We consider a class of semilinear elliptic equations of the form $$-\e^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $\e>0$, $a:\R\to\R$ is a 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 look for solutions which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$. We show via variational methods that if $\e$ is sufficiently small then the equation admits infinitely many of such solutions, distinct up to periodic translations, which are not solutions to the associated ODE problem $$-\e^{2}\ddot q(x)+a(x)W'(q(x))=0,\qquad \lim_{x\to\pm\infty}q(x)=\pm 1.$$

### Entire solutions in ℝ2 for a class of Allen-Cahn equations

#### Abstract

We consider a class of semilinear elliptic equations of the form $$-\e^{2}\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $\e>0$, $a:\R\to\R$ is a 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 look for solutions which verify the asymptotic conditions $u(x,y)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $y\in\R$. We show via variational methods that if $\e$ is sufficiently small then the equation admits infinitely many of such solutions, distinct up to periodic translations, which are not solutions to the associated ODE problem $$-\e^{2}\ddot q(x)+a(x)W'(q(x))=0,\qquad \lim_{x\to\pm\infty}q(x)=\pm 1.$$
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11566/34067