Entire solutions in  for a class of Allen-Cahn equations

ESAIM: Control, Optimisation and Calculus of Variations

ESAIM: Control, Optimisation and Calculus of Variations / Volume 11 / Issue 04 / October 2005, pp 633-672
© EDP Sciences, SMAI, 2005
Published online by Cambridge University Press: 22 February 2011
DOI: http://dx.doi.org/10.1051/cocv:2005023 (About DOI)

Table of Contents - October 2005 - Volume 11, Issue 04

Research Article

Entire solutions in ${\mathbb{R}}^{2}$ for a class of Allen-Cahn equations

Article author query
alessio f [PubMed] [Google Scholar]
montecchiari p [PubMed] [Google Scholar]

Alessio, Francesca^a1 and Montecchiari, Piero^a1

^a1 Dipartimento di Scienze Matematiche, Università Politecnica delle Marche, via Brecce Bianche, 60131 Ancona, Italy; alessio@dipmat.univpm.it;montecch@mta01.univpm.it

Abstract

We consider a class of semilinear elliptic equations of the form 15.7cm - $\varepsilon^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in{\mathbb{R}}^{2}$ where $\varepsilon>0$ , $a:{\mathbb{R}}\to{\mathbb{R}}$ is a periodic, positive function and $W:{\mathbb{R}}\to{\mathbb{R}}$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$ . We look for solutions to ([see full textsee full text]) which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in{\mathbb{R}}$ . We show via variational methods that if ε is sufficiently small and a is not constant, then ([see full textsee full text]) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.