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

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