Ground State Solutions of Critical Nonlinear Systems

Since the prediction of Bose-Einstein Condensation (abbreviated as BEC) had been validated in experiments 20 years ago, the problems related to BEC have attracted much more attentions. This book considers 3 kinds of critical nonlinear Schrödinger systems (abbreviated as NLSS), that is, usual NLSS, NLSS with magnetic fields and fractional NLSS. There is no compactness as the nonlinear terms in the three NLSS are critical. We mainly rescale and cut off the extremal functions of the corresponding limiting problems, by invariance of rescaling with respect to "gradient-norm" and "critical-norm" and a series of energy estimations to overcome the lack of compactness. Applying variational principle, methord of Nehari manifold, mountain-pass theorem, concentration-compactness principle and maximum principle et al, under proper conditions, the existence results of ground state solutions to the above three NLSS are established.