On linearly coupled systems of Schrödinger equations with critical growth

Abstract

In thisworkwestudytheexistenceofgroundstatesforthefollowingclassofcoupled systems involvingnonlinearSchrödingerequations 8<: 􀀀 u + V1(x)u = f1(x; u) + (x)v;x 2 RN; 􀀀 v + V2(x)v = f2(x; v) + (x)u; x 2 RN; where thepotentials V1 : RN ! R, V2 : RN ! R are nonnegativeandrelatedwith the couplingterm : RN ! R by j (x)j < pV1(x)V2(x), forsome 0 < < 1. In the case N = 2, thenonlinearities f1 e f2 havecriticalexponentialgrowthinthesense of Trudinger-Moserinequality.Inthecase N 3, thenonlinearitiesarepolynomials with subcriticalandcriticalexponentintheSobolevsense.Westudyalsothefollowing class ofnonlocalcoupledsystems 8<: (􀀀 )1=2u + V1(x)u = f1(u) + (x)v;x 2 R; (􀀀 )1=2v + V2(x)v = f2(v) + (x)u; x 2 R; where (􀀀 )1=2 denotes thesquarerootoftheLaplacianoperatorandthenonlinearities havecriticalexponentialgrowth.Ourapproachisvariationalandbasedon minimization techniqueovertheNeharimanifold