Anisotropic singular double Phase Dirichlet problems
Let $\Omega \subseteq \mathbb{R}^N$ be a bounded domain with a $C^2$-boundary $\partial \Omega$. The authors study a parametric Dirichlet problem of the form \[\begin{cases} -\Delta_{p(z)}u(z)- \Delta_{q(z)}u(z)= \lambda u(z)^{-\eta(z)}+ f(z,u(z)) \mbox{ in } \Omega,\\ u|_{\partial \Omega}=0, \quad u> 0 \quad (\lambda>0 \mbox{ being the parameter)}, \end{cases}\tag{\(P_\lambda\)}\] where the function $p,q :\overline{\Omega}\to (1,+\infty)$ are Lipschitz continuous, satisfying the condition $1<q_-\leq q_+<p_-\leq p_+$ (set $r_-:=\min_{\overline{\Omega}}r$ and $r_+:=\max_{\overline{\Omega}}r$), and $\Delta_r(z) u= \operatorname{div}(|\nabla u|^{r(z)-2}\nabla u)$ for all $u \in W_0^{1,r(z)}(\Omega)$ is the $r(z)$-Laplace operator. In the right-hand side, $f:\Omega \times \mathbb{R} \to \mathbb{R}$ is a Caratheodory function satisfying useful properties. In particular $f$ is $(p_+-1)$-superlinear in the $z$-variable, but without satisfying the Ambrosetti-Rabinowitz condition. Moreover, $\lambda u(z)^{-\eta(z)}$ is a suitable parametric singular term. The authors propose a variational approach to study positive solutions of $(P_\lambda)$. A bifurcation-type result, 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 7451506
scientific article