We consider a class of non autonomous Allen-Cahn equations

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.

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. We consider a class of semilinear elliptic system of the formwhere W : R 2 → R is a double well nonnegative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system −q(x) + ∇W (q(x)) = 0, x ∈ R, which connect the two minima of W as x → ±∞ has a discrete structure, then (0.1) has infinitely many layered solutions.

We consider the pendulum type equationwhere the functions Wand h satisfy (HI) WE C 2 (JR. x JR.; JR.) is I-periodic in t and T-periodic in u, (H2) hE C(JR.;JR.) is I-periodic and fol h(t)dt = 0, (H3) Vet, u) = Wet, u) -h(t)u is even in t.