On a semilinear variational problem

Bernd Schmidt

ESAIM: Control, Optimisation and Calculus of Variations (2011)

  • Volume: 17, Issue: 1, page 86-101
  • ISSN: 1292-8119

Abstract

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We provide a detailed analysis of the minimizers of the functional u ↦ ∫ ℝ n | ∇ u | 2 + D ∫ ℝ n | u | γ , γ ∈ ( 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.

How to cite

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Schmidt, Bernd. "On a semilinear variational problem." ESAIM: Control, Optimisation and Calculus of Variations 17.1 (2011): 86-101. <http://eudml.org/doc/272775>.

@article{Schmidt2011,
abstract = {We provide a detailed analysis of the minimizers of the functional $u \mapsto \int _\{\mathbb \{R\}^n\} |\nabla u|^2 + D \int _\{\mathbb \{R\}^n\} |u|^\{\gamma \}$, $\gamma \in (0, 2)$, subject to the constraint $\Vert u\Vert _\{L^2\} = 1$. 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.},
author = {Schmidt, Bernd},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {nonlinear minimum problem; parabolic Anderson model; variational methods; gamma-convergence; ground state solutions; Nonlinear minimum problem; Parabolic Anderson model; Variational methods; Gamma-convergence; Ground state solutions},
language = {eng},
number = {1},
pages = {86-101},
publisher = {EDP-Sciences},
title = {On a semilinear variational problem},
url = {http://eudml.org/doc/272775},
volume = {17},
year = {2011},
}

TY - JOUR
AU - Schmidt, Bernd
TI - On a semilinear variational problem
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2011
PB - EDP-Sciences
VL - 17
IS - 1
SP - 86
EP - 101
AB - We provide a detailed analysis of the minimizers of the functional $u \mapsto \int _{\mathbb {R}^n} |\nabla u|^2 + D \int _{\mathbb {R}^n} |u|^{\gamma }$, $\gamma \in (0, 2)$, subject to the constraint $\Vert u\Vert _{L^2} = 1$. 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.
LA - eng
KW - nonlinear minimum problem; parabolic Anderson model; variational methods; gamma-convergence; ground state solutions; Nonlinear minimum problem; Parabolic Anderson model; Variational methods; Gamma-convergence; Ground state solutions
UR - http://eudml.org/doc/272775
ER -

References

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  8. [8] J. Gärtner and S.A. Molchanov, Parabolic problems for the Anderson model. I. Intermittency and related topics. Comm. Math. Phys. 132 (1990) 613–655. Zbl0711.60055MR1069840
  9. [9] W. König, Große Abweichungen, Techniken und Anwendungen. Vorlesungsskript Universität Leipzig, Germany (2006). 
  10. [10] M.K. Kwong, Uniqueness of positive solutions of Δu - u +up = 0 in ℝ n . Arch. Rational Mech. Anal.105 (1989) 243–266. Zbl0676.35032MR969899
  11. [11] E.H. Lieb and M. Loss, Analysis, AMS Graduate Studies 14. Second edition, Providence, USA (2001). Zbl0966.26002MR1817225
  12. [12] P. Pucci, M. García-Huidobro, R. Manásevich and J. Serrin, Qualitative properties of ground states for singular elliptic equations with weights. Ann. Mat. Pura Appl.185 (2006) 205–243. Zbl1115.35050MR2187761

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