ESAIM: Control, Optimisation and Calculus of Variations

Table of Contents - January 2011 - Volume 17, Issue 01  

Research Article

On a semilinear variational problem

Schmidt, Bernd

Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching, Germany. schmidt@ma.tum.de

Abstract

We provide a detailed analysis of the minimizers of the functional $u \mapsto \int_{\Bbb R^n} |\nabla u|^2 + D \int_{\Bbb R^n} |u|^{\gamma}$ , $\gamma \in (0, 2)$ , subject to the constraint $\|u\|_{L^2} = 1$ . This problem, e.g., describes the long-time behavior of the parabolic Anderson in probability theory or ground state solutions of a nonlinear Schrödinger equation. While existence can be proved with standard methods, we show that the usual uniqueness results obtained with PDE-methods can be considerably simplified by additional variational arguments. In addition, we investigate qualitative properties of the minimizers and also study their behavior near the critical exponent 2.

(Received February 22 2009)

(Revised July 27 2009)

(Online publication October 9 2009)

Key Words:

  • Nonlinear minimum problem;
  • parabolic Anderson model;
  • variational methods;
  • Gamma-convergence;
  • ground state solutions

Mathematics Subject Classification:

  • 35J20;
  • 49J45;
  • 35Q55