Existence and Multiplicity of Positive Solutions for Kirchhoff-Type Equations with the Critical Sobolev Exponent

which was proposed by Kirchhoff in [6]. In fact, (3) is an extension and generalization of the classical D’Alembert wave equation in some ways. .is model is widely used in many fields, such as non-Newtonian mechanics, cosmophysics, elastic theory, and electromagnetics. It is worth noting that equation (2) has a nonlocal term 􏽒L0 |zu/zx| 2dx. Only after Lions [7] proposed an abstract functional analysis framework about the following equation:

Over the past decades, the following Kirchhoff equation: has been extensively considered. With various assumptions about the nonlinearity g(x, u), the existence and multiplicity of solutions for system (2) are obtained by variational methods, see [1][2][3][4][5] and the references therein. To our best knowledge, system (2) is related to the stationary analogue of the equation which was proposed by Kirchhoff in [6]. In fact, (3) is an extension and generalization of the classical D'Alembert wave equation in some ways. is model is widely used in many fields, such as non-Newtonian mechanics, cosmophysics, elastic theory, and electromagnetics. It is worth noting that equation (2) has a nonlocal term L 0 |zu/zx| 2 dx. Only after Lions [7] proposed an abstract functional analysis framework about the following equation: problem (4) received much attention, and we refer the readers to [8][9][10][11][12][13][14][15] for more details and the references therein. More precisely, Bisci and Pizzimenti [13] studied the existence of infinitely many solutions for a class of Kirchhofftype problems involving the p-Laplacian by using variational methods. In [15], Bisci considered the existence of (weak) solutions for some Kirchhoff-type problems on a geodesic ball of the hyperbolic space and the main technical approach is based on variational and topological methods. In [16], the authors firstly used the variation method to study the existence of positive solution of the Kirchhoff-type problems with the Sobolev critical exponent. After that, there are many works on the existence and multiplicity of solutions for Kirchhoff-type problems with the Sobolev critical exponent (one can see [17][18][19][20][21] and the references therein).
In [17], the author considered the following Kirchhofftype elliptic equation: and by using the variation method, the existence of positive solutions of system (5) is obtained. To our best knowledge, a nonlinear elliptic boundary value problem has a critical term, which is a difficulty to prove the existence of solutions for the problem. e difficulty is caused by the lack of compactness of the embedding H 1 0 (Ω) ↪ L 2 * (Ω), which makes the PS condition cannot be checked directly. In [16], the authors make the parameter μ large enough to make a critical value below a certain level. In [17], under the AR condition, the authors restored the compactness of the embedding by using the second concentration compactness lemma, which is an extension of the work in [8].
In [18], the authors used the variational method to consider system (5) with μ � 1 and the existence and multiplicity of solutions for the system are obtained.
Moreover, problems on the unbounded domain R N have also been widely studied by some researchers, for example, [22][23][24][25][26]. More precisely, in [25], Liu and He used the variant version of fountain theorem to get the existence of infinitely many high energy solutions of the system. In [26], the authors studied the concentration behavior of positive solutions. For more information about this problem, we refer the readers to [11,20,27,28] and the reference therein.
Motivated by the above facts, we want to consider the positive solutions of system (1). By using the mountain pass theorem and Brézis-Lieb lemma, the existence and multiplicity of positive solutions of system (1) are obtained.
To show our main results, we introduce some conditions on nonlinearity g(x, u) and V(x).
(V1) V(x) is 1-periodic in each of x i (i � 1, 2, . . . , N), and there exists a positive constant V 0 such that where In the next section, we will present our main results.

then system (1) has a positive ground state solution.
Remark 1. In [17], the author used a condition which is stronger than (F3), that is, Theorem 2. Let N � 4 and 0 < μ < bS 2 . If (G1) and (G2) hold, then there exists a constant λ * > 0 (we will give in the proof of eorem 2 in Section 3) such that ∀λ > λ * , and system (1) has at least two positive solutions.
Remark 2. In reference [21], the authors considered system (1) as f(x) ≡ 1, 1 < q < 2, V(x) � 0, and μ � 1. Underlying the condition λ < λ 0 (a constant the authors given in their paper), two positive solutions are obtained. However, our results are very different from those in [21]. In our paper, 2 ≤ q < 2 * , and if λ > λ * (a constant we give in the proof of eorem 2) is sufficiently large, two positive solutions are obtained. Besides, in [21], N � 3; in our paper, for N � 4, the multiple solutions of higher dimensional space are obtained.
Remark 3. When a � 1, b � 0, and V(x) � 0, system (1) degenerates to a classical semilinear elliptic problem. eorem 2 can be the generalization of the corresponding results in [8] of Kirchhoff-type problems. e reminder of this paper is organized as follows. In Section 2, some preliminary results are presented. e proof of main results will be given in Section 3.

Preliminaries
In this paper, we make some notations as follows: 2 Mathematical Problems in Engineering (i) e space H 1 0 (Ω) (denoted by E) is equipped with the norm ‖u‖ � ( Ω |∇u| 2 dx) 1/2 , and we also define the equivalent norm by We say that if the functional satisfies (PC) c condition for any (PS) c sequence, it has a convergent subsequence.
Now, we give the energy functional corresponding to problem (1), that is, It is obvious that I ∈ C 1 (E, R) and has the following derivative: Using the continuity of g(x, u) and V(x), it shows that u ∈ E is a critical point of I, if it is a solution of problem (1). (1) ere exist constants ρ, α > 0, such that (2) ere exists u ∈ E such that Proof. From (F1) and (F2), it shows that there has a constant C 1 > 0 such that By (10), (15), (V1), and Sobolev inequality, it follows that (1) Taking ρ > 0 small enough, there exists a constant α > 0 such that (2) If we take v 0 ∈ H 1 0 (Ω) and v 0 ≡ 0, then one gets the following: Since ‖·‖ and the ‖·‖ E are equivalent, then It is obvious that erefore, we can find a positive constant t 0 , and ‖t 0 v 0 ‖ > ρ, such that Let u � t 0 v 0 and the conclusion is satisfied.
then I satisfies the (PS) c condition.

Mathematical Problems in Engineering
Proof. By (F1) and (F2), there exists a constant C 5 > 0, such that ∀(x, s) ∈ Ω × R, and it has 1 5 g(x, s)s − G(x, s) ≤ 1 30 Suppose u n is a (PS) c sequence, c ∈ (0, Λ), e next work is to prove the boundness of u n . Clearly, one has at is to say ‖u n ‖ is bounded in E. Going necessary to a subsequence, it has u n ⇀u, By (V1) and (F2), one has Let v n � u n − u, then we can claim ‖v n ‖ ⟶ 0 as n ⟶ ∞. Otherwise, there exist a subsequence (for convenience, we still denote it by v n ) such that where l > 0; then By using Brézis-Lieb lemma in [29], it has Because I ′ (u n ) ⟶ 0 in (E) * , one has which shows that It also has lim n⟶∞ I ′ u n , u � a‖u‖ 2 + bl‖u‖ 2 + b‖u‖ 4 Combining (32), (33), and (10), one gets which implies that By (35), (36) From (34) and (37), it has From the above inequality, it has On the other hand, from (F3) and (33), which deduce bl 3 which is a contradiction with (39). So l � 0, that is to say, u n ⟶ u in E as n ⟶ ∞. us, I satisfies the (PS) c condition. Proof. If u n ⊂ H 1 0 (Ω) is a (PS) c sequence of I, that is, I u n ⟶ c, I ′ u n ⟶ 0(n ⟶ ∞).

Proof of Main Results
Now we will prove eorem 1 and eorem 2.
e above inequality can deduce that there exists a constant C > 0, such that ‖u‖ ≥ C, ∀u ∈ M. (66) We claim that if C 8 > 0, such that Ω u + 6 dx ≥ C 8 , ∀u ∈ M.
Otherwise, we assume that u n ∈ M, such that In addition, we can calculate which is a contradiction. erefore, the assertion is established. erefore, ∀u ∈ M, we have