The question of interior blow-up points for an elliptic Neumann problem: the critical case.
type: Item
instance of: scholarly article
title: The question of interior blow-up points for an elliptic Neumann problem: the critical case.
zbMATH Open document ID: 1066.35033
DOI: 10.1016/S0021-7824(01)01251-X
author: Olivier Rey
published in: Journal de Mathématiques Pures et Appliquées. Neuvième Série
publication date: 2003-09-07T00:00:00Z
review text: The author considers, given a regular open bounded domain \(\Omega\subset {\mathbb R}^{3}\), the Neumann elliptic problem with critical nonlinearity \[ -\Delta u+\mu u= u^{5},\quad u>0\text{ in }\Omega;\quad{\partial u\over\partial\nu}=0 \text{ on }\partial\Omega.\eqno(P_{\mu}) \] In the paper a point \(y\in\bar\Omega\) is said to be a blow-up point for a family \((u_{\mu})_{\mu\geq\mu_{0}}\) of solutions of \((P_{\mu})\) when \[ \liminf_{r\to 0}\limsup_{\mu\to 0}\int_{\{x\in\Omega\,/\, \| x-y\| <r\}}| \nabla u_{\mu}| ^{2}+u_{\mu}^{6}\, dx\, >0. \] The main result proved in the paper is: Theorem: Let \((u_{\mu})_{\mu\geq\mu_{0}}\) a sequence, bounded in \(H^{1}(\Omega)\) of solutions of \((P_{\mu})\). There exists at least one blow-up point which lies on the boundary of \(\Omega\).
reviewed by: Piero Montecchiari
Mathematics Subject Classification ID: 35J60
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 1975948
zbMATH Keywords: nonlinear PDE
zbMATH Keywords: boundary value problem
zbMATH Keywords: blow-up analysis
MaRDI profile type: Publication
cites work: Q4863664
cites work: Geometry and topology of the boundary in the critical Neumann problem.
cites work: The role of the mean curvature in semilinear neumann problem involving critical exponent
cites work: Interaction between the geometry of the boundary and positive solutions of a semilinear Neumann problem with critical nonlinearity
cites work: Characterization of concentration points and \(L^ \infty\)-estimates for solutions of a semilinear Neumann problem involving the critical Sobolev exponent
cites work: Existence and nonexistence of positive radial solutions of Neumann problems with critical Sobolev exponents
cites work: Q3992582
cites work: On a nonlinear elliptic equation involving the critical sobolev exponent: The effect of the topology of the domain
cites work: On a variational problem with lack of compactness: The topological effect of the critical points at infinity
cites work: Equilibria with many nuclei for the Cahn-Hilliard equation
cites work: Convergence of solutions of H-systems or how to blow bubbles
cites work: Asymptotic behaviour of solutions of elliptic equations with critical exponents and Neumann boundary conditions
cites work: Asymptotic symmetry and local behavior of semilinear elliptic equations with critical sobolev growth
cites work: Existence and nonexistence of interior-peaked solution for a nonlinear Neumann problem.
cites work: Bifurcation and multiplicity results for nonlinear elliptic problems involving critical Sobolev exponents
cites work: Q4212663
cites work: Multipeak solutions for a singularly perturbed Neumann problem
cites work: ON A SINGULARLY PERTURBED NEUMANN PROBLEM WITH THE CRITICAL EXPONENT
cites work: On the effect of critical points of distance function in superlinear elliptic problems
cites work: Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory
cites work: Multipeak solutions for a semilinear Neumann problem
cites work: Multi-peak solutions for a semilinear Neumann problem involving the critical Sobolev exponent
cites work: Estimates for boundary-bubbling solutions to an elliptic Neumann problem
cites work: Multiple interior peak solutions for some singularly perturbed Neumann problems.
cites work: On Multiple Mixed Interior and Boundary Peak Solutions for Some Singularly Perturbed Neumann Problems
cites work: Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold
cites work: Prescribing scalar curvature on \(\mathbb{S}^ n\) and related problems. I
cites work: On a singularly perturbed equation with neumann boundary condition
cites work: YAMABE TYPE EQUATIONS ON THREE DIMENSIONAL RIEMANNIAN MANIFOLDS
cites work: Large amplitude stationary solutions to a chemotaxis system
cites work: On neumann problems for semilinear elliptic equations with critical nonlinearity:existance and symmetry of multi—peaked solutions1
cites work: Singular behavior of least-energy solutions of a semilinear Neumann problem involving critical Sobolev exponents
cites work: On the shape of least‐energy solutions to a semilinear Neumann problem
cites work: Locating the peaks of least-energy solutions to a semilinear Neumann problem
cites work: Boundary effect for an elliptic neumann problem with critical nonlinearity
cites work: The topological impact of critical points at infinity in a variational problem with lack of compactness: The dimension 3
cites work: AN ELLIPTIC NEUMANN PROBLEM WITH CRITICAL NONLINEARITY IN THREE DIMENSIONAL DOMAINS
cites work: Q3359340
cites work: Q4023938
cites work: Neumann problems of semilinear elliptic equations involving critical Sobolev exponents
cites work: Q4320573
cites work: The effect of the domain geometry on the number of positive solutions of Neumann problems with critical exponents
cites work: High-energy and multi-peaked solutions for a nonlinear Neumann problem with critical exponents
cites work: Construction of multi-peaked solutions for a nonlinear Neumann problem with critical exponent in symmetric domains
cites work: On the interior spike layer solutions to a singularly perturbed Neumann problem
cites work: Stationary solutions for the Cahn-Hilliard equation
cites work: Uniqueness and a priori estimates for some nonlinear elliptic Neumann equations in \(\mathbb R^3\)
cites work: On the number of interior multipeak solutions for singularly perturbed Neumann problems
cites work: Uniqueness results through a priori estimates. I: A three dimensional Neumann problem
full work available at URL: s0021-7824(01)01251-x
OpenAlex ID: W1996131456
description: scientific article; zbMATH DE number 1975948