Electronic Journal of Differential Equations,
Vol. 2000(2000), No. 32, pp. 1-14.

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

Author: Gregory S. Spradlin (U.S. Military Academy, West Point, NY, USA) 
         
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.  

Submitted February 4, 2000. Published May 2, 2000.
Math Subject Classifications: 35J50.
Key Words: Nonlinear Schrodinger Equation;  variational methods; 
          singularly perturbed elliptic equation;  mountain-pass theorem; 
          concentration compactness; degenerate critical points.