We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(\varepsilon x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $\e>0$, $a:\R\to\R$ is an almost periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show via variational methods that if $\e$ is sufficiently small and $a$ is not constant then the equation admits infinitely many two dimensional entire solutions verifying the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$.
Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations / Alessio, FRANCESCA GEMMA; Montecchiari, Piero. - In: ADVANCED NONLINEAR STUDIES. - ISSN 1536-1365. - STAMPA. - 5:4(2005), pp. 515-549.
Multiplicity of entire solutions for a class of almost periodic Allen-Cahn type equations
ALESSIO, FRANCESCA GEMMA;MONTECCHIARI, Piero
2005-01-01
Abstract
We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(\varepsilon x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $\e>0$, $a:\R\to\R$ is an almost periodic, positive function and $W:\R\to\R$ is modeled on the classical two well Ginzburg-Landau potential $W(s)=(s^{2}-1)^{2}$. We show via variational methods that if $\e$ is sufficiently small and $a$ is not constant then the equation admits infinitely many two dimensional entire solutions verifying the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
