Entire solutions in 2 for a class of Allen-Cahn equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, pp. 633-672.

We consider a class of semilinear elliptic equations of the form

-ε 2 Δu(x,y)+a(x)W ' (u(x,y))=0,(x,y) 2
where ε>0, a: is a periodic, positive function and W: is modeled on the classical two well Ginzburg-Landau potential W(s)=(s 2 -1) 2 . We look for solutions to (1) which verify the asymptotic conditions u(x,y)±1 as x± uniformly with respect to y. We show via variational methods that if ε is sufficiently small and a is not constant, then (1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

DOI: 10.1051/cocv:2005023
Classification: 34C37, 35B05, 35B40, 35J20, 35J60
Keywords: heteroclinic solutions, elliptic equations, variational methods
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     author = {Alessio, Francesca and Montecchiari, Piero},
     title = {Entire solutions in $\mathbb {R}^{2}$ for a class of {Allen-Cahn} equations},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {633--672},
     publisher = {EDP-Sciences},
     volume = {11},
     number = {4},
     year = {2005},
     doi = {10.1051/cocv:2005023},
     mrnumber = {2167878},
     zbl = {1084.35020},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv:2005023/}
}
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Alessio, Francesca; Montecchiari, Piero. Entire solutions in $\mathbb {R}^{2}$ for a class of Allen-Cahn equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 11 (2005) no. 4, pp. 633-672. doi : 10.1051/cocv:2005023. http://www.numdam.org/articles/10.1051/cocv:2005023/

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