Multiple positive solutions for the critical Kirchhoff type problems involving sign-changing weight functions

Abstract

This paper is concerned with the multiplicity of positive solutions for the critical Kirchhoff type problems involving indefinite weight functions(0.1){−M(∫Ω|∇u|2dx)Δu=Qλ(x)|u|q−2u+K(x)|u|2⁎−2u,x∈Ω,u=0,x∈∂Ω, where Ω is a smooth bounded domain in RN(N≥3), 1<q<2, M(s)=a+bsβ with β>0,a>0 and b>0, the weight functions Qλ and K are continuous and changing-sign. Using the Nehari manifold, fibering maps and Ljusternik-Schnirelmann category, we prove that at least two positive solutions for (0.1) exist provided that β=1 and 2⁎≥4. Furthermore, by the mountain pass theorem and Ekeland's variational principle, it is shown that (0.1) possesses at least three positive solutions whenever β>2N−2, including the case that β=1 and 2⁎<4. Our results generalize some recent results in the literature.