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

#### 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 $y\in\R$.
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2005
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/53379
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