We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\R^{3}, \end{equation} where $a:\R\to\R$ is a periodic, positive, even function and, in the simplest case, $W:\R\to\R$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the associated one dimensional heteroclinic problem we show, via variational methods the existence of infinitely many geometrically distinct solutions $u$ of (\ref{eq:abs}) verifying $u(x,y,z)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $(y,z)\in\R^{2}$ and such that $\partial_{y}u\not\equiv0$, $\partial_{z}u\not\equiv0$ in $\R^{3}$.

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in R^{3} / Alessio, FRANCESCA GEMMA; Montecchiari, Piero. - In: CALCULUS OF VARIATIONS AND PARTIAL DIFFERENTIAL EQUATIONS. - ISSN 0944-2669. - STAMPA. - 46:3(2013), pp. 591-622. [10.1007/s00526-012-0495-2]

Layered solutions with multiple asymptotes for non autonomous Allen–Cahn equations in R^{3}

ALESSIO, FRANCESCA GEMMA;MONTECCHIARI, Piero
2013-01-01

Abstract

We consider a class of semilinear elliptic equations of the form \begin{equation}\label{eq:abs} -\Delta u(x,y,z)+a(x)W'(u(x,y,z))=0,\quad (x,y,z)\in\R^{3}, \end{equation} where $a:\R\to\R$ is a periodic, positive, even function and, in the simplest case, $W:\R\to\R$ is a double well even potential. Under non degeneracy conditions on the set of minimal solutions to the associated one dimensional heteroclinic problem we show, via variational methods the existence of infinitely many geometrically distinct solutions $u$ of (\ref{eq:abs}) verifying $u(x,y,z)\to\pm 1$ as $x\to\pm\infty$ uniformly with respect to $(y,z)\in\R^{2}$ and such that $\partial_{y}u\not\equiv0$, $\partial_{z}u\not\equiv0$ in $\R^{3}$.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11566/66072
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