Nontrivial solution for the fractional \(p\)-Laplacian equations via perturbation methods
type: Item
instance of: scholarly article
title: Nontrivial solution for the fractional \(p\)-Laplacian equations via perturbation methods
zbMATH Open document ID: 1401.35136
DOI: 10.1155/2017/5317213
author: Huxiao Luo
author: Shengjun Li
author: X. H. Tang
published in: Advances in Mathematical Physics
publication date: 2018-10-23T00:00:00Z
review text: Summary: We study the existence of nontrivial solution of the following equation without compactness: \[ (- \Delta)_p^\alpha u + \left|u\right|^{p - 2} u = f(x, u),\quad x \in \mathbb{R}^N, \] where \(N, p \geq 2\), \(\alpha \in(0,1)\), \((- \Delta)_p^\alpha\) is the fractional \(p\)-Laplacian, and the subcritical \(p\)-superlinear term \(f \in C(\mathbb{R}^N \times \mathbb{R})\) is 1-periodic in \(x_i\) for \(i = 1,2, \ldots, N\). Our main difficulty is that the weak limit of (PS) sequence is not always the weak solution of fractional \(p\)-Laplacian type equation. To overcome this difficulty, by adding coercive potential term and using mountain pass theorem, we get the weak solution \(u_\lambda\) of perturbation equations. And we prove that \(u_\lambda \rightarrow u\) as \(\lambda \rightarrow 0\). Finally, by using vanishing lemma and periodic condition, we get that \(u\) is a nontrivial solution of fractional \(p\)-Laplacian equation.
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35R11
zbMATH DE Number: 6962641
zbMATH Keywords: fractional \(p\)-Laplacian
zbMATH Keywords: quasilinear equation
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1155/2017/5317213
OpenAlex ID: W2770924997
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description: scientific article; zbMATH DE number 6962641