A prescribed scalar curvature-type equation: Almost critical manifolds and multiple solutions.
type: Item
instance of: scholarly article
title: A prescribed scalar curvature-type equation: Almost critical manifolds and multiple solutions.
zbMATH Open document ID: 1173.35425
author: Pierpaolo Esposito
author: Gianni Mancini
published in: Journal of Functional Analysis
publication date: 2004-02-03T00:00:00Z
review text: Let \(\Omega\) be a smooth bounded open set in \(\mathbb R^N\), \(N\geq 3\), \(a\in C^2(\overline{\Omega})\). The authors study the asymptotic behavior and existence of multiple solutions, as \(\delta\to 0\), for the perturbed boundary value problem: \(-\Delta u=(1+\delta a)u^{(N+2)/(N-2)}\) and \(u>0\) in \(\Omega\), \(u=0\) on \(\partial \Omega\). Among other things it is established the non-existence in low dimensions of one-peak solutions (i. e. with energy close to \(S^{N/2}\), where \(S\) stands for the best Sobolev constant). The approach relies on an extension of Lyapunov-Schmidt-type finite-dimensional reduction.
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 53C21
Mathematics Subject Classification ID: 35B33
zbMATH DE Number: 2037131
zbMATH Keywords: nonlinear ellptic boundary value problem
zbMATH Keywords: critical growth
zbMATH Keywords: finite-dimensional reduction
zbMATH Keywords: scalar curvature
zbMATH Keywords: asymptotic analysis
reviewed by: Dumitru Motreanu
MaRDI profile type: Publication
full work available at URL: j.jfa.2003.07.004
OpenAlex ID: W2167957515
cites work: Geometry and topology of the boundary in the critical Neumann problem.
cites work: A sharp solvability condition in higher dimensions for some Brezis-Nirenberg type equation
cites work: The role of the mean curvature in semilinear neumann problem involving critical exponent
cites work: Perturbation of \(\Delta u+u^{(N+2)/(N-2)}=0\), the scalar curvature problem in \(\mathbb{R}^N\), and related topics
cites work: Homoclinics: Poincaré-Melnikov type results via a variational approach
cites work: Symmetry breaking in Hamiltonian systems
cites work: On the symmetric scalar curvature problem on \(S^n\)
cites work: A note on the scalar curvature problem in the presence of symmetries
cites work: Q3657016
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: Homoclinics and chaotic behaviour for perturbed second order systems
cites work: Non-compactness and multiplicity results for the Yamabe problem on \(S^n\)
cites work: Positive solutions of nonlinear elliptic equations involving critical sobolev exponents
cites work: Q4735483
cites work: Q5748244
cites work: Multiple solutions of nonhomogeneous elliptic equation with critical nonlinearity
cites work: Concentration of solutions for a nonlinear elliptic problem with nearly critical exponent
cites work: A perturbation result in prescribing scalar curvature on \(S^ n\)
cites work: The scalar curvature equation on 2- and 3-spheres
cites work: Some continuation and variational methods for positive solutions of nonlinear elliptic eigenvalue problems
cites work: On the elliptic equation \(\Delta u+Ku^{(n+2)/(n-2)}=0\) and related topics
cites work: Q4827374
cites work: Perturbations of Paneitz-Branson operators on \(S^n\)
cites work: Symmetry and related properties via the maximum principle
cites work: Asymptotic approach to singular solutions for nonlinear elliptic equations involving critical Sobolev exponent
cites work: Asymptotics for some quasilinear elliptic equations
cites work: Asymptotic behavior of positive solutions of quasilinear elliptic equations with critical Sobolev growth
cites work: Prescribing scalar curvature on \(\mathbb{S}^ n\) and related problems. I
cites work: Some existence results for the scalar curvature problem via Morse theory
cites work: Q3961209
cites work: Q5516356
cites work: Proof of two conjectures of H. Brezis and L. A. Peletier
cites work: The role of the Green's function in a nonlinear elliptic equation involving the critical Sobolev exponent
cites work: Bifurcation from infinity in a nonlinear elliptic equation involving the limiting Sobolev exponent
cites work: Concentration of solutions to elliptic equations with critical nonlinearity
cites work: A global compactness result for elliptic boundary value problems involving limiting nonlinearities
cites work: Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent: the radial case.
cites work: Asymptotic behaviour of a nonlinear elliptic equation with critical Sobolev exponent: the radial case. II
DOI: 10.1016/J.JFA.2003.07.004
description: scientific article; zbMATH DE number 2037131