Spike solutions of a nonlinear Schrodinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation epsilon(2) Deltau(x, y) - V(y)u(x, y) + f(u(x, y)) = 0 for (x, y) in an unbounded domain R-N1 x Omega(2) subject to the boundary condition u = 0 whenever partial derivativeOmega(2) is nonempty. Our potential V depends only on the y variable and Omega(2) is a bounded or unbounded domain which may coincide with R-N2. The positive parameter epsilon is tending to zero and our solutions u(epsilon) concentrate along minimum points of the unbounded manifold of critical points of V. (C) 2004 Elsevier Inc. All rights reserved.