Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in [formula omitted

Abstract

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities:(0.1){−(a+b∫R3|Du|2)Δu+V(x)u=|u|p−1u,x∈R3,u∈H1(R3),u>0,x∈R3, where a,b>0 are constants, 2<p<5 and V:R3→R. Under certain assumptions on V, we prove that (0.1) has a positive ground state solution by using a monotonicity trick and a new version of global compactness lemma.Our main results especially solve problem (0.1) in the case where p∈(2,3], which has been an open problem for Kirchhoff equations and can be viewed as a partial extension of a recent result of He and Zou in [14] concerning the existence of positive solutions to the nonlinear Kirchhoff problem{−(ε2a+εb∫R3|Du|2)Δu+V(x)u=f(u),x∈R3,u∈H1(R3),u>0,x∈R3, where ε>0 is a parameter, V(x) is a positive continuous potential and f(u)∼|u|p−1u with 3<p<5 and satisfies the Ambrosetti–Rabinowitz type condition. Our main results extend also the arguments used in [7,33], which deal with Schrödinger–Poisson system with pure power nonlinearities, to the Kirchhoff type problem.