Multiplicity of solutions for Kirchhoff type equations with critical growth in ℝ N

Abstract

In this paper, we consider the following Kirchhoff type problem with critical growth: { − ( a + b ∫ R N | ∇u | 2 d x ) Δu = λk ( x ) | u | q − 2 u + μ | u | 2 ∗ − 2 u , x ∈ R N , u ∈ D 1 , 2 ( R N ) , where N ≥ 3 , a ≥ 0 , b>0, 1<q<2, λ , μ > 0 , 2 ∗ = 2 N N − 2 is the critical Sobolev exponent and k ( x ) ∈ L 2 ∗ / ( 2 ∗ − q ) ( R N ) is a positive function. Using variational methods and the concentration-compactness principle, we demonstrate the existence and multiplicity of solutions.