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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 Pino is used to handle the lack of compactness and the absence of the Palais-Smale condition in the vibrational framework.

Publication Title

Electronic Journal of Differential Equations


Southwest Texas State University

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Dr. Spradlin was not affiliated with Embry-Riddle Aeronautical University at the time this article was published.

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