On critical double phase Kirchhoff problems with singular nonlinearity
type: Item
instance of: scholarly article
title: On critical double phase Kirchhoff problems with singular nonlinearity
author: Alessio Fiscella
author: Patrick Winkert
author: Rakesh Arora
author: Tuhina Mukherjee
publication date: 2022-10-24T00:00:00Z
review text: Let $\Omega\subset\mathbb{R}^N$ be a bounded domain with Lipschitz boundary $\partial\Omega$. This work combines the effects of a nonlocal Kirchhoff coefficient $m:[0,+\infty)\to [0,+\infty)$ given as \[ m(t)=a_0+b_0t^{\theta-1}\text{ for all }t\geq0,\text{ with }a_0\geq 0, b_0>0\text{ and }\theta \in [1,p^\ast/q) \] ($p^\ast=Np/(N-p)$ is the critical Sobolev exponent to $p$, $1<p<N$ and $p<q<p^\ast$), and a double phase operator \[ \operatorname{div}(|\nabla u|^{p-2}\nabla u+a(x)|\nabla u|^{q-2}\nabla u)\text{ for all }u\in W_0^{1,\mathcal{H}}(\Omega) \] ($W_0^{1,\mathcal{H}}(\Omega)$ is the homogeneous Musielak-Orlicz Sobolev space) with a parametric singular term $\lambda u^{-\gamma}$ ($\lambda>0$ and $0<\gamma<1$) and a critical term $u^{p^\ast-1}$. The proof of main result is based on a minimization argument on the Nehari manifold. Under appropriate hypotheses on the data, the authors obtain the existence of at least a weak solution $u_\lambda \in W_0^{1,\mathcal{H}}(\Omega)$ to the Dirichlet problem for appropriate values of $\lambda$.
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J15
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35J62
Mathematics Subject Classification ID: 35J75
zbMATH DE Number: 7606144
zbMATH Keywords: critical growth
zbMATH Keywords: double phase operator
zbMATH Keywords: fibering method
zbMATH Keywords: Nehari manifold
zbMATH Keywords: nonlocal Kirchhoff term
zbMATH Keywords: singular problem
MaRDI profile type: Publication
full work available at URL: s12215-022-00762-7
OpenAlex ID: W4281696000
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DOI: 10.1007/S12215-022-00762-7
published in: Rendiconti del Circolo Matematico di Palermo
description: scientific article; zbMATH DE number 7606144