The initial value problem for some coupled nonlinear Schrödinger equations in two space dimensions with exponential growth is investigated. In the defocusing case, global well-posedness and scattering are obtained. In the focusing sign, global and nonglobal existence of solutions are discussed via potential well-method.
"Well-posedness issues for a class of coupled nonlinear Schrödinger equations with critical exponential growth." Adv. Oper. Theory 3 (3) 551 - 581, Summer 2018. https://doi.org/10.15352/aot.1709-1227