Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms
type: Item
instance of: scholarly article
title: Symmetry and nonexistence of positive solutions of elliptic equations and systems with Hardy terms
zbMATH Open document ID: 1235.35018
author: Tianling Jin
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2011-12-28T00:00:00Z
review text: The author considers the rotationally symmetric cone \(\Omega_a = \{\, (x', x_n) \in \mathbb R^{n - 1} \times \mathbb R : x_n > a \, |x'| \,\}\), \(a \geq 0\), \(n \geq 3\), and proves that every solution \(u(x) = u(x',x_n)\) to \[ \begin{cases} -\Delta u(x) = \frac{u^{2^*(t) - 1}(x) \,}{|x|^t} &\text{in \(\Omega_a\),}\\ u \geq 0 &\text{in \(\Omega_a\),}\\ u = 0 &\text{on \(\partial\Omega_a\),} \end{cases} \eqno{(1)} \] belonging to an appropriate function space, is also rotationally symmetric in the sense that \(u(x',x_n) = u(\tilde x', x_n)\) if \(|x'| = |\tilde x'|\). Moreover \(u(x',x_n) \leq u(\tilde x', x_n)\) if \(|x'| \geq |\tilde x'|\). Here \(2^*(t) = 2 \, (n - t)/(n - 2)\). Problem~\((1)\) is known to possess at least one solution for \(t \in (0,2)\). Rotational symmetry was proved by \textit{N. Ghoussoub} and \textit{F. Robert} [Geom.\ Funct.\ Anal.\ 16, No. 6, 1201--1245 (2006; Zbl 1232.35044)] in the special case \(a = 0\) (half-space) and under an assumption on the behavior of~\(u(x)\) as \(|x| \to +\infty\). The interest of problem~\((1)\) is motivated in connection to the Hardy-Sobolev inequality \[ C \left( \int_\Omega \frac{\, |u|^{2^*(t)} \,}{|x|^t} \, dx \right) ^{\frac2{2^*(t)}} \leq \int_\Omega |\nabla u|^2 \, dx , \] which reduces to the Sobolev inequality for \(t = 0\) and to the Hardy inequality for \(t = 2\). The same symmetry result holds if the equation in~\((1)\) is replaced by \[ -\Delta u(x) = b \, \frac{u^{2^*(t) - 1}(x)}{|x|^t} + \frac{u^{2^*(t) - 1}(x)}{|x|^s}, \] where \(0 \leq s < t \leq 2\) and \(b > 0\). The last equation arises when searching for minimizers to the Caffarelli-Kohn-Nirenberg inequality. Lane-Emden systems with Hardy weights are also considered. Namely, the author proves that if \(u,v \geq 0\) are (entire) solutions to \[ \begin{cases} -\Delta u(x) = |x|^{-s} \, v^p(x), &x \in \mathbb R^n,\\ -\Delta v(x) = |x|^{-t} \, u^q(x), &x \in \mathbb R^n, \end{cases} \eqno{(2)} \] where \(0 \leq s,t < 2\), \((s,t) \neq (0,…
reviewed by: Antonio Greco
Mathematics Subject Classification ID: 35B06
Mathematics Subject Classification ID: 35B08
Mathematics Subject Classification ID: 35B09
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35J61
Mathematics Subject Classification ID: 35A01
zbMATH DE Number: 5992249
zbMATH Keywords: Hardy-Sobolev inequality
zbMATH Keywords: Caffarelli-Kohn-Nirenberg inequality
zbMATH Keywords: Lane-Emden system
zbMATH Keywords: nonexistence
zbMATH Keywords: moving sphere method
zbMATH Keywords: Kelvin transformation
zbMATH Keywords: Pohozaev identity
MaRDI profile type: Publication
full work available at URL: j.anihpc.2011.07.003
OpenAlex ID: W2102348157
cites work: Problèmes isoperimetriques et espaces de Sobolev
cites work: Existence and non-existence of solutions to elliptic equations related to the Caffarelli-Kohn-Nirenberg inequalities
cites work: On the method of moving planes and the sliding method
cites work: Q3676658
cites work: Radial and non radial solutions for Hardy-Hénon type elliptic systems
cites work: Results on positive solutions of elliptic equations with a critical Hardy-Sobolev operator
cites work: On the Caffarelli-Kohn-Nirenberg inequalities: sharp constants, existence (and nonexistence), and symmetry of extremal functions.
cites work: Minimizers of Caffarelli-Kohn-Nirenberg inequalities with the singularity on the boundary
cites work: Q4319815
cites work: The critical hyperbola for a Hamiltonian elliptic system with weights
cites work: Positive Solutions of Semilinear Equations in Cones
cites work: Hardy-Sobolev critical elliptic equations with boundary singularities
cites work: The effect of curvature on the best constant in the Hardy-Sobolev inequalities
cites work: Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents
cites work: Symmetry and related properties via the maximum principle
cites work: Q4097979
cites work: Revisiting an idea of Brézis and Nirenberg
cites work: Liouville-type theorems and harnack-type inequalities for semilinear elliptic equations
cites work: Uniqueness theorems through the method of moving spheres
cites work: Sharp constants in the Hardy-Littlewood-Sobolev and related inequalities
cites work: Nontrivial solutions of Hardy-Hénon type elliptic systems
cites work: Nonexistence of positive solutions of semilinear elliptic systems in \(\mathbb{R}^ N\)
cites work: Ground state solutions of a critical problem involving cylindrical weights
cites work: A symmetry problem in potential theory
cites work: Non-existence of positive solutions of Lane-Emden systems
cites work: Q4393225
cites work: Q4219729
cites work: Q3251547
cites work: Variational Methods
cites work: Best constant in Sobolev inequality
DOI: 10.1016/J.ANIHPC.2011.07.003
description: scientific article; zbMATH DE number 5992249