On the inter-critical inhomogeneous generalized Hartree equation.

It is the purpose of this work to study the Choquard equation i u ˙ - (- Δ) s u = ± | x | γ (I α ∗ | · | γ | u | p ) | u | p - 2 u in the space H ˙ s ∩ H ˙ s c , where 0 < s c < s corresponds to the scale invariant homogeneous Sobolev norm. Here, one considers to two separate cases. The first one is the classical case s = 1 and the second one is the fractional regime 0 < s < 1 with radial data. One tries to develop a local theory using a new adapted sharp Gagliardo–Nirenberg estimate. Moreover, one investigates the concentration of non-global solutions in L T ∗ ∞ (H ˙ s c) . One needs to deal with the lack of a mass conservation, since the data are not supposed to be in L 2 . This note gives a complementary to the previous works about the same problem in the energy space H 1 . [ABSTRACT FROM AUTHOR]