Energy scattering for a 3D Hartree equation with inverse square potential.

This work studies the focusing inhomogeneous nonlinear equation of Hartree type і∂tu - Кλu +|х|-τ-|u|p-2)(Jα*|.|-τ|u|P)u = 0, u(t.x): R x R³ → C. Here, the linear Schrödinger operator reads Kλ:= - Δ+ λ/|x|² for some λ > - 1/4. The Riesz potential is Jα (x) = Cα|x|-(3-α) for certain 0 <α< 3. The singular decaying term |x|-τ for some τ > 0 gives an inhomogeneous non-linearity. One considers the inter-critical regime, namely, 1 +2 (1-τ)+α /3<p<1+2(1 – τ) + α. Moreover, one assumes that p≥ 2 in order to avoid a singular 3 term |u |p-2. Furthermore, one restricts λ > 0 because there is no dispersive estimate L¹→ L∞ for λ<0. Contrarily to the homogeneous case τ = 0, for λ> 0, there is a ground state which minimizes the associated Gagliardo-Nirenberg-type estimate. The purpose is to investigate the energy scattering of global solutions under the ground state threshold. One uses the method of Dodson-Murphy based on Tao's scattering criteria and Morawetz estimates. The decay of the inhomogeneous term |x|-τ avoids any spherically symmetric assumption. [ABSTRACT FROM AUTHOR]