Brake orbits type solutions to some class of semilinear elliptic equations

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Abstract

We consider a class of semilinear elliptic equations of the form
$$\label{eq:abs} -\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\mathbb{R}^{2}$$
where \({a:\mathbb{R}\to\mathbb{R}}\) 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 show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem
$$-\ddot q(x)+a(x)W'(q(x))=0,\quad x\in\mathbb{R},\quad q(\pm\infty)=\pm 1,$$
has a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions \({u(x,y)\to\pm 1}\) as \({x\to\pm\infty}\) uniformly with respect to \({y\in\mathbb{R}}\) .

Mathematics Subject Classification (2000)

35J60 35B05 35B40 35J20 34C37 
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Copyright information

© Springer-Verlag 2007

Authors and Affiliations

  1. 1.Dipartimento di Scienze MatematicheUniversità Politecnica delle MarcheAnconaItaly

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