Multiplicity of nontrivial solutions of a class of fractional \(p\)-Laplacian problem
type: Item
instance of: scholarly article
title: Multiplicity of nontrivial solutions of a class of fractional \(p\)-Laplacian problem
zbMATH Open document ID: 1322.35157
DOI: 10.4171/ZAA/1541
author: Ghanmi Abdeljabbar
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2015-09-22T00:00:00Z
review text: Summary: In this paper, we deal with existence of nontrivial solutions to the fractional \(p\)-Laplacian problem of the type \[ \begin{cases} (-\Delta)^\alpha_pu=\frac{1}{r}\frac{\partial F(x,u)}{\partial u}+\lambda a(x)|u|^{q-2}u\quad & \text{ in }\Omega, \\ u=0\quad & \text{ in }\mathbb R^n\setminus\Omega,\end{cases} \] where \(\Omega\) is a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial \Omega\), \(a\in C(\Omega)\), \(p\geq 2\), \(\alpha\in (0,1)\) such that \(p\alpha<n\), \(1<q<p<r<\frac{np}{n-\alpha p}\), and \(F\in C^1(\overline{\Omega}\times\mathbb R,\mathbb R)\). Using the decomposition of the Nehari manifold, we prove that the non-local elliptic problem has at least two nontrivial solutions.
Mathematics Subject Classification ID: 35R11
Mathematics Subject Classification ID: 35A15
Mathematics Subject Classification ID: 35J92
zbMATH DE Number: 6484444
zbMATH Keywords: nontrivial solutions
zbMATH Keywords: sign-changing weight function
zbMATH Keywords: Nehari manifold
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 6484444