Electron. J. Diff. Eqns., Vol. 2000(2000), No. 32, pp. 1-14.

An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential

Gregory S. Spradlin

We consider the equation $-\epsilon^2 \Delta u + V(z)u = f(u)$ which arises in the study of nonlinear Schrodinger equations. We seek solutions that are positive on ${\Bbb R}^N$ and that vanish at infinity. Under the assumption that f satisfies super-linear and sub-critical growth conditions, we show that for small $\epsilon$ 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
e-mail: gregory-spradlin@usma.edu spradlig@erau.edu

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