A singularly perturbed elliptic partial differential equation with an almost periodic term
type: Item
instance of: scholarly article
title: A singularly perturbed elliptic partial differential equation with an almost periodic term
zbMATH Open document ID: 0952.35022
author: Gregory S. Spradlin
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-01-04T00:00:00Z
full work available at URL: https://works.bepress.com/cgi/viewcontent.cgi
review text: Driven by an analogy with a result in Hamiltonian systems \textit{E. Serra, M. Tarallo} and \textit{S. Terracini} [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 13, No.~6, 783-812 (1996; Zbl 0873.58032)] the author considers partial differential equations of the form \(-\varepsilon^2\Delta u +V(x)u=f(u)\) for \(x\in\mathbb{R}^n\), \(V\) almost periodic and \(f\) satisfying certain growth conditions. He shows that for sufficiently small \(\varepsilon\) there exits a nontrivial homoclinic solution (\(|u(x)|+|\nabla u|\rightarrow 0\) as \(|x|\rightarrow \infty\)) under certain other technical restrictions, which are automatic in the two-dimensional case. The analogy with the Hamiltonian situation in one dimension does not supply a general proof for the higher dimensional case for topological reasons. The result follows from elliptic regularity theory applied to an appropriate functional for the equation.
reviewed by: Christopher Athorne
Mathematics Subject Classification ID: 35J20
Mathematics Subject Classification ID: 35B25
Mathematics Subject Classification ID: 35B05
Mathematics Subject Classification ID: 35A05
zbMATH DE Number: 1383817
zbMATH Keywords: existence
zbMATH Keywords: growth conditions
zbMATH Keywords: homoclinic solution
Wikidata QID: Q115387760
MaRDI profile type: Publication
OpenAlex ID: W2029766434
DOI: 10.1007/S005260050138
description: scientific article; zbMATH DE number 1383817