On a fourth-order elliptic Kirchho type problem with critical Sobolev exponent

This work is concerned with a class of fourth-order elliptic Kirchho type problems involving the critical term. By means of the truncation and the concentration compact argument, for each positive integer k, the existence of k pairs nontrivial solutions is established.


Introduction and main result
Let Ω be a smooth bounded domain of R N , with N ≥ 5. Consider the problem where λ > 0, 2 := 2N N −4 and M, f are continuous functions satisfying some hypothesis which will be given later.
The presence of the nonlocal term M Ω |∇u| 2 dx in (1) causes some mathematical diculties and so the study of such a class of problems is of much interest. This type of problems are closely related to the following hyperbolic equation which was proposed by Kirchho [9] as a model to describe the transversal vibrations of a stretched string by considering the subsequent change in string length during the vibrations. Recently, the Kirchho type problems with or without critical growth have been investigated by many researchers, we cite here [1,2,5,6,7,8,10,13,14,16]. In [17], Wand and An have used the mountain pass theorem to prove the existence of solutions for the problem A more general problem was considered in [4,12]. Particularly, the critical case is studied in our previous paper [8]. By the concentration compactness principle of Lions [11] and the ideas of Brezis and Nirenberg [3], sucient conditions were obtained to the existence of a least one nontrivial solution of the perturbed problem (1) for λ large enough. To our knowledge, the existence of multiple solutions for problem (1) has not studied until now. Motivated by the above results, in this note we are interested in nding multiple solutions by using the variational method, the truncation technique and the concentration compact argument. Throughout the paper, we assume the following conditions on the Kirchho function and the nonlinearity: ( (f 3 ) There exists q ∈ (2, 2 ) such that lim |t|→+∞ f (x, t) |t| q−2 t = 0, uniformly in Ω; (f 4 ) There exists θ ∈ (2, 2 ) such that The main result is the following theorem. Theorem 1.1. Suppose that (m 1 ) and (f 1 ) − (f 4 ) hold. Then for each positive integer k, there exists λ k > 0 such that problem (1) admits a least k pairs nontrivial solutions provided that λ ≥ λ k . .

Auxiliary Results
We look for solutions in the Hilbert space Replacing M with M a , problem (1) turns into The energy functional associated to (3) is given by where M a (t) = t 0 M a (s)ds. By the above assumptions, I λ,a ∈ C 1 (H) and for all u, v ∈ H This shows that {u n } is bounded in H. Then up to subsequence, for some u ∈ H, u n u in H, u n → u a.e. in Ω, |∆u n | 2 µ weakly in the sense of measures, |u n | 2 ν weakly in the sense of measures, where µ and ν are nonnegative bounded measures on Ω. Applying concentration compact result due to Lions [11], we can nd at most countable index set J and elements {x j } j∈J of Ω such that We claim that ν j ≥ S N 4 for all j ∈ J. Let j ∈ J be xed and for an arbitrary Observe that Set A 1 n,ε := Ω u n ∇u n ∇φ ε dx, A 2 n,ε := Ω ∆u n ∇u n ∇φ ε dx, A 3 n,ε := Ω u n ∆u n ∆φ ε dx.
By the Hölder inequality, we have where w N is the volume of B(0, 1). In the same way → 0 as ε → 0 and lim sup By (4), (6) - (7), continuity of f and From (8)-(10), we see that Letting ε → 0 in (11), we obtain µ j ≤ ν j . Therefore (5) implies S N 4 ≤ ν j . Now we prove that J is empty. Assume by contradiction that there is some j ∈ J. Then which is impossible and hence J = ∅. It follows that u n → u in L 2 (Ω), thus On the other hand, it not dicult to see that Since I (u n ), u n − u = o n (1), by continuity of M , (7) and (12)-(13) we deduce that Similarly, we also obtain lim n→∞ Ω ∆u∆(u n − u)dx = 0.
So that ∆u n → ∆u in L 2 (Ω). From this and (7) we conclude that ||u n || H → ||u|| H . Finally u n → u in H.

Proof of Theorem 1.1
To this end, we need to ensure that I λ,a satises the conditions of the following version of Symmetric Mountain Pass theorem [15]. (ii) there exists a subspace E ⊂ H such that dimV < dimE and max u∈E I(u) ≤ β for some β > 0; (iii) the functional I satises (P S) c for every c ∈ (0, β).
Then I admits at least dimE − dimV pairs nontrivial critical points.
Proof. By (f 2 ) − (f 3 ) and the continuity of f, for any ε > 0 there is C ε > 0 such that It follows from Sobolev's embeddings that Since 2 < q < 2 , the desired result follows by choosing ε small enough. Lemma 3.2. Assume that (f 1 ) − (f 4 ) hold. Then, for each positive integer k and β > 0, there exists λ k > 0 such that for any λ ≥ λ k , there is a k-dimensional subspace E k,λ ⊂ H satisfying max u∈E k,λ I(u) < β.