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 $u↦{\int }_{{ℝ}^{n}}{|\nabla u|}^{2}+D{\int }_{{ℝ}^{n}}{|u|}^{\gamma }$, $\gamma \in \left(0,2\right)$, subject to the constraint ${\parallel u\parallel }_{{L}^{2}}=1$. 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/

[1] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Existence of a ground state. Arch. Rational Mech. Anal. 82 (1983) 313-345. | MR 695535 | Zbl 0533.35029

[2] M. Biskup and W. König, Long-time tails in the parabolic Anderson model with bounded potential. Ann. Probab. 29 (2001) 636-682. | MR 1849173 | Zbl 1018.60093

[3] A. Braides, Γ-Convergence for Beginners. Oxford University Press, Oxford, UK (2002). | MR 1968440 | Zbl 1198.49001

[4] J.E. Brother and W.P. Ziemer, Minimal rearrangements of Sobolev functions. J. Reine Angew. Math. 384 (1988) 153-179. | MR 929981 | Zbl 0633.46030

[5] C.C. Chen and C.S. Lin, Uniqueness of the ground state solutions of Δu + f(u) = 0 in ${ℝ}^{n}$, n ≥ 3. Comm. Partial Diff. Eq. 16 (1991) 1549-1572. | MR 1132797 | Zbl 0753.35034

[6] C. Cortazar, M. Elgueta and P. Felmer, On a semilinear elliptic problem in ${ℝ}^{N}$ with a non-Lipschitzian nonlinearity. Adv. Diff. Eq. 1 (1996) 199-218. | Zbl 0845.35031

[7] C. Cortazar, M. Elgueta and P. Felmer, Uniqueness of positive solutions of Δu + f(u) = 0 in ${ℝ}^{n}$, N ≥ 3. Arch. Rational Mech. Anal. 142 (1998) 127-141. | MR 1629650 | Zbl 0912.35059

[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. | MR 1069840 | Zbl 0711.60055

[9] W. König, Große Abweichungen, Techniken und Anwendungen. Vorlesungsskript Universität Leipzig, Germany (2006).

[10] M.K. Kwong, Uniqueness of positive solutions of Δu - u +up = 0 in ${ℝ}^{n}$. Arch. Rational Mech. Anal. 105 (1989) 243-266. | MR 969899 | Zbl 0676.35032

[11] E.H. Lieb and M. Loss, Analysis, AMS Graduate Studies 14. Second edition, Providence, USA (2001). | MR 1817225 | Zbl 0966.26002

[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. | MR 2187761 | Zbl 1115.35050