Ground State Solution And Ground State Normalized Solution For Schr(?)dinger-Korteweg-de Vries System

This paper mainly studies the following Schrodinger-Korteweg-de Vries system where N ≤3,β∈R and Vi(x)is potential function,i=1,2.When Vi(x)are different functions,by using the variational method,we obtain the existence of ground state solution and the ground state normalized solution for the system(0.0.1).Firstly,we consider the case where Vi(x)is an asymptotically periodic poten-tial.We use Lions lemma to overcome the lack of compactness of the Palais-Smale sequence,and then apply Nehari manifold and Ekeland variational principle to ob-tain the existence of nontrivial ground state solutions for the system(0.0.1)with periodic potential,the existence of nontrivial ground state solutions for the system(0.0.1)with asymptotic periodic potential is further proved.Secondly,assume that Vi(x)=λi∈R,we consider the following system satisfying the L2 normalization conditions∫RN|u|2dx=a,∫RN|v|2dx=b,where N=1,2,β∈R and a,b>0.By constrained minimization method and com-bining Lions lemma and Schwarz rearrangement,we obtain the existence of positive ground state normalized solution for the system(0.0.2).