Gregory S. Spradlin
We consider the equation which arises in the study of nonlinear Schrodinger equations. We seek solutions that are positive on and that vanish at infinity. Under the assumption that f 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.
Submitted February 4, 2000. Published May 2, 2000.
Math Subject Classifications: 35J50.
Key Words: Nonlinear Schrodinger Equation, variational methods, singularly perturbed elliptic equation, mountain-pass theorem, concentration compactness, degenerate critical points.
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| Gregory S. Spradlin |
Department of Mathematical Sciences
United States Military Academy
West Point, New York 10996, USA
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