Multiple solutions of Kirchhoff type equations involving Neumann conditions and critical growth

In this paper, we consider a Neumann problem of Kirchhoff type equation $ \begin{equation*} \begin{cases} -\left(a+b\int_{\Omega}|\nabla u|^2dx\right)\Delta u+u = Q(x)|u|^4u+\lambda P(x)|u|^{q-2}u, &\rm \mathrm{in}\ \ \Omega, \\ \frac{\partial u}{\partial v} = 0, &\rm \mathrm{on}\ \ \partial\Omega, \end{cases} \end{equation*} $ where $ \Omega $ $ \subset $ $ \mathbb{R}^3 $ is a bounded domain with a smooth boundary, $ a, b > 0 $, $ 1 0 $ is a real parameter, $ Q(x) $ and $ P(x) $ satisfy some suitable assumptions. By using the variational method and the concentration compactness principle, we obtain the existence and multiplicity of nontrivial solutions.

In recent years, the following Dirichlet problem of Kirchhoff type equation has been studied extensively by many researchers which is related to the stationary analogue of the equation proposed by Kirchhoff in [13] as an extension of the classical D'Alembert's wave equation for free vibrations of elastic strings. Kirchhoff's model takes into account the changes in length of the string produced by transverse vibrations. In (1.2) and (1.3), u denotes the displacement, b is the initial tension and f (x, u) stands for the external force, while a is related to the intrinsic properties of the string (such as Youngs modulus). We have to point out that such nonlocal problems appear in other fields like biological systems, such as population density, where u describes a process which depends on the average of itself (see Alves et al. [2]). After the pioneer work of Lions [18], where a functional analysis approach was proposed. The Kirchhoff type Eq (1.2) with critical growth began to call attention of researchers, we can see [1,9,14,17,23,24,28,30] and so on. Recently, the following Kirchhoff type equation has been well studied by various authors There has been much research regarding the concentration behavior of the positive solutions of (1.4), we can see [10-12, 25, 33]. Many papers studied the existence of ground state solutions of (1.4), for example [5,8,15,16,21,22,24]. In addition, the authors established the existence of sign-changing solutions of (1.4) in [20,31]. In papers [27,32] proved the existence and multiplicity of nontrivial solutions of (1.4) by using mountain pass theorem.
In particular, Chabrowski in [6] studied the solvability of the Neumann problem where Ω ⊂ R N is a smooth bounded domain, 2 * = 2N N−2 (N ≥ 3) is the critical Sobolev exponent, λ > 0 is a parameter. Assume that Q(x) ∈ C(Ω) is a sign-changing function and Ω Q(x)dx < 0, under the condition of f (x, u). Using the space decomposition H 1 (Ω) = span1 ⊕ V, where V = {v ∈ H 1 (Ω) : Ω vdx = 0}, the author obtained the existence of two distinct solutions by the variational method.
In [14], Lei et al. considered the following Kirchhoff type equation with critical exponent where Ω ⊂ R 3 is a smooth bounded domain, a, b > 0, 1 < q < 2, λ > 0 is a parameter. They obtained the existence of a positive ground state solution for 0 ≤ β < 2 and two positive solutions for 3 − q ≤ β < 2 by the Nehari manifold method.
In [34], Zhang obtained the existence and multiplicity of nontrivial solutions of the following equation where Ω is an open bounded domain in R 3 , a, b > 0, 1 < q < 2, λ ≥ 0 is a parameter, f (x, u) and Q(x) are positive continuous functions satisfying some additional assumptions. Moreover, f (x, u) ∼ |u| p−2 u with 4 < p < 6. Comparing with the above mentioned papers, our results are different and extend the above results to some extent. Specially, motivated by [34], we suppose Q(x) changes sign on Ω and f (x, u) ≡ 0 for (1.5). Since (1.1) is critical growth, which leads to the cause of the lack of compactness of the embedding H 1 (Ω) → L 6 (Ω), we overcome this difficulty by using P.Lions concentration compactness principle [19]. Moreover, note that Q(x) changes sign on Ω, how to estimate the level of the mountain pass is another difficulty.
Throughout this paper, we make use of the following notations: • The space H 1 (Ω) is equipped with the norm u 2 H 1 (Ω) = Ω (|∇u| 2 + u 2 )dx, the norm in L p (Ω) is denoted by · p .
. . denote various positive constants, which may vary from line to line.
• We denote by S ρ (respectively, B ρ ) the sphere (respectively, the closed ball) of center zero and radius ρ,

Proofs of theorems
In this section, we firstly show that the functional I λ (u) has a mountain pass geometry. Lemma 2.1. There exist constants r, ρ, Λ 0 > 0 such that the functional I λ satisfies the following conditions for each λ ∈ (0, Λ 0 ): There exists e ∈ H 1 (Ω) with e > ρ such that I λ (e) < 0.
Proof. (i) From (P 1 ), by the Hölder inequality and the Sobolev inequality, for all u ∈ H 1 (Ω) one has and there exists a constant C > 0, we get Hence, combining (2.1) and (2.2), we have the following estimate So we obtain I λ (tu) < 0 for every u 0 and t small enough. Therefore, for u small enough, one has Then we have the following compactness result. Lemma 2.2. Suppose that 1 < q < 2. Then the functional I λ satisfies the (PS ) c λ condition for every Proof. Let {u n } ⊂ H 1 (Ω) be a (PS ) c λ sequence for It follows from (2.1), (2.3) and the Hölder inequality that Therefore {u n } is bounded in H 1 (Ω) for all 1 < q < 2. Thus, we may assume up to a subsequence, still denoted by {u n }, there exists u ∈ H 1 (Ω) such that in Ω, as n → ∞. Next, we prove that u n → u strongly in H 1 (Ω). By using the concentration compactness principle (see [19]), there exist some at most countable index set J, δ x j is the Dirac mass at x j ⊂Ω and positive numbers {ν j }, {µ j }, j ∈ J, such that Moreover, numbers ν j and µ j satisfy the following inequalities For ε > 0, let φ ε, j (x) be a smooth cut-off function centered at x j such that 0 ≤ φ ε, j ≤ 1, |∇φ ε, j | ≤ 2 ε , and There exists a constant C > 0 such that Since |∇φ ε, j | ≤ 2 ε , by using the Hölder inequality and L 2 (Ω)-convergence of {u n }, we have where C 1 > 0, and we also derive that Noting that u n φ ε, j is bounded in H 1 (Ω) uniformly for n, taking the test function ϕ = u n φ ε, j in (2.3), from the above information, one has which shows that {u n } can only concentrate at points x j where Q(x j ) > 0. If ν j > 0, by (2.5) we get (2.6) From (2.5) and (2.6), we have (2.7) To proceed further we show that (2.7) is impossible. To obtain a contradiction assume that there exists