Existence of infinitely many stationary layered solutions in R2 for a class of periodic Allen-Cahn equations

We consider a class of periodic Allen-Cahn equations -Deltau(x, y) +a(x, y)W'(u(x, y)) = 0, (x, y) is an element of R-2 (1) where a : R-2 --> R-2 is an even, periodic, positive function and W : R --> R is modeled on the classical two well Ginzburg-Landau potential W(s) = (s(2) -b(2))(2). We show, via variational methods, that there exist infinitely many solutions, distinct up to periodic translations, of (1) asymptotic as x --> +/-infinity to the pure states +/-b, i.e., solutions satisfying the boundary conditions lim(x-->+/-infinity) u(x, y) = +/-b, uniformly in y is an element of R. (2) In fact, we prove the existence of solutions of (1)-(2) which are periodic in the y variable and if such solutions are finite modulo periodic translations, we can prove the existence of infinitely many (modulo periodic translations) solutions of (1)-(2) asymptotic to different periodic solutions as y greater than or equal to infinity.