On the Hénon equation with a Neumann boundary condition: asymptotic profile of ground states
type: Item
instance of: scholarly article
title: On the Hénon equation with a Neumann boundary condition: asymptotic profile of ground states
zbMATH Open document ID: 1436.35193
author: Jaeyoung Byeon
author: Zhi-Qiang Wang
published in: Journal of Functional Analysis
publication date: 2018-04-23T00:00:00Z
review text: This paper is concerned with the asymptotic behaviour of ground state solutions to a Neumann boundary value problem for the Hénon equation \[ \begin{cases} - \Delta u + u = |x|^\alpha u^p&\text{in}\quad\Omega,\\ \frac{\partial u}{\partial \nu} = 0&\text{on}\quad\partial\Omega,\end{cases}\tag{$*$} \] where \(\Omega\) is a bounded domain in \(\mathbb{R}^N\), \(N\ge 2\), with \(C^2\)-smooth boundary \(\partial\Omega\) and \(\nu\) is the outward unit normal on \(\partial\Omega\). Assuming that \(\Omega\) lies entirely in the unit ball \(B(0,1)\) and \(\partial^* \Omega = \partial\Omega \cap \partial B(0,1) \ne \emptyset\), the authors show that the ground state solutions to (\(*\)) have different limiting behaviour as \(\alpha\to\infty\) depending on the interplay between the geometric properties of \(\partial^* \Omega\) and the growth rate \(p\in (1,2^* - 1)\), where \(2^*\) is the critical Sobolev exponent of the \(H^1(\Omega)\)-embedding. More precisely, the localization phenomenon of ground state solutions occurs for large \(\alpha\) in the following two situations: (i) if \(p\in (2_*-1,2^* - 1)\) and \(\partial^* \Omega\) is arbitrary, or (ii) if \(p\in (1,2_*-1]\) and \(\partial^* \Omega\) has measure zero, where \(2_*\) is the critical exponent of the Sobolev trace embedding.
Mathematics Subject Classification ID: 35J91
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35B33
Mathematics Subject Classification ID: 35B40
zbMATH DE Number: 6863567
zbMATH Keywords: least energy solution
zbMATH Keywords: Hénon equation
zbMATH Keywords: Neumann boundary condition
zbMATH Keywords: limiting profile
reviewed by: Oleh Omel'chenko
MaRDI profile type: Publication
full work available at URL: j.jfa.2018.03.015
OpenAlex ID: W2800814972
cites work: Q5318420
cites work: Multiplicity Results for the Supercritical Hénon Equation
cites work: Existence of large positive solutions of some nonlinear elliptic $si:equations on singularly perturbed doamins
cites work: On the location of a peak point of a least energy solution for Hénon equation
cites work: On the Hénon equation: asymptotic profile of ground states. II
cites work: On the Hénon equation: asymptotic profile of ground states. I.
cites work: The asymptotic behaviour of the ground state solutions for Hénon equation.
cites work: Asymptotic behaviour of ground state solutions for the Henon equation
cites work: ALGORITHMS AND VISUALIZATION FOR SOLUTIONS OF NONLINEAR ELLIPTIC EQUATIONS
cites work: Q3818662
cites work: The Neumann problem for the Hénon equation, trace inequalities and Steklov eigenvalues
cites work: Q5187597
cites work: Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
cites work: The concentration-compactness principle in the calculus of variations. The limit case. II
cites work: Q3767853
cites work: On the existence of an extremal function in critical Sobolev trace embedding theorem
cites work: Q3661978
cites work: SYMMETRY BREAKING RESULTS FOR PROBLEMS WITH EXPONENTIAL GROWTH IN THE UNIT DISK
cites work: Non radial positive solution for the Hénon equation with critical growth
cites work: NON-RADIAL GROUND STATES FOR THE HÉNON EQUATION
cites work: Partial symmetry and asymptotic behavior for some elliptic variational problems
cites work: Infinitely many nonradial solutions for the Hénon equation with critical growth
DOI: 10.1016/J.JFA.2018.03.015
description: scientific article; zbMATH DE number 6863567