Multiple solutions for a Kirchhoff equation with critical growth

We consider the problem - m ∫ Ω | ∇ u | 2 d x Δ u = λ f ( x , u ) + μ | u | 2 ∗ - 2 u , x ∈ Ω , u ∈ H 0 1 ( Ω ) , where Ω ⊂ R N , N ≥ 3 , is a bounded smooth domain, 2 ∗ = 2 N / ( N - 2 ) , λ , μ > 0 and m is an increasing positive function. The function f is odd in the second variable and has superlinear growth. In our first result we obtain, for each k ∈ N , the existence of k pairs of nonzero solutions for all μ > 0 fixed and λ large. Under weaker assumptions of f , we also obtain a similar result if N = 3 , λ > 0 is fixed and μ is close to 0. In the proofs, we apply variational methods.