In this talk I will focus on the asymptotic behavior of unbounded radial solutions of semilinear Schr\"odinger equations with a barely supercritical nonlinearity (i.e a nonlinearity that grows faster than the critical power but not faster than a logarithm). It is known that we have scattering of bounded radial solutions of defocusing loglog energy-supercritical Schr\"odinger equations. I will recall the techniques used to prove this result. Then I will explain how we can use Jensen-type inequalities to prove scattering of unbounded radial solutions of defocusing loglog energy-supercritical Schr\"odinger equations and unbounded radial solutions below ground state of focusing size-dependent log energy-supercritical Schr\"odinger equations.