Existence of solution for an asymptotically linear Schrödinger-Kirchhoff equation
type: Item
instance of: scholarly article
title: Existence of solution for an asymptotically linear Schrödinger-Kirchhoff equation
zbMATH Open document ID: 1423.35094
author: Alex M. Batista
author: Marcelo F. Furtado
published in: Potential Analysis
publication date: 2019-04-30T00:00:00Z
review text: The authors deal with the Kirchhoff equation \[ -\left(1+\lambda \int_{\mathbb{R}^N}|\nabla u|^2\,dx\right)\Delta u + V(x)u=f(u),\ x\in \mathbb{R}^N,\ N\in \{3,4\},\ \lambda\ge 0, \] where the potential \(V\) is radial and \(f\) can be superlinear or asymptotically linear at infinity. The main result is the theorem on the existence and the asymptotic behavior of a solution. \par In the case \(N=4\) there exists \(\lambda^* > 0\) such that for any \(\lambda \in (0,\lambda^*)\) the equation has a ground state solution \(u_\lambda \in H^1_{\mathrm{rad}}(\mathbb{R}^4)\). Moreover \(u_\lambda \to u_0\) in \(H^1_{\mathrm{rad}}(\mathbb{R}^4)\) as \(\lambda \to 0^+\), where \(u_0\) is a weak solution of \(\Delta u_0+V(x)u_0=f(u_0)\), \( x\in \mathbb{R}^4\). \par The same result holds for \(N=3\) with no restriction on \(\lambda\).
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J60
zbMATH DE Number: 7049469
zbMATH Keywords: Kirchhoff equation
zbMATH Keywords: nonlocal problems
zbMATH Keywords: radial problems
zbMATH Keywords: ground state solution
MaRDI profile type: Publication
full work available at URL: s11118-018-9697-3
OpenAlex ID: W2801961667
cites work: Positive solutions for a quasilinear elliptic equation of Kirchhoff type
cites work: The elliptic Kirchhoff equation in \(\mathbb {R}^{N}\) perturbed by a local nonlinearity.
cites work: Multiple critical points for a class of nonlinear functionals
cites work: A note on the elliptic Kirchhoff equation in ℝNperturbed by a local nonlinearity
cites work: Nonlinear scalar field equations. I: Existence of a ground state
cites work: Multiple solutions for the asymptotically linear Kirchhoff type equations on \(\mathbb{R}^N\)
cites work: A priori estimates and existence of positive solutions of semilinear elliptic equations
cites work: Ground states for Kirchhoff equations without compact condition
cites work: Existence and concentration behavior of positive solutions for a Kirchhoff equation in \(\mathbb R^3\)
cites work: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^3\)
cites work: Q4189540
cites work: Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations
cites work: Positive solutions of asymptotically linear equations via Pohozaev manifold
cites work: Bound states of a nonhomogeneous nonlinear Schrödinger equation with non symmetric potential
cites work: Existence of a positive solution to Kirchhoff-type problems without compactness conditions
cites work: Existence and multiplicity of solutions for asymptotically linear Schrödinger-Kirchhoff equations
cites work: On a class of nonlinear Schrödinger equations
cites work: Positive high energy solution for Kirchhoff equation in \(\mathbb{R}^{3}\) with superlinear nonlinearities via Nehari-Pohožaev manifold
DOI: 10.1007/S11118-018-9697-3
Recommended article: Existence and multiplicity of solutions for asymptotically linear Schrödinger-Kirchhoff equations
Recommended article: Q5127882
Recommended article: Existence of solution for Schrödinger equation with discontinuous nonlinearity and asymptotically linear
Recommended article: Existence and multiplicity of solutions for generalized asymptotically linear Schrödinger-Kirchhoff equations
Recommended article: Q5741527
Recommended article: Existence of solutions for a class of quasilinear Schrödinger equation with a Kirchhoff-type
Recommended article: Existence of nontrivial solutions for asymptotically linear periodic Schrödinger equations
Recommended article: Positive solutions for asymptotically linear Schrödinger-Kirchhoff-type equations
Recommended article: Existence of ground state solutions for asymptotically linear Schrödinger-Poisson systems
description: scientific article; zbMATH DE number 7049469