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| Abstract |
Year: 2000 Volume: 2000 - Issue: 32
| Title: | An elliptic equation with spike solutions concentrating at local minima of the Laplacian of the potential |
| Author: | Gregory S. Spradlin |
| Abstract: | We consider the equation $-epsilon^2 Delta u + V(z)u = f(u)$ which arises in the study of nonlinear Schr"odinger equations. We seek solutions that are positive on ${mathbb R}^N$ 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. |
| Journal: | Electronic Journal of Differential Equations |
| Issn: | 10726691 |
| EIssn: | |
| Year: | 2000 |
| Volume: | 2000 |
| Issue: | 32 |
| pages/rec.No: | 1-14 |
| Key words | Nonlinear Schrodinger Equation ; variational methods ; singularly perturbed elliptic equation ; mountain-pass theorem ; concentration compactness ; degenerate critical points. |
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