Long-time dynamics for the radial focusing fractional INLS

We consider the following fractional NLS with focusing inhomogeneous power-type nonlinearity: i partial derivative(t)u - (-Delta)(s)u + |x|(-b)|u|(p-1)u = 0, (t, x) is an element of R x R-N, where N >= 2, 1/2 < s < 1, 0 < b < 2s, and 1 + 2(2s-b) / N < p < 1 + 2(2s-b) / N-2s. We prove the ground state threshold of global existence and scattering versus finite time blowup of energy solutions in the inter-critical regime with spherically symmetric initial data. The scattering is proved by the new approach of Dodson-Murphy. This method is based on Tao's scattering criteria and Morawetz estimates. We describe the threshold using some non-conserved quantities in the spirit of the recent paper by Dinh. The radial assumption avoids a loss of regularity in Strichartz estimates. The challenge here is to overcome two main difficulties. The first one is the presence of a non-local fractional Laplacian operator. The second one is the presence of a singular weight in the nonlinearity. The greater part of this paper is devoted to the scattering of global solutions in H-s(R-N). The Lorentz spaces and the Strichartz estimates play crucial roles in our approach.