Papageorgiou, Nikolaos S.; Rǎdulescu, Vicenţiu D.; Zhang, Youpei Anisotropic singular double Phase Dirichlet problems. (English) Zbl 1482.35101 Discrete Contin. Dyn. Syst., Ser. S 14, No. 12, 4465-4502 (2021). 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. Reviewer: Calogero Vetro (Palermo) Cited in 26 Documents MSC: 35J92 Quasilinear elliptic equations with \(p\)-Laplacian 35J25 Boundary value problems for second-order elliptic equations 35B09 Positive solutions to PDEs 35J20 Variational methods for second-order elliptic equations Keywords:anisotropic \(p\)-Laplacian; Dirichlet problem; positive solutions; bifurcation-type result PDFBibTeX XMLCite \textit{N. S. 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