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

We provide a detailed analysis of the minimizers of the functional u n |u| 2 +D 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 : https://doi.org/10.1051/cocv/2009038
Classification:  35J20,  49J45,  35Q55
Keywords: nonlinear minimum problem, parabolic Anderson model, variational methods, gamma-convergence, ground state solutions
@article{COCV_2011__17_1_86_0,
     author = {Schmidt, Bernd},
     title = {On a semilinear variational problem},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     publisher = {EDP-Sciences},
     volume = {17},
     number = {1},
     year = {2011},
     pages = {86-101},
     doi = {10.1051/cocv/2009038},
     zbl = {1213.35222},
     mrnumber = {2775187},
     language = {en},
     url = {http://www.numdam.org/item/COCV_2011__17_1_86_0}
}
Schmidt, Bernd. On a semilinear variational problem. ESAIM: Control, Optimisation and Calculus of Variations, Volume 17 (2011) no. 1, pp. 86-101. doi : 10.1051/cocv/2009038. http://www.numdam.org/item/COCV_2011__17_1_86_0/

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