# On a semilinear variational problem

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

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

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topSchmidt, 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 -

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