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

Spradlin, Gregory S.

Electronic Journal of Differential Equations, Tome 2000 (2000).

Résumé

Summary: 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.

Détail

Classification : 35J50
Keywords: nonlinear Schrödinger equation, variational methods, singularly perturbed elliptic equation, mountain-pass theorem, concentration compactness, degenerate critical points