The Lieb-Yau conjecture for ground states of pseudo-relativistic Boson stars

Abstract

It is known that ground states of the pseudo-relativistic Boson stars exist if and only if the stellar mass N > 0 satisfies N < N * , where the finite constant N * is called the critical stellar mass. Lieb and Yau conjecture in [Comm. Math. Phys., 1987] that ground states of the pseudo-relativistic Boson stars are unique for each N < N * . In this paper, we prove that the above uniqueness conjecture holds for the particular case where N > 0 is small enough.By making full use of (1.3), Lenzmann in [26] established the following interesting existence and analytical characters of minimizers for e(N ):Theorem A ([26, Theorem 1]) Under the assumption m > 0, the following results hold for e(N ):1. e(N ) has minimizers if and only if 0 < N < N * , where the finite constant N * is independent of m. Any minimizer3. Any nonnegative minimizer of e(N ) must be strictly positive and radially symmetricdecreasing, up to phase and translation.We remark that the existence of the critical constant N * > 0 stated in Theorem A, which is called the critical stellar mass of boson stars, was proved earlier in [14,29]. Further, the dynamics and some other analytic properties of minimizers for e(N ) were also investigated by Lenzmann and his collaborators in [12,13,14,15,25,26] and references therein. Stimulated by [20,22], the related limit behavior of minimizers for e(N ) as N ր N * were studied more recently in [23,33,37], where the Gagliardo-Nirenberg type inequality (1.3) also played an important role. Note also from the variational theory that any minimizer u of e(N ) satisfies the Euler-Lagrange equation −∆ + m 2 − m u − 1 |x| * |u| 2 u = −µu in R 3 , (1.5)

Validity of Bogoliubov's approximation for translation-invariant Bose gases

“…The chain of operator inequalities in Assumption 1.1 holds as long as the coupling is below the critical value 4 π , see [11,13]. By restricting the attention to possibly smaller couplings g ∈ (0, g * ) it has been shown in [15,10] that minimizers u 0 are unique in the sense of Assumption 1.1. Furthermore it follows from the results in [15,10] that the Hessian is non-degenerate in the sense of Assumption 1.3 for couplings g below a critical value.…”

“…By restricting the attention to possibly smaller couplings g ∈ (0, g * ) it has been shown in [15,10] that minimizers u 0 are unique in the sense of Assumption 1.1. Furthermore it follows from the results in [15,10] that the Hessian is non-degenerate in the sense of Assumption 1.3 for couplings g below a critical value. We will verify this explicitly in Appendix A, using an argument similar to the one in [7] for non-relativistic systems.…”

Normalized solutions for Schrödinger equations with critical Sobolev exponent and mixed nonlinearities

“…In these references, we would like to highlight [36,49,50] to the readers for their detail introductions and references on normalized solutions of (1.4) and new directions on the study of normalized solutions of autonomous problems. We also would like to point out [26][27][28][29][30][31][32] and the references therein for the studies on normalized solutions of problems with trapping potentials.…”

Many-body blow-up profile of boson stars with external potentials

“…The uniqueness (up to translations and dilations) of the optimizers of (1.2), as well as the uniqueness (up to translations) of the positive solutions of (1.4), are open problems (see [34,17,25] for related discussions). Note that the translations and dilations can be fixed by using (1.3) and the radial properties.…”