Singular and superlinear perturbations of the eigenvalue problem for the Dirichlet \(p\)-Laplacian
The authors study the following nonlinear parametric Dirichlet problem with singular and superlinear terms \[ \begin{cases} - \Delta_p u = \lambda u^{p-1}+\theta u^{-\eta}+f(z,u) \quad \mbox{in }\Omega, \\ \quad u\Big|_{\partial \Omega}=0, \, u \geq0,\, 1<p<\infty, \, \lambda\in \mathbb{R}, \, \theta >0, \, 0<\eta<1,\end{cases}\tag{\(P_{\lambda,\theta}\)}\] where $\Omega \subseteq \mathbb{R}^N$ is a bounded domain with a $C^2$-boundary $\partial \Omega$ and $ \Delta_p u:= \mbox{div}\, (|\nabla u|^{p-2}\nabla u)$ for all $W_0^{1,p}(\Omega)$ is the $p$-Laplacian differential operator. The authors establish a nonexistence and multiplicity theorem for problem $(P_{\lambda,\theta})$ in terms of the principal eigenvalue $\widehat{\lambda}_1>0$ of the Dirichlet $p$-Laplacian $(-\Delta_p,W_0^{1,p}(\Omega))$. A main feature of the problem is that the singular term $u \to \theta u^{-\eta}$ in the reaction changes the geometry of the problem and leads to an energy functional which is not $C^1$. By varying the parameter $\theta>0$, the authors are able to control the singular term and eventually satisfy the underlying geometry of the minimax theorems.
scientific article; zbMATH DE number 7362294
scientific article