Nonlinear singular problems with convection
The authors study the following nonlinear Dirichlet problem with singular and gradient dependent terms \begin{align*} -\Delta_pu=u^{-\eta}+f(z,u,\nabla u)\text{ in }\Omega,\\ u|_{\partial\Omega}=0, \, u>0, \, 1<p<N,\, 0<\eta, \end{align*} where \(\Omega\subseteq\mathbb{R}^N\) is a bounded domain with a \(C^2\)-boundary \(\partial\Omega\) and \(\Delta_pu:=\operatorname{div}(|\nabla u|^{p-2}\nabla u)\) for all \(u\in W^{1,p}_0(\Omega)\) is the \(p\)-Laplacian differential operator. A main feature of the problem is that the singular term \(u\to u^{-\eta}\) leads to an energy functional, associated to the problem, which is not \(C^1\). By using the so-called frozen variable method and working on appropriate auxiliary problems, the authors establish the existence of a smallest element of the set of positive solutions to the main problem. Finally, as the given minimal solution map possesses certain regularities, the authors apply the Leray-Schauder alternative principle and conclude that the main problem admits a positive solution.
scientific article; zbMATH DE number 7370842
scientific article