Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R3

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: {-(a + b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = vertical bar u vertical bar(p-1)u, x is an element of R-3, (0.1) u is an element of H-1 (R-3), u > 0, x is an element of R-3, where a, b > 0 are constants, 2 < p < 5 and V : R-3 -> 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 is an element of (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 {-(epsilon(2)a + epsilon b integral(R3) vertical bar Du vertical bar(2)) Delta u + V(x)u = f(u), x is an element of R-3, u is an element of H-1 (R-3), u > 0, x is an element of R-3, where epsilon > 0 is a parameter, V(x) is a positive continuous potential and f (u) similar to vertical bar u vertical bar(p-1)u 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 Schrodinger-Poisson system with pure power nonlinearities, to the Kirchhoff type problem. (C) 2014 Elsevier Inc. All rights reserved.