Normalized solutions of nonautonomous Kirchhoff equations: sub- and super-critical cases
type: Item
instance of: scholarly article
title: Normalized solutions of nonautonomous Kirchhoff equations: sub- and super-critical cases
zbMATH Open document ID: 1473.35149
author: Sitong Chen
author: Vicenţiu D. Rădulescu
author: X. H. Tang
published in: Applied Mathematics and Optimization
publication date: 2021-08-11T00:00:00Z
review text: The authors study a nonautonomous Kirchhoff-type equation involving a non-constant potential \(K \in C(\mathbb{R}^3,\mathbb{R}^+)\) with some regularities. Precisely, they consider the equation \[(P_\lambda) \quad -\left(a+b\int_{\mathbb{R}^3}|\nabla u|^2 dx\right)\Delta u-\lambda u=K(x)f(u), \, x \in \mathbb{R}^3, \, u \in H^1(\mathbb{R}^3).\] Here \(a,b>0\), \(\lambda \in \mathbb{R}\), \(0< \lim_{|y|\to +\infty}K(y)\leq \inf_{\mathbb{R}^3}K\) and \(f \in C(\mathbb{R},\mathbb{R})\) satisfies certain \(L^2\)-supercritical or \(L^2\)-subcritical assumptions. The authors look at the critical points of a suitable energy functional associated to \((P_\lambda)\), on the manifold \(S_c=\{u \in H^1(\mathbb{R}^3) : \|u\|^2_2=c \}\), that is, the set of solutions \(u \in H^1(\mathbb{R}^3)\) having a prescribed \(L^2\)-norm, as \(c>0\) is given. The parameter \(\lambda \in \mathbb{R}\) is unknown and appears as a Lagrange multiplier. Thus it is shown that each critical point \(u_c \in S_c\) of the energy functional corresponds to a Lagrange multiplier \(\lambda_c \in \mathbb{R}\) so that \((u_c,\lambda_c)\) solves \((P_\lambda)\).
reviewed by: Calogero Vetro
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35J65
zbMATH DE Number: 7381020
zbMATH Keywords: Kirchhoff problem
zbMATH Keywords: normalized solution
zbMATH Keywords: concentration-compactness
zbMATH Keywords: indefinite potential
zbMATH Keywords: \(L^2\)-supercritical growth
zbMATH Keywords: \(L^2\)-subcritical growth
MaRDI profile type: Publication
full work available at URL: s00245-020-09661-8
OpenAlex ID: W3003870137
cites work: On the Well-Posedness of the Kirchhoff String
cites work: A note on the elliptic Kirchhoff equation in ℝNperturbed by a local nonlinearity
cites work: Scaling properties of functionals and existence of constrained minimizers
cites work: Existence and instability of standing waves with prescribed norm for a class of Schrödinger-Poisson equations
cites work: Q5780514
cites work: Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation
cites work: Multiple solutions for the nonhomogeneous Kirchhoff equation on \(\mathbb R^N\)
cites work: Berestycki-Lions conditions on ground state solutions for a nonlinear Schrödinger equation with variable potentials
cites work: On the planar Schrödinger-Poisson system with the axially symmetric potential
cites work: Semiclassical ground state solutions for critical Schrödinger-Poisson systems with lower perturbations
cites work: Stationary Waves with Prescribed $L^2$-Norm for the Planar Schrödinger--Poisson System
cites work: Global solvability for the degenerate Kirchhoff equation with real analytic data
cites work: Existence and asymptotic behavior of nodal solutions for the Kirchhoff-type problems in \(\mathbb{R}^3\)
cites work: Existence of solutions with prescribed norm for semilinear elliptic equations
cites work: Sharp nonexistence results of prescribed \(L^2\)-norm solutions for some class of Schrödinger-Poisson and quasi-linear equations
cites work: Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in \(\mathbb{R}^3\)
cites work: Existence of positive solutions for nonlinear Kirchhoff type problems in R3 with critical Sobolev exponent
cites work: Q4189540
cites work: Nonlocal Kirchhoff diffusion problems: local existence and blow-up of solutions
cites work: Fractional Kirchhoff problems with critical Trudinger-Moser nonlinearity
cites work: Variational Methods for Nonlocal Fractional Problems
cites work: Nonlinear Analysis - Theory and Methods
cites work: Ground state solutions of Nehari-Pohozaev type for Kirchhoff-type problems with general potentials
cites work: Singularly perturbed Choquard equations with nonlinearity satisfying Berestycki-Lions assumptions
cites work: Regularity for a more general class of quasilinear equations
cites work: Minimax theorems
cites work: Existence and multiplicity of normalized solutions for the nonlinear Kirchhoff type problems
cites work: The sharp existence of constrained minimizers for a class of nonlinear Kirchhoff equations
cites work: The existence of normalized solutions for \(L^2\)-critical constrained problems related to Kirchhoff equations
cites work: The mass concentration phenomenon for \(L^2\)-critical constrained problems related to Kirchhoff equations
cites work: The existence of least energy nodal solutions for some class of Kirchhoff equations and Choquard equations in \(\mathbb R^N\)
cites work: Existence and uniqueness of normalized solutions for the Kirchhoff equation
DOI: 10.1007/S00245-020-09661-8
description: scientific article; zbMATH DE number 7381020