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An Elliptic Equation With Spike Solutions Concentrating At Local Minima Of The Laplacian Of The Potential, Gregory S. Spradlin

Publications

We consider the equation −ԑ2∆u + V (z)u = f(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 f satisfies super-linear and sub-critical growth conditions, we show that for small ԑ 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 vibrational mountain-pass and concentration compactness arguments. A penalization technique developed by Felmer and del …