Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance
type: Item
instance of: scholarly article
title: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems at resonance
zbMATH Open document ID: 1360.35068
author: Chun-Lei Tang
author: Jiu Liu
author: Peng Zhang
author: Jia-Feng Liao
published in: Discrete and Continuous Dynamical Systems. Series S
publication date: 2016-12-22T00:00:00Z
review text: The authors study a class of Kirchhoff type problems with resonance \[ \begin{cases} -\left(a+b\int_\Omega |\nabla u|^2\, dx \right)\Delta u=\nu u^3+\lambda |u|^{q-1}u & \text{in } \Omega,\\ u=0 & \text{on } \partial\Omega \end{cases} \] where \(\Omega\subset \mathbb R^3\) is a bounded domain, \(a,b,\nu,\lambda>0\) and \(0<q<1.\) Using the minimizing method, it is obtained existence of positive ground state solutions for all \(0<\nu\leq b \nu_1\) and \(\lambda>0.\) Using the Nehari method, the authors obtain two positive solutions for all \(\nu> b\nu_1\) and \(0<\lambda<\bar\lambda,\) where \(\nu_1\) is the first eigenvalue of the considered problem, \(\bar\lambda\) is a positive constant and one of the two positive solutions is a ground state solution.
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 6667928
zbMATH Keywords: Kirchhoff type equation
zbMATH Keywords: resonance
zbMATH Keywords: positive solution
zbMATH Keywords: Nehari method
reviewed by: Lubomira Softova
MaRDI profile type: Publication
full work available at URL: dcdss.2016080
OpenAlex ID: W2556815545
cites work: Combined effects of concave and convex nonlinearities in some elliptic problems
cites work: A uniqueness result for a nonlocal equation of Kirchhoff type and some related open problem
cites work: The Nehari manifold for a Kirchhoff type problem involving sign-changing weight functions
cites work: The critical problem of Kirchhoff type elliptic equations in dimension four
cites work: Infinitely many positive solutions for Kirchhoff-type problems
cites work: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb R^3\)
cites work: Multiple positive solutions for Kirchhoff type of problems with singularity and critical exponents
cites work: Existence of a positive solution to Kirchhoff-type problems without compactness conditions
cites work: Positive solutions to Kirchhoff type equations with nonlinearity having prescribed asymptotic behavior
cites work: A uniqueness result for Kirchhoff type problems with singularity
cites work: Existence and multiplicity of positive solutions for a class of Kirchhoff type problems with singularity
cites work: Q4189540
cites work: Multiple positive solutions for Kirchhoff type problems with singularity
cites work: Sign-changing solutions of a class of nonlocal quasilinear elliptic boundary value problems
cites work: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. condition
cites work: Nontrivial solutions of Kirchhoff-type problems via the Yang index
cites work: Existence and multiplicity of solutions for Kirchhoff type equations
cites work: Resonance problems for Kirchhoff type equations
cites work: Ground state solutions for an indefinite Kirchhoff type problem with steep potential well
cites work: Minimax theorems
cites work: Existence and multiplicity of solutions for Kirchhoff type problem with critical exponent
cites work: Ground states for the nonlinear Kirchhoff type problems
DOI: 10.3934/DCDSS.2016080
description: scientific article; zbMATH DE number 6667928