Spike solutions of a nonlinear Schrödinger equation with degenerate potential

We study the existence of positive solutions to the elliptic equation ε2Δu(x, y) − V (y)u(x, y) + f (u(x, y)) = 0 for (x, y) in an unbounded domain RN1 ×Ω2 subject to the boundary condition u = 0 whenever ∂Ω2 is nonempty. Our potential V depends only on the y variable and Ω2 is a bounded or unbounded domain which may coincide with RN2 . The positive parameter ε is tending to zero and our solutions uε concentrate along minimum points of the unbounded manifold of critical points of V .  2004 Elsevier Inc. All rights reserved