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We study scattering properties in non-isotropic Sobolev fractional spaces for small data solutions to the following system of N ≥ 2, nonlinear Schrödinger equations (NLS) having mass-supercritical power-type nonlinearity and posed on product spaces ℝd × ℳk, for d, k ≥ 1 and ℳk any k-dimensional compact Riemaniann manifold. [ABSTRACT FROM AUTHOR]

We perform a new inequality regarding the operator L+ = - Δ + 1- β(Q²) - 2β′(Q²) concerning the ground states Q associated to the nonlinear Schrödinger equations (NLS), ... with d ≥ 1. We present also some results straightforward consequence of an application of such an inequality. [ABSTRACT FROM AUTHOR]

We perform new Morawetz identities, interaction Morawetz identities and their corresponding inequalities for the system of N nonlinear defocusing Schrodinger equations (NLS), with N ≥ 2, ... Here, for all ... and .... are coupling parameters, moreover we require that the nonlinearity parameter p is both L²(Rd × T)-supercritical and H¹(Rd × T)-subcritical (or at most H¹(Rd × T)-critical). [ABSTRACT FROM AUTHOR]

In this work we are concerned with the minimization problem associated to the best constant Cn,α in the following interpolation inequality ∫∫|u(x)|α|u(y)|α|x-y|n-2dxdy≤Cn,α|∇ψ|L2nα-(n+2)|ψ|L2(n+2)-(n-2)α, u∈H1(Rn) for 2≤α<1+4/(n-2) and n≥3. The corresponding variational problem is equivalent to the problem of existence of a ground state solution to the nonlinear non-local elliptic equation -Δu+ωu-(|x|-(n-2)*|u|α)|u|α-2u = 0. We show that the ground state solution is unique and its mass |u|L22 determines the best constant Cn,α. As a consequence of the above results, we obtain sharp sufficient condition for global existence for the L2-critical Hartree equation. [ABSTRACT FROM AUTHOR]

In this work we prove the existence of solitary wave solutions and give conditions for formation of singularities for the focusing time-dependent Schrödinger-Hartree equation in Rn. [ABSTRACT FROM AUTHOR]