On a semilinear variational problem
ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 86-101.

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.

DOI : 10.1051/cocv/2009038
Classification : 35J20, 49J45, 35Q55
Mots-clés : nonlinear minimum problem, parabolic Anderson model, variational methods, gamma-convergence, ground state solutions 
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Schmidt, Bernd. On a semilinear variational problem. ESAIM: Control, Optimisation and Calculus of Variations, Tome 17 (2011) no. 1, pp. 86-101. doi : 10.1051/cocv/2009038. http://www.numdam.org/articles/10.1051/cocv/2009038/

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