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On a semilinear variational problem

Published online by Cambridge University Press:  09 October 2009

Bernd Schmidt
Bernd Schmidt*
Affiliation:
Zentrum Mathematik, Technische Universität München, Boltzmannstr. 3, 85747 Garching, Germany. schmidt@ma.tum.de
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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.

Keywords

Nonlinear minimum problemparabolic Anderson modelvariational methodsGamma-convergenceground state solutions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 17 , Issue 1 , January 2011 , pp. 86 - 101
Copyright
© EDP Sciences, SMAI, 2009

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