Anisotropic singular double Phase Dirichlet problems
type: Item
instance of: scholarly article
title: Anisotropic singular double Phase Dirichlet problems
zbMATH Open document ID: 1482.35101
author: Nikolaos S. Papageorgiou
author: Vicenţiu D. Rădulescu
author: Youpei Zhang
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2022-01-03T00:00:00Z
review text: 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.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 7451506
zbMATH Keywords: anisotropic \(p\)-Laplacian
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: positive solutions
zbMATH Keywords: bifurcation-type result
MaRDI profile type: Publication
full work available at URL: dcdss.2021111
OpenAlex ID: W3206898650
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Anisotropic singular double Phase Dirichlet problems (cites work)
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Anisotropic singular double Phase Dirichlet problems (DOI)
DOI: 10.3934/DCDSS.2021111