A singular eigenvalue problem for the Dirichlet \((p, q)\)-Laplacian
Let $\Omega \subseteq \mathbb{R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$, and $\lambda >0$ be a parameter. The authors study a parametric singular Dirichlet problem of the form \[\begin{cases} -\Delta_{p}u(z)- \Delta_{q}u(z)= \lambda [u(z)^{-\eta}+ f(z,u(z))] \mbox{ in } \Omega,\\ u|_{\partial \Omega}=0, \, u> 0 \, \mbox{ in }\Omega,\quad0<\eta<1,\quad 1<q<p,\end{cases}\tag{\(P_\lambda\)}\] where $\Delta_r u= \operatorname{div}(|D u|^{r-2}D u)$ for all $u \in W^{1,r}(\Omega)$ is the $r$-Laplace operator. In the right-hand side, $f:\Omega \times \mathbb{R} \to \mathbb{R}$ is a Caratheodory function satisfying suitable properties. Precisely, $f$ is $(p-1)$-superlinear near $+\infty$ in the $z$-variable, and does not satisfy the Ambrosetti-Rabinowitz condition. The authors assume that $ u \to \lambda u^{-\eta}$ is a parametric singular term. The authors present a variational approach to study positive solutions to problem $(P_\lambda)$. A bifurcation-type theorem, describing the dependence of the set of positive solutions as the parameter $\lambda>0$ varies, is the main result of the paper.
scientific article; zbMATH DE number 7463793
scientific article