Multiple solutions for the fractional \(p\)-Laplacian equation with Hardy-Sobolev exponents
type: Item
instance of: scholarly article
title: Multiple solutions for the fractional \(p\)-Laplacian equation with Hardy-Sobolev exponents
zbMATH Open document ID: 1473.35642
author: Chunyan Zhang
author: Jihui Zhang
published in: Rocky Mountain Journal of Mathematics
publication date: 2021-09-10T00:00:00Z
review text: The authors study the fractional \(p\)-Laplacian problem with Hardy-Sobolev exponents \[ \left\{ \begin{array}{ll} -\Delta_p^s u = \lambda |h|^{r-2} u + \frac{|u|^{p_{s, \alpha^{-2}}^*} u}{|x|^\alpha} & \text{in }\Omega, \\ u=0 & \text{on }\partial\Omega , \end{array} \right. \] where \(\Omega \subseteq \mathbb{R}^N\) be a bounded domain containing the origin, \(p\in (1,\infty)\), \(s\in (0,1)\), \(1<r<p\), \(0\leq \alpha < sp < N\), and \(\lambda > 0\) is a parameter, \(p^*_{s, \alpha} = ((N - \alpha)p)/(N- ps)\) is the fractional Hardy-Sobolev exponent. The authors prove that there is a \(\lambda_0 > 0\) such that for any \(\lambda \in (0, \lambda_0)\), the above problem possesses infinitely many solutions. They use variational methods, more specifically, the Nehari manifold and Ljusternik-Schnirelmann theory.
reviewed by: Youssef Jabri
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 58E30
Mathematics Subject Classification ID: 65Nxx
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35J20
zbMATH DE Number: 7393770
zbMATH Keywords: infinitely many solutions
zbMATH Keywords: Nehari manifold
zbMATH Keywords: Ljusternik-Schnirelmann theory
MaRDI profile type: Publication
full work available at URL: rmj.2021.51.363
OpenAlex ID: W3172710925
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DOI: 10.1216/RMJ.2021.51.363
description: scientific article; zbMATH DE number 7393770