We are interested in entire solutions of the Allen–Cahn equation $\Delta u-{F}^{\text{'}}\left(u\right)=0$ which have some special structure at infinity. In this equation, the function F is an even, double well potential. The solutions we are interested in have their zero set asymptotic to 4 half oriented affine lines at infinity and, along each of these half affine lines, the solutions are asymptotic to the one dimensional heteroclinic solution: such solutions are called 4-ended solutions. The main result of our paper states that, for any $\theta \in (0,\pi /2)$, there exists a 4-ended solution of the Allen–Cahn equation whose zero set is at infinity asymptotic to the half oriented affine lines making the angles θ, $\pi -\theta $, $\pi +\theta $ and $2\pi -\theta $ with the x-axis. This paper is part of a program whose aim is to classify all 2k-ended solutions of the Allen–Cahn equation in dimension 2, for $k\u2a7e2$.

Keywords: Allen–Cahn equation, Classification of solutions, Entire solutions of semilinear elliptic equations

@article{AIHPC_2012__29_5_761_0, author = {Kowalczyk, Micha\l\ and Liu, Yong and Pacard, Frank}, title = {The space of 4-ended solutions to the Allen--Cahn equation in the plane}, journal = {Annales de l'I.H.P. Analyse non lin\'eaire}, publisher = {Elsevier}, volume = {29}, number = {5}, year = {2012}, pages = {761-781}, doi = {10.1016/j.anihpc.2012.04.003}, zbl = {1254.35219}, mrnumber = {2971030}, language = {en}, url = {http://www.numdam.org/item/AIHPC_2012__29_5_761_0} }

Kowalczyk, Michał; Liu, Yong; Pacard, Frank. The space of 4-ended solutions to the Allen–Cahn equation in the plane. Annales de l'I.H.P. Analyse non linéaire, Volume 29 (2012) no. 5, pp. 761-781. doi : 10.1016/j.anihpc.2012.04.003. http://www.numdam.org/item/AIHPC_2012__29_5_761_0/

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