Existence of positive ground state solutions for the nonlinear Kirchhoff type equations in R~3

摘要

In this paper, we study the following nonlinear problem of Kirchhoff type with pure power nonlinearities: 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 e (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 then by (4.1),there exists λ0 = 1/4b(a-1)C~3 > 0 such that for all λ≥λ0 and t ≥ 0，g(t）≥0，then g⑴≥0,i.e. Since 1< p ≤2,then the function h(t) = t~2+t~3 - t~(p+1) is nonnegative for all t ≥ 0 and vanishes only if t = 0. Hence u≡0. The proof is completed.