Abstract
We study the existence of solutions u:R3→R2 for the semilinear elliptic systems(0.1)−Δu(x,y,z)+∇W(u(x,y,z))=0, where W: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± of W, that (0.1) has infinitely many geometrically distinct solutions u∈C2(R3,R2) which satisfy u(x,y,z)→a± as x→±∞ uniformly with respect to (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)|→+∞.
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