In this paper we study the multiplicity of non-trivial solutions to a class of nonlinear boundaryvalue problems of Kirchhoff type. We prove existence results when the problem has nonlinearities with subcritical and with critical Caffarelli-Kohn-Nirenberg exponent.Problem (1.1) with a = b = δ = β = 0 and p = 2, that is,is called non-local because of the presence of the term M ( Ω |∇u| 2 dx), which implies that (1.2) is no longer a pointwise identity. This phenomenon causes some mathematical difficulties, which makes the study of such a class of problem particularly interesting. Besides this, this class of problem has a physical motivation. Indeed, the operator M ( Ω |∇u| 2 dx)Δu appears in the Kirchhoff equation, which arises in nonlinear vibrations, namely,Proof . Fix k ∈ N and let X k be a k-dimensional subspace of D 1,p a . Thus, there exists C(k) > 0 such that C(k) u r C 1 u r L r (Ω,|x| −δ ) for all u ∈ X k . Consideringρ > 0 such that 0 < u =ρ and 0 < u p < t 1 , we obtain that J λ (u) = I λ (u). Arguing as in the proof of Theorem 1.1, we can take R > 0 such thatfor all u ∈ S = {u ∈ X k : u = s}, with 0 < s < min{R,ρ}. Hence, S ⊂ J −ε and, since J −ε is symmetric and closed, from Corollary 2.3,We define now, for each k ∈ N, the sets Γ k = {C ⊂ D 1,p a \{0} : C is closed, C = −C and γ(C) k}, K c = {u ∈ D 1,p a : J λ (u) = 0 and J λ (u) = c} and the number c k = inf C∈Γ k sup u∈C J λ (u).Lemma 4.6. Given k ∈ N, the number c k is negative.Proof . From Lemma 4.5, for each k ∈ N there exists ε > 0 such that γ(J −ε ) k. Moreover, 0 / ∈ J −ε and J −ε ∈ Γ k . On the other hand,The next lemma allows us to prove the existence of critical points of J.Lemma 4.7. If c = c k = c k+1 = · · · = c k+r for some r ∈ N, then γ(K c ) r + 1 for all λ ∈ (0, λ * ).
