Multiplicity of layered solutions for Allen–Cahn systems with symmetric double well potential

We study the existence of solutions u:R3→R2u:R3→R2 for the semilinear elliptic systemsequation(0.1)−Δu(x,y,z)+∇W(u(x,y,z))=0,−Δu(x,y,z)+∇W(u(x,y,z))=0, where W:R2→RW:R2→R is a double well symmetric potential. We use variational methods to show, under generic non-degenerate properties of the set of one dimensional heteroclinic connections between the two minima a±a± of W, that (0.1) has infinitely many geometrically distinct solutions u∈C2(R3,R2)u∈C2(R3,R2) which satisfy u(x,y,z)→a±u(x,y,z)→a± as x→±∞x→±∞ uniformly with respect to (y,z)∈R2(y,z)∈R2 and which exhibit dihedral symmetries with respect to the variables y and z  . We also characterize the asymptotic behavior of these solutions as |(y,z)|→+∞|(y,z)|→+∞.