Some bifurcation results and multiple solutions for the \(p\)-Laplacian equation
type: Item
instance of: scholarly article
title: Some bifurcation results and multiple solutions for the \(p\)-Laplacian equation
zbMATH Open document ID: 1494.35025
DOI: 10.12775/TMNA.2021.022
author: Jiabao Su
author: Leiga Zhao
author: Mingzheng Sun
published in: Topological Methods in Nonlinear Analysis
publication date: 2022-02-15T00:00:00Z
review text: In this paper, the authors first give some bifurcation results near the origin for the following quasilinear elliptic equation \[ \begin{cases} -\Delta_pu = \lambda|u|^{p-2}u + f(x, u) & \mbox{in}\;\; \Omega\,, \cr u = 0 & \mbox{on}\;\; \partial \Omega\,, \cr \end{cases} \eqno{(P)} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\) with smooth boundary \(\partial\Omega\), \(\Delta_p u=\) \(\mbox{div}(|\bigtriangledown u|^{p-2}\bigtriangledown u)\) with \(1 < p < \infty\), and \(\lambda \in\mathbb{R}\) is a parameter. Then they got multiple solutions to this problem. To obtain these results, the authors assume on \(f\): (\(f_1\)) \(f \in C(\Omega\times\mathbb{R},\mathbb{R}), f(x, 0) = 0\) and \[ \lim_{|t|\to 0} \frac{f(x,t)}{|t|^{p-2}t}= 0\,,\;\;\mbox{for}\;\; x\in \Omega\,; \] (\(f_2\)) there are \(c > 0\) and \(\gamma\in [1, p^\ast)\) such that \[ |f(x,t)| \le c\,(1 + |t|^{\gamma-1})\,,\;\;\mbox{for}\;\; x\in\Omega \;\;\mbox{and}\;\; t \in\mathbb{R}\,, \] where \(p^\ast = Np/(N-p)\) if \(p < N\) and \(p^\ast = \infty\) if \(N \le p\); (\(f_3\)) there exist constants \(\theta> p\) and \(M > 0\) such that \[ 0 < \theta F(x,t) := \theta \int^t_0 f(x, s)\,ds \le f(x,t)t\,\;\;\mbox{for}\;\; |t| \ge M\,,\; x\in\Omega\,; \] (\(f^-\)) there exists \(\alpha > 0\) such that \[ tf (x,t) < 0\,,\;\;\mbox{for}\;\; 0 < |t| \le \alpha\,, \;\; x\in \Omega\,. \] Let \(\lambda_1\) be the first eigenvalue of \(-\Delta_p\) in the space \(W^{1,p}_0 (\Omega)\). The authors are interested in the existence and multiplicity of nontrivial solutions of problem (P) when \(\lambda\) near \(\lambda_1\). The first result of this article is the following theorem. Theorem 1.2. Suppose that (\(f_1\)) and (\(f^-\)) hold, then there exists \(\delta > 0\) such that for \(\lambda\in (\lambda_1, \lambda_1 + \delta)\), problem (P) has at least two nontrivial solutions. Next, the authors aim to give some multiplicity results on existence of nontrivial solutions of problem (P) mainly under addition…
reviewed by: Petr Tomiczek
Mathematics Subject Classification ID: 35B32
Mathematics Subject Classification ID: 58E05
Mathematics Subject Classification ID: 35J92
Mathematics Subject Classification ID: 35P30
zbMATH DE Number: 7474118
zbMATH Keywords: Morse theory
zbMATH Keywords: multiple solutions
zbMATH Keywords: topological linking methods
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 7474118