Stationary layered solutions in R-2 for a class of non autonomous Allen-Cahn equations

We consider a class of non autonomous Allen-Cahn equations -Delta u(x, y) + a(x)W' (u(x, y)) = 0, (x, y) is an element of R-2, (0.1) where W is an element of C-2(R, R) is a multiple-well potential and a is an element of C(R, R) is a periodic, positive, non-constant function. We look for solutions to (0.1) having uniform limits as x --> +/-infinity, corresp onding to minima of W. We show, via variational methods, that under a nonde generacy condition on the set of heteroclinic solutions of the associated o rdinary differential equation -q(x) + a(x)W'(q(x)) = 0, x is an element of R, the equation (0.1) has solutions which depends on both the variables x: and y. In contrast, when a is constant such nondegeneracy condition is not satisfied and all such solutions are known to depend only on x. Mathematics Subject Classification (1991):35J60, 35J20, 34C37.