We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $a:\R\to\R$ 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 show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W'(q(x))=0,\ x\in\R,\qquad q(\pm\infty)=\pm 1,$$ has a discrete structure, then the equation 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 $y\in\R$.

Brake orbits type solutions to some class of semilinear elliptic equations / Alessio, FRANCESCA GEMMA; Montecchiari, Piero. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 30:1(2007), pp. 51-83. [10.1007/s00526-006-0078-1]

Brake orbits type solutions to some class of semilinear elliptic equations

ALESSIO, FRANCESCA GEMMA;MONTECCHIARI, Piero
2007-01-01

Abstract

We consider a class of semilinear elliptic equations of the form $$-\Delta u(x,y)+a(x)W'(u(x,y))=0,\quad (x,y)\in\R^{2}$$ where $a:\R\to\R$ 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 show, via variational methods, that if the set of solutions to the one dimensional heteroclinic problem $$-\ddot q(x)+a(x)W'(q(x))=0,\ x\in\R,\qquad q(\pm\infty)=\pm 1,$$ has a discrete structure, then the equation 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 $y\in\R$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/51894
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