EXISTENCE OF MULTIPLE NONTRIVIAL SOLUTIONS FOR A p-KIRCHHOFF TYPE ELLIPTIC PROBLEM INVOLVING SIGN-CHANGING WEIGHT FUNCTIONS

This paper deals with a p-Kirchhoff type problem involving signchanging weight functions. It is shown that under certain conditions, by means of variational methods, the existence of multiple nontrivial nonnegative solutions for the problem with the subcritical exponent are obtained. Moreover, in the case of critical exponent, we establish the existence of the solutions and prove that the elliptic equation possesses at least one nontrivial nonnegative solution.

1. Introduction and main theorems.The purpose of this article is to investigate the existence of multiple nontrivial nonnegative solutions to the following nonlocal boundary value problem of the p-Kirchhoff type where ∆ p u = div(|∇u| p−2 ∇u), Ω is a bounded domain in R N with a smooth boundary ∂Ω, 1 < q < p < r ≤ p * where p * = N p N −p if N > p and p * = ∞ if N ≤ p, M (s) = as + b and the parameters a, b, λ > 0, the weight functions f , g satisfy the following conditions: 884 YUANXIAO LI, MING MEI AND KAIJUN ZHANG (A1) f , g ∈ C(Ω), and f ± = max{±f, 0} ≡ 0, g ± = max{±g, 0} ≡ 0; (A2) Ω f |u| q dx > 0 and Ω g|u| r dx > 0 for u ∈ W 1,p 0 (Ω) \ {0}.Such problems are called nonlocal problems because of the expression of M ( Ω |∇u| p dx), which implies that the equation contains an integral over Ω, and is no longer pointwise identities.In the case p = 2, if we replace λf (x)|u| q−2 u + g(x)|u| r−2 u by function h(x, u), the problem (1) reduces to the following nonlocal Kirchhoff elliptic problem in Ω, u = 0 on ∂Ω. ( This is related to the stationary analogue of the Kirchhoff problem such a model was first proposed by Kirchhoff [19] in 1883 to describe transversal oscillations of a stretched string, particularly, taking into account the subsequent change in string length caused by oscillations.Nonlocal problems also arise in other fields, for example, physical and biological systems where u describes a process which depends on the average of itself.For more details of background, we refer to [1,6,7]. The study of Kirchhoff type problems is one of hot spots in nonlocal partial differential equations.The first frame work was given by Lions [20].Since then, the study of Kirchhoff type problems have been paid more attention.In [24], Ma and Muñoz Rivera proved the existence of positive solutions for the Kirchhoff elliptic problem (2) by the variational method and minimization arguments, under some restrictions on M (s) and h(x, u).Subsequently, by the truncation argument and uniform a priori estimates of Gidas and Spruck type [15], Alves, Corrêa and Ma [2] proved the existence of positive solutions if the nonlinear h(x, u) satisfies the socalled Ambrosetti-Rabinowitz condition, where M (s) is nonincreasing and does not grow too fast in a suitable interval near zero.When M (s) is increasing, the existence of positive solutions is also obtained by Ma [25] and Perera and Zhang [27], where in [27] the nontrivial solutions was established by the Yang index.Furthermore, Chen, Kuo and Wu [8] considered the problem (2) with h(x, u) = λf (x)|u| q−2 u + g(x)|u| r−2 u, where 1 < q < 2 < r < 2 * .By using the Nehari manifold and fibering map methods, they examined the multiplicity of positive solutions for the exponent r satisfying r > 4, r = 4, and r < 4, respectively.If the nonlinearity is critical, Figueiredo [14] obtained the existence of solutions by using the truncation argument.For more results, we refer to [3,4,9,16,28].
With regard to p-Kirchhoff type elliptic problems, Corrêa and Figueiredo [10] proved a result of existence and multiplicity of solutions by the Krasnoselskii's genus when the nonlinear term is nonnegative function and satisfies subcritical growth condition.Liu [22] established the existence of infinite many solutions by the Fountain theorem and Dual Fountain theorem.According to Morse theory and the local linking, Liu and Zhao [23] further proved the existence of two nontrivial solutions if M (s) is bounded.Recently, Huang, Chen and Xiu [17] studied the following quasilinear elliptic problem with concave-convex nonlinearities where M (s) = as k + b, 1 < q < p < r < p * , and proved that the problem has at least one positive solution when r > p(k + 1) and the functions h(x), H(x) are nonnegative.The approach adopted is the mountain pass lemma.In [18], Hamydy, Massar and Tsouli considered the following problem with critical exponent By the variational method, they obtained a nontrivial solution when the parameter λ is sufficiently large.After that, Ourraoui [26] showed the existence of at least one solution when the parameter λ = 1.However, when the weight functions f (x) and g(x) change their signs, the existence of solutions to the p-Kirchhoff elliptic equations is open, as we know.To attach this problem will be the main target of the present paper.Motivated by the results of above-mentioned papers, in this paper, we will discuss the existence of multiple nontrivial nonnegative solutions to the problem (1) by a variational method.There are three special features of this study.Firstly, the corresponding energy functional J λ,M (u) of the problem (1) is not bounded in W 1,p 0 (Ω) for r ≥ 2p, then we cannot take advantage of the standard variational argument directly.In order to overcome this difficulty and obtain the existence of nontrivial nonnegative solutions, we will adopt a variational method on the Nahari manifold which is similar to the fibering method (see [5,12] for details).Secondly, the problem (1) involves the p-Laplacian operator, which makes the uniform a prior estimates of Gidas and Spruck type for the case p = 2 failed.To overcome this shortage and to get the existence of two nontrivial nonnegative solutions for r < 2p, we need to compare the min-max levels of energy and use the truncation arguments.Such an idea originally comes from Corrêa and Figueiredo [11].Finally, when r = p * , due to the lack of compactness of the embedding of W 1,p 0 (Ω) → L p * (Ω), we prove the compactness of the extraction of the Palais-Smale sequences in the Nehari manifold by the Lions concentration-compactness principle.
Before stating our main theorems, let us have the following notations.Let W 1,p 0 (Ω) be the Sobolev space with norm u = Ω |∇u| p dx 1 p , and we denote by S l the best Sobolev constant for the embedding of Firstly, we give the definition of the weak solution to the problem (1).Definition 1.1.We say that a function u ∈ W 1,p 0 (Ω) is a weak solution of the problem (1) if for all ϕ ∈ W 1,p 0 (Ω).Thus, the corresponding energy functional of the problem (1) is defined by It is well known that the weak solutions to the problem (1) are the critical points of the energy functional J λ,M (u).However, from the expression of functional J λ,M (u), we know that it is not bounded in W 1,p 0 (Ω) when r ≥ 2p, so it is useful to discuss the functional J λ,M (u) on the Nehari manifold and N λ,M contains every nonzero solution of the problem (1).Define we have and K u,M (t) = 0 for u ∈ W 1,p 0 (Ω) \ {0}, t > 0 if and only if tu ∈ N λ,M .In particular, K u,M (1) = 0 if and only if u ∈ N λ,M .Now, we split N λ,M into three parts: < 0}.Thus, for each u ∈ N λ,M , one has The main results of this paper are the following theorems: Theorem 1.2.Assume 2p < r < p * and N < 2p.Then for each a > 0, there exists a positive number λ * = max{ q √ 2p λ 1 (a), q 2p λ 2 (a), q p λ 3 } such that the problem (1) has at least two nontrivial nonnegative solutions for 0 < λ < λ * , where then Λ > 0 is achieved by some φ Λ ∈ W 1,p 0 (Ω) with Ω g|u| 2p dx = 1.In particular, Theorem 1.3.Assume r = 2p and N < 2p.Then (i) for each a ≥ 1 Λ and λ > 0, the problem (1) has at least one nontrivial nonnegative solution u λ,M ∈ N + λ,M = N λ,M ; (ii) for each a < 1 Λ and 0 < λ < q p λ 0 (a), where Then for each a > 0 and has at least one nontrivial nonnegative solution u λ,M ∈ N + λ,M .
Theorem 1.6.Assume p < 2p 2 2p−q < r < 2p.Then for each ϑ > 0 and 0 < a < min{ b 2 (r−p) rL(ϑ) , A * }, there exists a positive number λ * ≤ min{ϑ, Λ, λ * } such that the problem (1) has at least two nontrivial nonnegative solutions u where The outline of this paper is as follows.In Section 2, we present some necessary preliminaries and some properties of Nehari manifold.Section 3 will be devoted to the proofs of Theorems 1.2, 1.3 and 1.4.In Section 4 and Section 5, we will prove Theorem 1.5 and Theorem 1.6, respectively.

2.
Preliminaries.We present some important properties of Nehari manifold.
Proof.The proof is similar to the proof of Theorem 2.3 in [5], we omit the details here.
Lemma 2.2.(i) If r ≥ 2p, then the energy functional J λ,M (u) is coercive and bounded in N λ,M ; (ii) If r < 2p, then the energy functional J λ,M (u) is coercive and bounded in W 1,p 0 (Ω).
Proof.(i) By the definition of N λ,M , the Sobolev imbedding theorem and Young's inequality, we find that Thus, J λ,M (u) is coercive and bounded in N λ,M .
(ii) Using the Sobolev imbedding theorem, we have then the energy functional J λ,M (u) is coercive and bounded in W 1,p 0 (Ω) by the Young's inequality.The proof of Lemma 2.2 is complete.Lemma 2.3.If r < p * , then each Palais-Smale sequence for J λ,M (u) in W 1,p 0 (Ω) has a strongly convergent subsequence.
Then for each u ∈ W 1,p 0 (Ω), there are unique Proof.Fix u ∈ W 1,p 0 (Ω), we define h a (t) = at 2p−q u 2p + bt p−q u p − t r−q Ω g|u| r dx for a, t ≥ 0, then it is easy to see that h a (0) = 0, lim t→+∞ h a (t) = −∞, h a (t) achieves its maximum at t = t a,max , increasing for t ∈ [0, t a,max ) and decreasing for t ∈ (t a,max , +∞).Now, we divide the proof into three cases: 2r−3p such that m a (t) achieves its maximum at t = t max , increasing for t ∈ [0, t max ) and decreasing for t ∈ (t max , +∞).Moreover, therefore, there are unique t + and t − such that 0 < t + < t a,max < t − , h a (t + ) = λ Ω f |u| q dx = h a (t − ) and h a (t + ) > 0 > h a (t − ).A bunch of computations yield and there is a unique t max = a(2p−q) u 2p (r−q) Ω g|u| r dx 1 r−2p such that n a (t) achieves its maximum at t = t max , increasing for t ∈ [0, t max ) and decreasing for t ∈ (t max , +∞).Moreover, then, there are unique t + and t − such that 0 < t + < t a,max < t − , h a (t + ) = λ Ω f |u| q dx = h a (t − ) and h a (t + ) > 0 > h a (t − ).Repeating the same argument of Case (i), we conclude that then, we see that h 0 (t) ≤ h a (t), h 0 (0) = 0, lim t→+∞ h 0 (t) = −∞ and there exists a unique t 0,max = ( b(p−q) u p (r−q) Ω g|u| r dx ) 1 r−p such that h 0 (t) achieves its maximum at t = t 0,max , increasing for t ∈ [0, t 0,max ) and decreasing for t ∈ (t 0,max , +∞).Moreover, On the other hand, since therefore, there are unique t + and t − such that 0 < t Repeating the same argument of Case (i), we conclude that t + u ∈ N + λ,M , t − u ∈ N − λ,M , and This completes the proof of Lemma 2.6.
Proof.Similar to the argument in Lemma 2.6, we can prove Lemma 2.10.Here, the details are omitted.
To prove Theorem 1.2 and Theorem 1.3, we need the following results.
Case (ii).λ * = q 2p λ 2 (a).Using (3) and the Sobolev imbedding theorem, we see that Then, we have Then, one has This completes the proof of Lemma 3.1.
Proof of Theorem 1.2.Applying Lemma 2.2 (i), Lemma 2.5 (i), Lemma 3.1 and the Ekeland variational principle [13], we obtain that there exist two minimizing sequences {u . Then, it follows from Lemma 2.3 that there exist subsequences still denoted by . Thus, the problem (1) has at least two nontrivial nonnegative solutions.This completes the proof of Theorem 1.2.
(ii) Similar to the proof of Theorem 1.2, we know that the problem (1) has at least two nontrivial nonnegative solutions u + λ,M ∈ N + λ,M , u − λ,M ∈ N − λ,M .Moreover, combining ( 6) with (7), we see that This completes the proof of Theorem 1.3.Before giving the proof of Theorem 1.4, we introduce the following lemmas.
The proof is similar to Lemma 2.5(i), we omit the details here.
-sequence satisfiying Similarly to the proof of Lemma 2.3, we know that {u n } is bounded in N + λ,M , and there exists a subsequence, still denoted by {u n } and u ∈ W 1,p 0 (Ω) such that u n u weakly in W 1,p 0 (Ω), u n → u strongly in L r (Ω) for 1 < r < p * , u n u weakly in L p * (Ω), u n → u almost everywhere in Ω.
By concentration-compactness principle [21], there exists at most set J, a set of different points {x j } j∈J ⊂ Ω, sets of nonnegative real numbers {µ j } j∈J , {ν j } j∈J such that where δ x is the Dirac mass at x, and the constants µ j , ν j satisfying Following, we claim that J is finite for any j ∈ J, either ν j = 0 or ν j ≥ p * −p .In fact, choosing ε > 0 sufficiently small such that B ε (x i ) ∩ B ε (x j ) = ∅ for i = j, i, j ∈ J. Let φ j ε (x) be a smooth cut off function centered at x j such that and Noting that and by (8), we have Thus, It follows from ( 9) and ( 10) that On the other hand, since u n ∈ N + λ,M , we have This implies which is a contradiction.Hence, µ j = ν j = 0 and we can obtain that u n → u strongly in L p * (Ω) and u n → u strongly in W 1,p 0 (Ω).Moreover, since u n ∈ N λ,M , we deduce which yields u is nonzero and u ∈ N λ,M .Next, we need show that u ∈ N + λ,M .Due to Hence, we have u ∈ N + λ,M .

4.
Proof of Theorem 1.5.First, we consider the following truncated problem: where k ∈ ( b(r−p) ar , b(r−p) pa ) and Then the solutions of truncated problem (11) are critical points of the energy functional where Mk (t) = t 0 M k (s)ds.Thus, we have the following lemma about the functional J λ,M k (u).
r > 0. Thus, J λ,M k (u) is coercive and bounded in N λ,M k by the Young's inequality.The proof of Lemma 4.1 is complete.
Note that by (3) and (4), if u ∈ N λ,M k with u p ≤ k, we see that and if u ∈ N λ,M k with u p > k, we have Subsequently, we have the following lemmas.
Furthermore, we can get Cr M ( u p ) .

Lemma 4 . 1 .
The energy functional J λ,M k (u) is coercive and bounded in N λ,M k .Proof.If u ∈ N λ,M k , then by the definition of N λ,M k and the Sobolev imbedding theorem, we find that