Nonexistence of local solutions to semilinear partial differential inequalities
type: Item
instance of: scholarly article
title: Nonexistence of local solutions to semilinear partial differential inequalities
zbMATH Open document ID: 1064.35220
published in: Annales de l'Institut Henri Poincaré. Analyse Non Linéaire
publication date: 2004-11-23T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/78626
review text: The authors investigate nonexistence of nonnegative solutions in any neighborhood of the origin, for semilinear elliptic inequalities of the following type: \[ -\Delta_x u\geq \lambda|x|^{-\mu}\cdot(x,\nabla u)+|x|^{-\alpha} u^q, \qquad u\geq 0\quad \text{in }\Omega,\tag{1} \] where \(\Omega\subset\mathbb{R}^e\), \(d\geq 3\) is a bounded smooth domain which contains the origin, \(q>1\) and \(\lambda\), \(\mu\), and \(\alpha\) are real parameters. Instead of using the maximum principle for the extended solutions, the authors take advantage of the local behavior of solutions near the origin by a direct bootstrap argument, which relies on a proper choice of the test functions.
Mathematics Subject Classification ID: 35R45
Mathematics Subject Classification ID: 35D05
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35K55
zbMATH DE Number: 2117891
zbMATH Keywords: nonexistence
zbMATH Keywords: local solutions
zbMATH Keywords: instantaneous blowup
zbMATH Keywords: semilinear elliptic inequalities
author: S. I. Pokhozhaev
author: Alberto Tesei
reviewed by: Messoud Efendiev
MaRDI profile type: Publication
OpenAlex ID: W2029475942
cites work: Q3842692
cites work: Blow up for \(u_ t- \Delta u=g(u)\) revisited
cites work: Q4378149
cites work: Weak eigenfunctions for the linearization of extremal elliptic problems
cites work: Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier
cites work: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
cites work: Some nonexistence results for quasilinear elliptic problems
cites work: A nonlinear elliptic PDE with the inverse square potential.
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Nonlinear PDE with vector fields
cites work: Quasilinear Dirichlet problems driven by positive sources
cites work: Refined asymptotics for constant scalar curvature metrics with isolated singularities
cites work: The absence of global positive solutions of quasilinear elliptic inequalities
cites work: A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities. Transl. from the Russian
cites work: Existence and nonexistence of solutions of nonlinear Dirichlet problems with first order terms
cites work: Q4467192
cites work: Blow-up of nonnegative solutions to quasilinear parabolic inequalities
cites work: Q4806981
cites work: Local behavior of solutions of quasi-linear equations
cites work: On a class of singular elliptic problems with first order terms.
cites work: On positive entire solutions to a class of equations with a singular coefficient and critical exponent
cites work: The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential
DOI: 10.1016/J.ANIHPC.2003.06.002
description: scientific article; zbMATH DE number 2117891