An Elliptic Equation with Spike Solutions Concentrating at Local Minima of the Laplacian of the Potential

Abstract

We consider the equation -∊2 ∆u + V(z)u = ƒ(u) which arises in the study of nonlinear Schrödinger equations. We seek solutions that are positive on ℝN and that vanish at infinity. Under the assumption that ƒ satisfies super-linear and sub-critical growth conditions, we show that for small ∊ there exist solutions that concentrate near local minima of V. The local minima may occur in unbounded components, as long as the Laplacian of V achieves a strict local minimum along such a component. Our proofs employ variational mountain-pass and concentration compactness arguments. A penalization technique developed by Felmer and del Pino is used to handle the lack of compactness and the absence of the Palais-Smale condition in the variational framework.