Multiple positive solutions for Kirchhoff-type equations involving critical-concave nonlinearities

Abstract

In this paper, we research the following Kirchhoff type problem with critical growth (1) 0, & u\\in H^1(\\mathbb{R}^{3}), \\end{cases} \\end{equation}$$]]> { − ( a + b ∫ R 3 | ∇ u | 2 d x ) Δ u + u = λ f ( x ) | u | q − 2 u + | u | 4 u , x ∈ R 3 , u > 0 , u ∈ H 1 ( R 3 ) , where 1<q<2, a>0 is a positive constant, 0 $ ]]> b , λ > 0 are real parameters and f ∈ L 6 6 − q ( R 3 ) is a nonzero nonnegative function. We prove the problem has at least two positive solutions by using concentration compactness principle and variational method when b is sufficiently small. Moreover, we also explore the asymptotic behaviour of the positive solutions as b n → n 0 + .