We consider a class of non autonomous Allen-Cahn equations

We consider a class of semilinear elliptic equations of the form −ε 2 ∆u(x, y) + a(x)W (u(x, y)) = 0, (x, y) ∈ R 2 (0.1) where ε > 0, a : R → R is a periodic, positive function and W : R → R is modeled on the classical two well Ginzburg-Landau potential W (s) = (s 2 − 1) 2. We look for solutions to (0.1) which verify the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∈ R. We show via variational methods that if ε is sufficiently small and a is not constant, then (0.1) admits infinitely many of such solutions, distinct up to translations, which do not exhibit one dimensional symmetries.

Abstract. This paper concerns the existence and asymptotic characterization of saddle solutions in R 3 for semilinear elliptic equations of the formDenoted with θ2 the saddle planar solution of (0.1), we show the existence of a unique solution θ3 ∈ C 2 (R 3 ) which is odd with respect to each variable, symmetric with respect to the diagonal planes, verifies 0 < θ3(x, y, z) < 1 for x, y, z > 0 and θ3 (x, y, z) →z→+∞ θ2(x, y) uniformly with respect to (x, y) ∈ R 2 .Mathematics Subject Classification (2010). 35J60, 35B05, 35B40, 35J20, 34C37.

A semilinear elliptic system of PDEs with a nonlinear term of double well potential type is studied in a cylindrical domain. The existence of solutions heteroclinic to the bottom of the wells as minima of the associated functional is established. Further applications are given, including the existence of multitransition solutions as local minima of the functional.

We consider a class of semilinear elliptic equations of the formwhere a : R → R is a periodic, positive function and W : R → 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 problemhas a discrete structure, then (0.1) has infinitely many solutions periodic in the variable y and verifying the asymptotic conditions u(x, y) → ±1 as x → ±∞ uniformly with respect to y ∈ R.