Semi-classical states for the Nonlinear Schr\"odinger Equation on saddle points of the potential via variational methods

Pietro d'Avenia; Alessio Pomponio; David Ruiz

Semi-classical states for the Nonlinear Schr\"odinger Equation on saddle points of the potential via variational methods

Analysis of PDEs 2012-03-12 v2

Authors: Pietro d'Avenia , Alessio Pomponio , David Ruiz

Abstract

In this paper we study semiclassical states for the problem $-\eps^2 \Delta u + V(x) u = f(u) \qquad \hbox{in} \RN,$ where $f(u)$ is a superlinear nonlinear term. Under our hypotheses on $f$ a Lyapunov-Schmidt reduction is not possible. We use variational methods to prove the existence of spikes around saddle points of the potential $V(x)$ .

Keywords

nonlinear Schrödinger equation semiclassical states saddle points variational methods spike solutions

Cite

@article{arxiv.1107.5652,
  title  = {Semi-classical states for the Nonlinear Schr\"odinger Equation on saddle points of the potential via variational methods},
  author = {Pietro d'Avenia and Alessio Pomponio and David Ruiz},
  journal= {arXiv preprint arXiv:1107.5652},
  year   = {2012}
}

Comments

pre-peer version, to appear in J. Funct. Anal

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