On the radiality of constrained minimizers to the Schrödinger–Poisson–Slater energy

Georgiev, Vladimir; Prinari, Francesca; Visciglia, Nicola

Annales de l'I.H.P. Analyse non linéaire, Tome 29 (2012) no. 3, pp. 369-376

We study the radial symmetry of minimizers to the Schrödinger–Poisson–Slater (S–P–S) energy:

$$\underset{\genfrac{}{}{0.0pt}{}{u\in H^1\left({ℝ}^3\right)}{\parallel u{\parallel }_{L^2\left({ℝ}^3\right)}=\rho }}{\inf }\frac{1}{2}\underset{{ℝ}^3}{\int }{\left|\nabla u\right|}^2+\frac{1}{4}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{{\left|u\right(x\left)\right|}^2{\left|u\right(y\left)\right|}^2}{|x-y|}dxdy-\frac{1}{p}\underset{{ℝ}^3}{\int }{\left|u\right|}^{p}dx$$

provided that $2<p<3$ and ρ is small. The main result shows that minimizers are radially symmetric modulo suitable translation.

On montre la radialité des minimiseurs de lʼénergie de Schrödinger–Poisson–Slater

$$\underset{\genfrac{}{}{0.0pt}{}{u\in H^1\left({ℝ}^3\right)}{\parallel u{\parallel }_{L^2\left({ℝ}^3\right)}=\rho }}{\inf }\frac{1}{2}\underset{{ℝ}^3}{\int }{\left|\nabla u\right|}^2+\frac{1}{4}\underset{{ℝ}^3}{\int }\underset{{ℝ}^3}{\int }\frac{{\left|u\right(x\left)\right|}^2{\left|u\right(y\left)\right|}^2}{|x-y|}dxdy-\frac{1}{p}\underset{{ℝ}^3}{\int }{\left|u\right|}^{p}dx$$

pourvu que $2<p<3$ et ρ est petit.

[1] A. Ambrosetti, G. Prodi, A Primer of Nonlinear Analysis, Cambridge Studies in Advanced Mathematics vol. 34, Cambridge University Press, Cambridge (1995) | Zbl

[2] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York (2011) | Zbl

[3] Benguria, H. Brezis, E. Lieb, The Thomas–Fermi–von Weizsäcker theory of atoms and molecules, Comm. Math. Phys. 79 no. 2 (1981), 167-180 | Zbl

[4] J. Bellazzini, G. Siciliano, Stable standing waves for a class of nonlinear Schrödinger–Poisson equations, Z. Angew. Math. Phys. 62 (2011), 267-280 | Zbl

[5] J. Bellazzini, G. Siciliano, Scaling properties of functionals and existence of constrained minimizers, J. Funct. Anal. 261 (2011), 2486-2507 | Zbl

[6] I. Catto, P.L. Lions, Binding of atoms and stability of molecules in Hartree and Thomas–Fermi type theories. I. A necessary and sufficient condition for the stability of general molecular systems, Comm. Partial Differential Equations 17 (1992), 1051-1110 | Zbl

[7] T. Cazenave, P.L. Lions, Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982), 549-561 | Zbl

[8] V. Coti-Zelati, M. Esteban, Symmetry breaking and multiple solutions for a Neumann problem in an exterior domain, Proc. Royal Soc. Edinburgh 116A (1990), 327-339 | Zbl

[9] M. Esteban, Nonsymmetric ground states of symmetric variational problems, Comm. Pure and Appl. Math. 44 (1991), 259-274 | Zbl

[10] M. Esteban, Rupture de symétrie pour des problémes de Neumann sur-linéaires dans les ouverts extérieur, C. R. Acad. Sci. Sér. I 308 (1989), 281-286 | Zbl

[11] B. Gidas, W. Ni, L. Nirenberg, Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), 209-243 | Zbl

[12] M.K. Kwong, Uniqueness of positive solutions of $\Delta u-u+u^p=0$ in $R^n$, Arch. Ration. Mech. Anal. 105 (1989), 243-266 | Zbl

[13] E. Lieb, Existence and uniqueness of the minimizing solution of Choquardʼs nonlinear equation, Stud. Appl. Math. 57 (1976/1977), 93-105

[14] E. Lieb, M. Loss, Analysis, Graduate Studies in Mathematics vol. 14, American Mathematical Society, Providence, RI (2001)

[15] O. Lopes, M. Maris, Symmetry of minimizers for some nonlocal variational problems, J. Funct. Anal. 254 (2008), 535-592 | Zbl

[16] O. Sanchez, J. Soler, Long-time dynamics of the Schrödinger–Poisson–Slater system, J. Statist. Phys. 114 (2004), 179-204 | Zbl

[17] J. Serrin, A symmetry problem in potential theory, Arch. Ration. Mech. 43 (1971), 304-318 | Zbl

[18] M.I. Weinstein, Modulational stability of ground states of nonlinear Schrödinger equations, SIAM J. Math. Anal. 16 (1985), 472-491 | Zbl