Peak solutions without non-degeneracy conditions
type: Item
instance of: scholarly article
title: Peak solutions without non-degeneracy conditions
zbMATH Open document ID: 1197.35121
published in: Journal of Differential Equations
publication date: 2009-05-05T00:00:00Z
review text: In this paper, a very important progress is made by the author in replacing the non-degeneracy hypothesis one imposes when using Lyapunov-Schmidt procedure to prove existence of peak solutions. Typically peak solutions are shown to exit for problems of type: \[ \varepsilon^2 \Delta u = g (x, u (x)) \quad\text{in } \Omega \] with either Dirichlet or Neumann boundrey conditions. He proves that solutions with single peak (multiple peaks) exist by considering a limit problem as the parameter goes to zero for certain rescaled equation and the non-degeneracy is asked for solutions of a problem posed on whole of \(\mathbb R^n\). In this paper, the non-degeneracy is replaced by appropriate degree and Conley index hypothesis. Since these are local the approach holds promise of extending to more general equations than discussed in the paper, where only the autonomous case is discussed.
reviewed by: P. N. Srikanth
Mathematics Subject Classification ID: 35J70
Mathematics Subject Classification ID: 35J25
zbMATH DE Number: 5551130
zbMATH Keywords: removal of non-degeneracy
zbMATH Keywords: Conley index
zbMATH Keywords: small diffusion
zbMATH Keywords: peak solutions on bounded domains
zbMATH Keywords: removal of invertibility assumptions
author: E. N. Dancer
MaRDI profile type: Publication
full work available at URL: j.jde.2009.01.030
OpenAlex ID: W1992774039
cites work: Multibump solutions and critical groups
cites work: Standing waves for nonlinear Schrödinger equations with a general nonlinearity
cites work: Solutions with multiple peaks for nonlinear elliptic equations
cites work: Infinite dimensional Morse theory and multiple solution problems
cites work: Degenerate critical points, homotopy indices and Morse inequalities.
cites work: Degenerate critical points, homotopy indices and Morse inequalities. II.
cites work: Degenerate critical points, homotopy indices and morse inequalities III
cites work: Real analyticity and non-degeneracy
cites work: A note on asymptotic uniqueness for some nonlinearities which change sign
cites work: Boundary-Value Problems for Ordinary Differential Equations on Infinite Intervals
cites work: On the uniqueness of the positive solution of a singularly perturbed problem
cites work: Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem
cites work: Q3402534
cites work: Peak solutions for the Dirichlet problem of an elliptic system
cites work: Sign-changing solutions for supercritical elliptic problems in domains with small holes
cites work: Q5643279
cites work: Nonspreading wave packets for the cubic Schrödinger equation with a bounded potential
cites work: Global and local behavior of positive solutions of nonlinear elliptic equations
cites work: Multiple interior peak solutions for some singularly perturbed Neumann problems.
cites work: Q4067778
cites work: Singularly perturbed elliptic problems with superlinear or asymptotically linear nonlinearities
cites work: Uniqueness of the positive solution of \(\Delta u+f(u)=0\) in an annulus
cites work: Radial symmetry of positive solutions of nonlinear elliptic equations in Rn
cites work: On the location and profile of spike‐layer solutions to singularly perturbed semilinear dirichlet problems
cites work: More indices and bifurcations of positive solutions of Delta u + f(u) = 0 on R^N
cites work: The homotopy index and partial differential equations
cites work: An Infinite Dimensional Version of Sard's Theorem
cites work: Q4746129
cites work: Q4192669
DOI: 10.1016/J.JDE.2009.01.030
description: scientific article; zbMATH DE number 5551130