Multiple Positive Solutions to a Kirchhoff Type Problem Involving a Critical Nonlinearity in ℝ3

Abstract

Abstract In this paper, we investigate a class of Kirchhoff type problems in ℝ 3 {\mathbb{R}^{3}} involving a critical nonlinearity, namely, - ( 1 + b ∫ ℝ 3 | ∇ u | 2 d x ) △ u = λ f ( x ) u + | u | 4 u , u ∈ D 1 , 2 ( ℝ 3 ) , -\biggl{(}1+b\int_{\mathbb{R}^{3}}\lvert\nabla u|^{2}\,dx\biggr{)}\triangle u=% \lambda f(x)u+|u|^{4}u,\quad u\in D^{1,2}(\mathbb{R}^{3}), where b > 0 {b>0} , λ > λ 1 {\lambda>\lambda_{1}} and λ 1 {\lambda_{1}} is the principal eigenvalue of - △ ⁢ u = λ ⁢ f ⁢ ( x ) ⁢ u {-\triangle u=\lambda f(x)u} , u ∈ D 1 , 2 ⁢ ( ℝ 3 ) {u\in D^{1,2}(\mathbb{R}^{3})} . We prove that there exists δ > 0 {\delta>0} such that the above problem has at least two positive solutions for λ 1 < λ < λ 1 + δ {\lambda_{1}<\lambda<\lambda_{1}+\delta} . Furthermore, we obtain the existence of ground state solutions. Our tools are the Nehari manifold and the concentration compactness principle. This paper can be regarded as an extension of Naimen’s work [21].