Scattering above energy norm of a focusing size-dependent log energy-supercritical Schrödinger equation with radial data below ground state

Given $$n \in \{ 3,4,5 \}$$ and $$k > 1$$ (resp. $$ \frac{4}{3}> k ) if 3,4 $$n=5$$ ), we prove scattering of the radial $${\tilde{H}}^{k} := {\dot{H}}^{k} ({\mathbb {R}}^{n}) \cap {\dot{H}}^{1} {R}}^{n})-$$ solutions focusing log energy-supercritical Schrödinger equation i \partial _{t} u + \triangle = -|u|^{\frac{4}{n-2}} \log ^{\gamma } ( 2 |u|^{2})$$ for a range positive $$\gamma \, s$$ depending on size initial data, critical energies below ground states’, potential below. In order to control barely supercritical nonlinearity in virial identity estimate growth energy nonsmooth solutions, i.e with data $${\tilde{H}}^{k}$$ , \le \frac{n}{2}$$ some Jensen-type inequalities, spirit Roy (Int Math Res Not 2020(8):2501–2541, 2020).