Ground State Solutions For Schr?odinger-KdV System

In this paper,by using the variational method,we study the following Schrodinger-KdV system where N=1,2,3,β∈R and λi(x)is a potential function,i=1,2.Firstly,we study system(0.0.1)with constraint potential,namely where N=1,2,3,/β∈R.And we assume that λi(x)satisfies the following condi-tions:(?),lim|x|→∞λi(x)=∞,i=1,2.By using the variational methods,the Nehari manifold and some analysis techniques,we obtained the existence result of nontrivial ground state solutions for system(0.0.2).Secondly,we consider the ground state solution of asymptotically autonomous system(0.0.1),namely.where N=1,2,3,β∈R.And we assume that λi(x)∈C(RN,R)satisfies the following conditions:We proved a pair of nontrivial ground state solutions of system(0.0.3)by using the variational methods,the Nehari manifold and Decomposition of bounded Palais-Smale sequence.Finally,we study the following Schrodinger-KdV system:where N=1,2,3,β∈R and λi is constant.i=1,2.We obtained a nontrivial ground state solutions of system(0.0.4)by using the variational methods,the Nehari manifold and Concentrated compactness lemma.