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

Abstract

Given n∈ { 3 , 4 , 5 } and k> 1 (resp. 43>k>1) if n∈ { 3 , 4 } (resp. n= 5), we prove scattering of the radial H~ k: = H˙ k(Rn) ∩ H˙ 1(Rn) - solutions of the focusing log energy-supercritical Schrödinger equation i∂tu+▵u=-|u|4n-2ulogγ(2+|u|2) for a range of positive γs depending on the size of the initial data, for critical energies below the ground states’, and for critical potential energies below. In order to control the barely supercritical nonlinearity in the virial identity and in the estimate of the growth of the critical energy for nonsmooth solutions, i.e solutions with data in H~ k, k≤n2, we prove some Jensen-type inequalities, in the spirit of Roy (Int Math Res Not 2020(8):2501–2541, 2020). © 2022, The Author(s), under exclusive licence to Springer Nature Switzerland AG.