KIRCHHOFF TYPE PROBLEMS INVOLVING P -BIHARMONIC OPERATORS AND CRITICAL EXPONENTS∗

In this paper, we study the existence of solutions for a class of Kirchhoff type problems involving p-biharmonic operators and critical exponents. The proof is essentially based on the mountain pass theorem due to Ambrosetti and Rabinowitz [2] and the Concentration Compactness Principle due to Lions [18,19].


Introduction and Preliminaries
In this paper, we are interested in the existence of solutions for the following Kirchhoff type problem M Ω |∆u| p dx ∆ |∆u| p−2 ∆u = λf (x, u) + |u| p * * −2 u, x ∈ Ω, u = ∂u ∂ν = 0, x ∈ ∂Ω, where Ω is a bounded domain in R N with C 2 boundary, N ≥ 2, ∆ is the Laplace operator and ∂u ∂ν is the outer normal derivative, λ is a positive parameter, f : Ω × R → R is a Carathéodory function, p * * is the critical exponent, i.e.
Since problem (1.1) contains integral over Ω, it is no longer a pointwise identity; therefore it is often called nonlocal problem. This problem models several physical and biological systems, where u describes a process which depends on the average of itself, such as the population density, see [10,17]. Kirchhoff type problems for p-Laplace operators have been studied by many mathematicians, see [4,5,9,13,14,21,22,24]. In a recent paper [11], Colasuonno and Pucci have introduced ppolyharmonic operators. Using variational methods, they studied the multiplicity of solutions for a class of p(x)-polyharmonic elliptic Kirchhoff equations. In [20], V.F. Lubyshev studied the existence of solutions for an even-order nonlinear problem with convex-concave nonlinearity. In [6], the author extended the previous results in [11] to nonlocal higher-order problems. Althought the Kirchhoff function M (t) in [6,11] was assumed to be degenerate at zero, the nonlinearity is not critical. Finally, we refer the readers to two related interesting papers [7,25]. In [7], G. Autuori et al. considered a class of Kirchhoff type problems involving a fractional elliptic operator and a critical nonlinearity while in [25], L. Zhao et al. studied the existence of solutions for a higher order Kirchhoff type problem with exponential critical growth.
In this paper, we study the existence of solutions for Kirchhoff type problems involving p-biharmonic operators and critical exponents. We are motivated by the results introduced in [1, 6,11] and some papers on p-biharmonic operators with critical exponents [3,8,12,15,16,23]. We also get a priori estimates of the obtained solution. We believe that this is the first contribution to the study of the existence of solutions for Kirchhoff type problems involving p-biharmonic operators with critical exponents. Due to the presence of the critical exponents, the problem considered here is lack of compactness. To overcome this difficulty, we use the Concentration Compactness Principle due to Lions [18,19]. The existence of a nontrivial solution is then obtained by the mountain pass theorem due to Ambrosetti and Rabinowitz in [2].
In order to state the main result, we assume that f : Ω×R → R is a Carathéodory function satisfying the following conditions: ∈ Ω × R, where p < q < p * * and p * * is defined by (1.2); |t| p−1 = 0 uniformly for x ∈ Ω; (F 3 ) There exists θ ∈ (p, p * * ) such that Let W 2,p 0 (Ω) be the usual Sobolev space with respect to the norm We then have that W 2,p 0 (Ω) is continuously and compactly embedded into the Lebesgue space L r (Ω) endowed the norm |u| r = Ω |u| r dx 1 r , 2 < r < p * * . Denote by S r the best constant for this embedding, that is, S r |u| r ≤ u for all u ∈ W 2,p 0 (Ω). Definition 1.1. We say that u ∈ W 2,p 0 (Ω) is a weak solution of problem (1.1) if The main result of this paper can be stated as follows.

Proof of the main result
Here we are assuming, without loss of generality, that the Kirchhoff function M (t) is unbounded. Contrary case, the truncation on M (t) is not necessary. Since we are intending to work with N ≥ 2, we shall make a truncation on M as follows.
As we shall see, the proof of Theorem 1.1 is based on a careful study of the solutions of the following auxiliary problem where f, N, p, λ are as in Section 1. We shall prove the following auxiliary result. We recall that u ∈ W 2,p 0 (Ω) is a weak solution of problem for all v ∈ W 2,p 0 (Ω). Hence, we shall look for nontrivial solutions of (2.3) by finding critical points of the C 1 − functional I a,λ : W 2,p 0 (Ω) → R given by the formula for all u, v ∈ W 2,p 0 (Ω). We say that a sequence {u n } ⊂ W 2,p 0 (Ω) is a Palais-Smale sequence for the functional I a,λ at level c ∈ R if where (W 2,p 0 (Ω)) * is the dual space of W 2,p 0 (Ω). If every Palais-Smale sequence of I a,λ has a strong convergent subsequence, then one says that I a,λ satisfies the Palais-Smale condition ((P S) condition for short).
Lemma 2.4. Let {u n } ⊂ W 2,p 0 (Ω) be a sequence such that Then {u n } is bounded.
Proof. Assuming by contradiction that {u n } is not bounded in W 2,p 0 (Ω), up to a subsequence, we may assume that u n → +∞ as n → ∞. It follows from (2.1), (M 0 ) and (F 3 ) that for n large enough where C 4 is a positive constant. Since a < m0 p θ and θ < p * * , the sequence {u n } is bounded.
Using the Concentration Compactness Principle due to Lions [18,19], if |∆u n | p µ, |u n | p * * ν weakly- * in the sense of measures, where µ and ν are bounded nonnegative measures on R N , then there exist an at most countable set J, sequences (x j ) j∈J ⊂ Ω and (ν j ) j∈J , (µ j ) j∈J nonnegative numbers such that for all j ∈ J, where δ xj is the Dirac mass at x j ∈ Ω. Now, we claim that J = ∅. Arguing by contradiction, assume that J = ∅ and fix j ∈ J. For > 0, consider φ j, ∈ C ∞ (R N ) such that φ j, ≡ 1 in B (x j ), φ j, ≡ 0 on Ω\B 2 (x j ) and |∇φ j, | ∞ ≤ 2C , and |∆φ j, | ≤ 2C 2 , where x j ∈ Ω belongs to the support of ν. Since {φ j, u n } is bounded in the space W 2,p 0 (Ω), it then follows from (2.1) that I a,λ (u n )(φ j, u n ) → 0 as n → ∞, that is, (2.14) Using the Hölder inequality and the boundedness of the sequence {u n }, we have Ω |∆u n | p−2 ∆u n (∇u n ∇φ j, ) dx  On the other hand, by (F 2 ) and the boundedness the sequence {u n } in W 2,p 0 (Ω) we also have lim From (2.14)-(2.18), letting n → ∞, we deduce that Letting → 0 and using the standard theory of Radon measures, we conclude that ν j ≥ M a (t p 1 )µ j ≥ m 0 µ j . Using (2.13) we have where S is given by (2.5). Now, we shall prove that (2.19) cannot occur, and therefore the set J = ∅. Indeed, arguing by contradiction, let us suppose that ν j ≥ S N 2p for some j ∈ J. Since {u n } is a (P S) c a,λ sequence for the functional I a,λ , from the conditions (F 3 ) and (M 0 ), and m 0 < a < θ p m 0 we have c a,λ = I a,λ (u n ) − 1 θ I a,λ (u n )(u n ) + o n (1) We also have u n (x) → u(x) a.e. x ∈ Ω as n → ∞, so by the condition (F 1 ) and the Dominated Convergence Theorem, we deduce that On the other hand, by (2.12), and the boundedness of u n −u we have I a,λ (u n )(u n − u) → 0 as n → ∞, that is, − Ω |u n | p * * −2 u n (u n − u) dx → 0 as n → ∞. and Ω |u n | p * * −2 u n (u n − u) dx ≤ Ω |u n | p * * −1 |u n − u| dx ≤ |u n | p * * −1 L p * * (Ω) |u n − u| L p * * (Ω) → 0 as n → ∞. By standard arguments, we can show that {u n } converges stronlgy to u in W 2,p 0 (Ω). This completes the proof of Theorem 2.1. Now, we are in the position to prove Theorem 1.1.