An Elliptic Equation with Spike Solutions Concentrating at Local Minima of the Laplacian of the Potential
Date
2000-05-02
Authors
Spradlin, Gregory S.
Journal Title
Journal ISSN
Volume Title
Publisher
Southwest Texas State University, Department of Mathematics
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.
Description
Keywords
Nonlinear Schrodinger equation, Variational methods, Singularly perturbed elliptic equation, Mountain-pass theorem, Concentration compactness, Degenerate critical points
Citation
Spradlin, G. S. (2000). An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential. Electronic Journal of Differential Equations, 2000(32), pp. 1-14.
Rights
Attribution 4.0 International