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Article
An Elliptic Equation With Spike Solutions Concentrating at Local Minima of the Laplacian of the Potential
Electronic Journal of Differential Equations (2000)
  • Gregory S. Spradlin, Embry-Riddle Aeronautical University - Daytona Beach
Abstract
We consider a singularly perturbed elliptic PDE that arises in the study of nonlinear Schrodinger equations. We seek solutions that are positive on the entirety of Euclidean space and that vanish at infinity. Under the assumption that the nonlinear term of the PDE satisfies super-linear and sub-critical growth conditions, we show that for small values of the epsilon parameter in the PDE, there solutions that concentrate near local minima of V (a coefficient function in the PDE) . 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 failure 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
Publication Date
2000
Citation Information
Gregory S. Spradlin. "An Elliptic Equation With Spike Solutions Concentrating at Local Minima of the Laplacian of the Potential" Electronic Journal of Differential Equations Vol. 2000 Iss. 32 (2000) p. 1 - 14 ISSN: 1072-6691
Available at: http://works.bepress.com/greg_spradlin/11/