Entire solutions in for a class of Allen-Cahn equations

Issue		ESAIM: COCV Volume 11, Number 4, October 2005


Page(s)		633 - 672
DOI		10.1051/cocv:2005023

ESAIM: COCV, October 2005, Vol. 11, pp. 633-672
DOI: 10.1051/cocv:2005023

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

Francesca Alessio and Piero Montecchiari

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

(Received September 10, 2004.)

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)=(s²-1)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 $\varepsilon$ 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.

Mathematics Subject Classification. 34C37, 35B05, 35B40, 35J20, 35J60.

Key words: Heteroclinic solutions, elliptic equations, variational methods.

© EDP Sciences, SMAI 2005

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