Abstract. We study the Cauchy problem for the fractional Schrö dinger equation, where n b 1, m b 0, 1 < a < 2, and F stands for the nonlinearity of Hartree type F ðuÞ ¼ lðcðÁÞj Á j Àg à juj 2 Þu with l ¼G1, 0 < g < n,and 0 a c A L y ðR n Þ. We prove the existence and uniqueness of local and global solutions for certain a, g, l, c. We also remark on finite time blowup of solutions when l ¼ À1.

We consider the orbital stability of standing waves of the nonlinear Schrdinger equationby the approach that was laid down by Cazenave and Lions in 1992. Our work covers several situations that do not seem to be included in previous treatments, namely,(i) g(x, s) − g(x, 0) → 0 as |x| → ∞ for all s ≥ 0. This includes linear problems.(ii) g(x, s) is a periodic function of x ∈ ℝ(iii) g(x, s) is asymptotically periodic in the sense that g(x, s) − gFurthermore, we focus attention on the form of the set that is shown to be stable and may be bigger than what is usually known as the orbit of the standing wave.

The inequalities of Hardy-Littlewood and Riesz say that certain integrals involving products of two or three functions increase under symmetric decreasing rearrangement. It is known that these inequalities extend to integrands of the form F (u 1 , . . . , u m ) where F is supermodular; in particular, they hold when F has nonnegative mixed second derivatives * i * j F for all i = j . This paper concerns the regularity assumptions on F and the equality cases. It is shown here that extended Hardy-Littlewood and Riesz inequalities are valid for supermodular integrands that are just Borel measurable. Under some nondegeneracy conditions, all equality cases are equivalent to radially decreasing functions under transformations that leave the functionals invariant (i.e., measure-preserving maps for the Hardy-Littlewood inequality, translations for the Riesz inequality). The proofs rely on monotone changes of variables in the spirit of Sklar's theorem.

Keywords:Fractional Laplacian Gagliardo-Nirenberg inequality Sobolev inequality Logarithmic Sobolev inequality Hardy inequality a b s t r a c tIn this paper, we establish the Gagliardo-Nirenberg inequality under Lorentz norms for fractional Laplacian. Based on special cases of this inequality under Lebesgue norms, we prove the L p -logarithmic Gagliardo-Nirenberg and Sobolev inequalities. Motivated by the L 2 -logarithmic Sobolev inequality, we obtain a fractional logarithmic Sobolev trace inequality in terms of the restriction τ k u of u from R n to R n−k . Finally, we prove the fractional Hardy inequality under Lorentz norms.

ABSTRACT. We study the stability of the standing wave solutions of a Gross-Pitaevskii equation describing Bose-Einstein condensation of dipolar quantum gases and characterize their orbit. As an intermediate step, we consider the corresponding constrained minimization problem and establish existence, symmetry and uniqueness of the ground state solutions. INTRODUCTIONSince the experimental realization of the first Bose-Einstein condensate (BEC) by Eric Cornell and Carl Wieman in 1995, tremendous efforts have been undertaken by mathematicians to exploit this achievement especially in atomic physics and optics. In the last years, a new kind of quantum gases with dipolar interaction, which acts between particles as a permanent magnetic or electric dipole moment has attracted the attention of a lot of scientists. The interactions between particles are both long-range and non-isotropic. Describing the corresponding BEC via Gross Pitaevskii approximation, one gets the following nonlinear Schrödinger equation (1.1) i ∂ t ψ = − 2 2m ∆ψ + g|ψ| 2 ψ + d 2 (K * |ψ| 2 )ψ + V (x)ψ, t ∈ R, x ∈ R 3 , where |g| = 4π 2 N |a| m , N ∈ N is the number of particles, m denotes the mass of individual particles and a its corresponding scattering length. The external potential V (x) describes the electromagnetic trap and has the following harmonic confinementThe factor d 2 denotes the strength of the dipole moment in Gaussian units andwhere θ = θ(x) is the angle between x ∈ R 3 and the dipole axis n ∈ R 3 . The local term g|ψ| 2 ψ describes the short-range interaction forces between particles, while the non-local potential K * |ψ| 2 describes their long-range dipolar interactions. For the mathematical analysis, it is more convenient to rescale (1.1) into the following dimensionless formThis work was supported by the French ANR projects SchEq (ANR-12-JS01-0005-01) and BoND (ANR-13-BS01-0009-01). , and a 0 = m .In the following, we assume that λ 1 and λ 2 are two given real-valued parameters.In [5], the authors have studied the existence and uniqueness of the equation (1.3) with initial condition ψ 0 ∈ H 1 (R 3 ),They have established that (1.4) has a unique, global solution if λ 1 4 3 πλ 2 0. They called this situation stable regime, referring to the fact that no singularity in formed in finite time. In this paper, we study another notion of stability, that is, the stability of standing waves. They have also showed that in the unstable regime (λ 1 < 4 3 πλ 2 ), finite time blow up may occur, hence the denomination.The evidence of blow-up relies on a function for which the corresponding energy is strictly negative ([5, Lemma 5.1]). They concluded using the virial approach of Zakharov and Glassey. Some refinements of the above result have been discussed in [5, Proposition 5.4].The most important issue in view of the applications of (1.4) in atomic physics and quantum optics seems to be the study of ground state solutions of (1.4). These solutions are the "only" observable states in experiments. A standing wave solution of (1.4) is a wave func...