Existence and instability of spike layer solutions to singular perturbation problems
type: Item
instance of: scholarly article
title: Existence and instability of spike layer solutions to singular perturbation problems
zbMATH Open document ID: 1010.47036
DOI: 10.1016/S0022-1236(02)00013-7
author: Peter W. Bates
author: Junping Shi
published in: Journal of Functional Analysis
publication date: 2003-03-26T00:00:00Z
review text: The authors propose an abstract framework to establish existence and to compute the Morse index of spike layer solutions of singularly perturbed semilinear elliptic equations. A nonlinear Lyapunov-Schmidt scheme is used to reduce the problem to one on a normally hyperbolic manifold and the related linearized problem is also analyzed by this reduction. As an application, it is shown the existence of a multi-peak spike layer solution with peaks on the boundary of the domain. Precise estimates of the small eigenvalues of an operator obtained by linearizing a spike layer solution are also established.
reviewed by: Dian K. Palagachev
Mathematics Subject Classification ID: 47H14
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 35P15
Mathematics Subject Classification ID: 46N20
Mathematics Subject Classification ID: 35B25
zbMATH DE Number: 1888372
zbMATH Keywords: spike layer solutions
zbMATH Keywords: singular perturbation problems
zbMATH Keywords: semilinear elliptic equations
zbMATH Keywords: instability
MaRDI profile type: Publication
cites work: Critical points of a singular perturbation problem via reduced energy and local linking
cites work: Multi-spike stationary solutions of the Cahn-Hilliard equation in higher-dimension and instability.
cites work: The Dynamics of Nucleation for the Cahn–Hilliard Equation
cites work: Equilibria with many nuclei for the Cahn-Hilliard equation
cites work: Q4519619
cites work: Critical point theorems for indefinite functionals
cites work: Q3672366
cites work: Q3993198
cites work: Homoclinic type solutions for a semilinear elliptic PDE on ℝn
cites work: Degenerate critical points, homotopy indices and Morse inequalities.
cites work: A note on asymptotic uniqueness for some nonlinearities which change sign
cites work: Multipeak solutions for a singularly perturbed Neumann problem
cites work: A singularly perturbed elliptic problem in bounded domains with nontrivial topology
cites work: Q5690466
cites work: Local mountain passes for semilinear elliptic problems in unbounded domains
cites work: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
cites work: Q3923023
cites work: Q4040640
cites work: Multipeak solutions for a semilinear 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 boundary peak solutions for some singularly perturbed Neumann problems
cites work: Matrix Analysis
cites work: Perturbation theory for linear operators.
cites work: Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold
cites work: Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus
cites work: Q3847203
cites work: On a singularly perturbed equation with neumann boundary condition
cites work: Large amplitude stationary solutions to a chemotaxis system
cites work: The concentration-compactness principle in the calculus of variations. The locally compact case. I
cites work: Q4221182
cites work: On the Neumann Problem for Some Semilinear Elliptic Equations and Systems of Activator-Inhibitor Type
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: On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems
cites work: Existence of Semiclassical Bound States of Nonlinear Schrödinger Equations with Potentials of the Class (V)a
cites work: On positive multi-lump bound states of nonlinear Schrödinger equations under multiple well potential
cites work: Exact multiplicity of positive solutions for a class of semilinear problem. II
cites work: Q3749777
cites work: On a class of nonlinear Schrödinger equations
cites work: On the existence of multiple, single-peaked solutions for a semilinear Neumann problem
cites work: On concentration of positive bound states of nonlinear Schrödinger equations
cites work: On the boundary spike layer solutions to a singularly perturbed Neumann problem
cites work: Uniqueness and critical spectrum of boundary spike solutions.
cites work: Stationary solutions for the Cahn-Hilliard equation
cites work: On the Modular Representation Theory of the Two-Dimensional Special Linear Group Over an Algebraically Closed Field
full work available at URL: s0022-1236(02)00013-7
OpenAlex ID: W2008177756
description: scientific article; zbMATH DE number 1888372