Existence of positive ground state solutions for Kirchhoff type problems

In this paper, we study the existence of positive ground state solutions for the nonlinear Kirchhoff type problem {−(a+b∫R3|∇u|2)△u+V(x)u=f(u)in  R3,u∈H1(R3),u>0in  R3, where a,b>0a,b>0 are constants, f∈C(R,R)f∈C(R,R) is subcritical near infinity and superlinear near zero and satisfies the Berestycki–Lions condition. By using an abstract critical point theorem established by Jeanjean and a new global compactness lemma, we show that the above problem has at least a positive ground state solution. Our result generalizes the results of Li and Ye (2014) concerning the nonlinearity f(u)=|u|p−1uf(u)=|u|p−1u with p∈(2,5)p∈(2,5).