \(p\)-Laplacian problem
In this paper, the authors deal with the existence and multiplicity of solutions to the following boundary value problem: \[ -\Delta_pu = \lambda a(x)u^{-\gamma}+\lambda f(x,u),\,\,\,x\in\Omega, \;u>0\,\,\,\text{in}\,\,\,\Omega, \;u=0\,\,\, \text{on}\,\,\,\partial\Omega, \tag{1} \] where \(\Omega\) is a bounded smooth open domain of \(\mathbb R^N\), \(\Delta_pu=\text{div}(|\nabla u|^{p-2}\nabla u)\) is the \(p\)-Laplacian, \(N<p<+\infty\), \(\gamma>0\) is a constant, \(\lambda>0\) is a parameter, \(f\) is a Carathéodory function on \(\Omega\times[0,+\infty)\), and \(a(x)\geqslant 0\) is a nontrivial measurable function satisfying the condition: there are \(\varphi_0\geqslant 0\) in \(C_0^1(\overline\Omega)\) and \(q>N\) such that \(a\varphi_0^{-\gamma}\in L^q(\Omega)\). Under some additional assumptions on the function \(f\), the authors establish that there exist an open interval \(\Sigma\subset (0,+\infty)\) and a constant \(\mu>0\) such that for every \(\lambda\in\Sigma\) problem (1) has at least three distinct positive solutions in \(W_0^{1,p}(\Omega)\), with their \(W_0^{1,p}(\Omega)\) norms less than \(\mu\).
scientific article; zbMATH DE number 5930967