Ground state sign-changing solutions for Kirchhoff type problems in bounded domains

Abstract

In the present paper, we consider the existence of ground state sign-changing solutions for a class of Kirchhoff-type problems(0.1){−(a+b∫Ω|∇u|2dx)△u=f(u),x∈Ω;u=0,x∈∂Ω, where Ω⊂RN is a bounded domain with a smooth boundary ∂Ω, N=1,2,3, a>0, b>0 and f∈C(R,R). Under some weak assumptions on f, with the aid of some new analytical skills and Non-Nehari manifold method, we prove that (0.1) possesses one ground state sign-changing solution ub, and its energy is strictly larger than twice that of the ground state solutions of Nehari-type. Furthermore, we establish the convergence property of ub as the parameter b↘0. Our results improve and generalize some results obtained by W. Shuai (2015) [34].