Existence of a ground state and blow-up problem for a nonlinear Schrodinger equation with critical growth

TL;DR

This paper investigates the existence and non-existence of ground-state solutions for energy-critical nonlinear Schrödinger equations with perturbations across different dimensions, and demonstrates finite-time blow-up for certain initial data.

Contribution

It establishes the existence of ground states in dimensions four and higher with perturbations, and shows non-existence in three dimensions under small perturbations, along with blow-up results.

Findings

Existence of ground states in dimensions ≥4 with subcritical perturbations.

Non-existence of ground states in dimension 3 for small perturbations.

Finite-time blow-up for radially symmetric solutions below ground state energy.

Abstract

In this paper we show the existence of ground-state solutions for the energy-critical NLS perturbed with subcritical terms when the space dimension $d\geq4$. However in dimension three, we show that when the perturbation is small enough, then such solution does not exist. For the evolution equation, we show the existence of finite time blow up of solutions with radially symmetric data with energy below the one of the ground state.