Multiplicity of solutions for Kirchhoff type equations involving critical Sobolev exponents in high dimension

In this paper, we study the following Kirchhoff‐type elliptic problem− ( a + b ∫ Ω | ∇ u | 2 dx ) Δ u = λ u q − 1 + μ u2 ∗ − 1 , u > 0in Ω ,u = 0 ,on ∂ Ω ,where Ω ⊂ R N ( N ≥ 4 ) is a bounded domain with smooth boundary ∂ Ω, a , b , λ , μ > 0 and 1 < q < 2 ∗ =2 N /( N − 2). When N = 4, we obtain that there is a ground state solution to the problem for q ∈(2,4) by using a variational methods constrained on the Nehari manifold and also show the problem possesses infinitely many negative energy solutions for q ∈(1,2) by applying usual Krasnoselskii genus theory. In addition, we admit that there is a positive solution to the equations for N ≥5 under some suitable conditions. Copyright © 2016 John Wiley & Sons, Ltd.