Long‐time dynamics for the radial focusing fractional INLS.

We consider the following fractional NLS with focusing inhomogeneous power‐type nonlinearity: i∂tu−(−Δ)su+|x|−b|u|p−1u=0,(t,x)∈ℝ×ℝN,$$ i{\partial}_tu-{\left(-\Delta \right)}&#x0005E;su&#x0002B;{\left&#x0007C;x\right&#x0007C;}&#x0005E;{-b}{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-1}u&#x0003D;0,\kern0.30em \left(t,x\right)\in \mathrm{\mathbb{R}}\times {\mathrm{\mathbb{R}}}&#x0005E;N, $$where N≥2$$ N\ge 2 $$, 1/2<s<1$$ 1/2<s<1 $$, 0<b<2s$$ 0<b<2s $$, and 1+2(2s−b)N<p<1+2(2s−b)N−2s$$ 1&#x0002B;\frac{2\left(2s-b\right)}{N}<p<1&#x0002B;\frac{2\left(2s-b\right)}{N-2s} $$. 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 Hs(ℝN)$$ {H}&#x0005E;s\left({\mathrm{\mathbb{R}}}&#x0005E;N\right) $$. The Lorentz spaces and the Strichartz estimates play crucial roles in our approach. [ABSTRACT FROM AUTHOR]