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

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.

Keywords

• Nonlinear Schrodinger Equation
• variational methods
• singularly perturbed elliptic equation
• mountain-pass theorem
• concentration compactness
• degenerate critical points.