This paper is concerned with a focusing nonlinear Schrödinger equation whose nonlinearity consists of the energy-critical local interaction term with a perturbation of the $L^2$-super and energy-subcritical term. We prove the existence of a ground state (= a standing-wave solution of minimal action) when the space dimension is four or higher and prove the nonexistence of any ground state when the space dimension is three and the perturbation is small. Once we have a ground state, a so-called potential-well scenario works well, so that we can give a sufficient condition for the nonexistence of global-in-time solutions.
Takafumi Akahori. Slim Ibrahim. Hiroaki Kikuchi. Hayato Nawa. "Existence of a ground state and blow-up problem for a nonlinear Schrödinger equation with critical growth." Differential Integral Equations 25 (3/4) 383 - 402, March/April 2012. https://doi.org/10.57262/die/1356012740