Multiple positive solutions for a class of Kirchhoff type equations with indefinite nonlinearities

: We study the following Kirchhoff type equation: in R N , where N ≥ 3, a , b > 0, 1 < q < 2 < p < min { 4, 2 * } , 2 * = 2 N /( N − 2), k ∈ C ( R N ) is bounded and m ∈ L p /( p − q ) ( R N ). By imposing some suitable conditions on functions k ( x ) and m ( x ), we firstly introduce some novel techniques to recover the compactness of the Sobolev embedding H 1 ( R N ) (cid:44) → L r ( R N )(2 ≤ r < 2 * ); then the Ekeland variational principle and an innovative constraint method of the Nehari manifold are adopted to get three positive solutions


Introduction
We investigate the existence of multiple positive solutions to the Kirchho type equation with inde nite nonlinearities: where N ≥ , a, b > , < q < < p < min{ , * }, * = N/(N − ), and functions k(x) and m(x) satisfy the following conditions. (H ) k ∈ C(R N ) is a bounded function in R N ; (H ) k is sign-changing in R N and Ω = {x ∈ R N : k(x) > } is a bounded domain; (H ) m ∈ L q * (R N ) and m + = max{m(x), } ≢ , where q * = p p − q .
Problem (P) is a variant type of the Kirchho problem as follows: in Ω. u = , on ∂Ω, (1.1) which is related to the stationary analogue of the equation: where Ω is a bounded domain in R N . Such problems are nonlocal owing to the appearance of  2), one can be referred to [2,4,12,20].
In [22], Lions rstly proposed an abstract framework of Eq.(1.1), from then on, lots of researchers began to study Eq.(1.1) in general dimension, see [11,16,25,26,28,39] and the references therein. More precisely, Zhang and Perera [39] obtained the existence of a positive solution, a negative solution and a sign-changing solutions for Eq.(1.1) with N ≥ by using invariant sets of descent ow. Pei and Ma [26] got three positive solutions for Eq.(1.1) with N = , , via the minimax method and the Morse theory. By using the theory developed in [27], Ricceri [28] obtained three positive solutions for Eq.(1.1) with N ≥ .
Recently, many papers [6,8,15,[29][30][31][32]38] study the Kirchho equation on the whole space: When the potential function V(x) satis es the following assumptions: (V )V ∈ C(R N ) with V(x) ≥ in R N and there exists c > such that the set {V < c } := {x ∈ R N | V(x) < c } has nite positive Lebesgue measure, where | · | is the Lebesgue measure; (V )Ω = int{x ∈ R N | V(x) = } is nonempty and has smooth boundary with Ω = V − ( ). Sun and Wu [29] obtained the existence, nonexistence and concentration of nontrivial solutions for Eq. (1.3) with N = and V being replaced by λV , λ > . Later, when N ≥ , V is replaced by λV , λ > , and f satis es the superlinear condition, Sun et al. [30] proved that Eq.(1.3) possessed two positive solutions. When N = , V satis es (V ) − (V ) and f satis es the classical Ambrosetti-Rabinowitz type condition, Xie and Ma [38] proved that Eq.(1.3) has at least a positive solution. Moreover, the concentration behavior is also studied by the authors. In [32], Sun and Zhang obtained the uniqueness of the positive ground state solution for Eq. (1.3) with N = , f (x, u) = d|u| p− u, < p < , and V(x) ≡ c, where c and d are positive constants. Besides, by using the uniqueness result and the concentration-compactness lemma [23], the authors also got the existence and concentration theorems for the following Kirchho problem: In [15], by using the Schwartz symmetric arrangement, Guo proved that Eq.(1.3) possesses a positive ground state solution when V is continuous and f does not satis es the classical Ambrosetti-Rabinowitz type condition.
For the elliptic problems with concave-convex nonlinearities, there are also several results. For example, one can be referred to [1,24,[35][36][37] and the references therein. Indeed, in [1], Ambrosetti et al. rstly introduced the elliptic problem involving the concave-convex nonlinearities: With the aid of variational methods, the authors established the existence, nonexistence and multiplicity of solutions for Eq.(1.5). Later, Wu [37] obtained the existence of multiple positive solutions for the following problem: To the best of our knowledge, for the Kirchho problem with concave-convex terms, there are few results [5,[8][9][10]21]. In [9], Chen et al. considered a class of Kirchho problem on the bounded domain Ω ⊂ R N (N ≥ ): . By giving di erent scopes on a and λ, the authors proved that Eq.(1.7) possesses multiple positive solutions. By using the Nehari manifold technique, Liao et al. [21] obtained two nonnegative solutions for Eq.(1.7). When f (x) = g(x) ≡ in Eq.(1.7), by constraining the energy functional of the problem on a subset of the nodal Nehari set, Chen and Ou [10] proved that there exists a constant λ * > such that for any λ < λ * , Eq.(1.7) has a nodal solution u λ with positive energy. In [8], we obtained three positive solutions for the Kirchho type problem with steep potential well. Motivated by the above facts, more precisely by [9], it is quite natural for us to ask that whether Eq.(P) can have multiple positive solutions when < q < < p ≤ min{ , * } and the domain is unbounded? As far as we know, such problem has never been discussed in the available literature. In our paper, we will give a de nite answer to the above question. Under some suitable conditions on functions k and m, we will obtain three positive solutions for Eq.(P). It is worthy pointing out, in [8], we considered Eq.(P) with steep potential well and positive nonlinearities, which we overcome the compactness of the Sobolev embedding by giving some inequalities. While, in the present paper, since the potential function is a constant and the nonlinearities are sigh-changing, the standard method of recovering the compactness does not work, motivated by [17], a novel method will be used to get the Palais-Smale(PS for short) condition.
Before introducing our main conclusions, we rst recall a well known result (c.f. [34]). Suppose that w Ω is the positive ground state solution to the nonlinear elliptic equation: where Ω is de ned by the condition (H ). Obviously, and inf u∈M Ω where J Ω is the energy functional of problem (E Ω ), de ned by (1.9) and the corresponding Nehari manifold is given by: In order to simplify the calculation, in the present paper, we hypothesize that a = in problem (P).
. Let Sr, S r,Ω and S be the best Sobolev constants for the For any ≤ r ≤ +∞, we shall also denote by | · |r the L r -norm. If we take a subsequence of a sequence {un}, we may denote it again by {un}, We now summarize our main results below.

Theorem 1.1. Assume that functions k and m satisfy hypotheses (H )−(H ). Then there exists a constant Π > such that for each
where I b is the corresponding energy functional of Eq.(P) de ned in Section 2.

Theorem 1.2. Assume that functions k and m satisfy hypotheses (H ) − (H ). In addition, we assume that
Remark 1.1. Theorem 1.2 seems to be the rst result about the Kirchho type equation with constant potential and inde nite nonlinearities, which has three positive solutions in the whole space R N , N ≥ . We also remark that, in the previous papers, to obtain three positive solutions for the Kirchho problem, the domain is usually required to be bounded. For the unbounded domain R N , N ≥ , as far as we are concerned, there are few results except [8], which we obtained three positive solutions for the Kirchho type equation with steep potential well.
In order to obtain our main results, we will make use of variational methods. Since we consider Eq.(P) in the whole space R N , the embedding from Besides, the weight function k is sign-changing, which also makes it much more complicated to recover the compactness. Note that in order to overcome the lack of compactness of the Sobolev embedding, there are some existed strategies, such as constructing a convergent Cerami sequence [18]. While in this paper, inspired by [17,20], we will make use of a novel method to verify that the (PS) condition holds. It is worthy pointing out, due to < p < min{ , * }, the energy functional of Eq.(P) is usually not coercive and bounded below, while, in this paper, under the hypotheses (H )-(H ), we can prove that I b is coercive and bounded below in R N , thus, two positive solutions are obtained. To get the third positive solution for Eq.(P), motivated by [31], the ltration of the Nehari manifold will be utilized: for some c i > , i = , . By a simple computation, we know that I b is bounded from below on M ( ) b (c) under the assumptions (H )-(H ), and then the third positive solution for Eq.(P) is obtained.
The remainder of this paper is organized as follows. In Section 2, we recall some preliminaries. In Section 3, we rst propose some novel techniques to recover the compactness of the Sobolev embedding, then obtain two positive solutions with negative energy for Eq.(P). After introducing the ltration of Nehari manifold in Section 4, we obtain the third positive solution with positive energy for Eq.(P) and prove our results in Section 5.

Preliminaries
For any u ∈ H (R N ), let Then I b is a well de ned C functional with the following derivative: De ne the Nehari manifold Thus u ∈ M b if and only if Note that M b is closely related to the behavior of the bering map [3,13]: it is easy to know that A straightforward calculation shows that Obviously, tu ∈ M b holds if and only if h ′ b,u (t) = . Hence, points in M b correspond to the stationary points of the maps h b,u , then it is natural to divide M b into three parts corresponding to the local minima, local maxima and points of in ection. Following [33], we de ne Similar to the arguments of Brown and Zhang [3, Theorem 2.3], we may get the following conclusion. For each u ∈ M b , there holds For each u ∈ M − b , in view of (2.4), there holds provided that (2.7) Therefore, we may have the following result.
Proof. For any u ∈ H (R N ) with u H = ρ , in view of (2.1) and the Hölder inequality, there holds De ne a function l : [ , +∞) → R as follows: A straightforward calculations implies that and Thus, for every u ∈ H (R N ) satisfying u H = ρ , there hold Consequently, the proof is complete.

Lemma 3.2. Assume that hypotheses (H )−(H ) hold. Then I b is bounded from below and coercive on H
Proof. Denote |u| r r,Ω i = Ω i |u| r dx for i = , and ≤ r ≤ +∞, where Ω and Ω are given by conditions (H ) and (H ). From the Sobolev and Hölder inequalities, for any ≤ r ≤ * and i = , , we obtain In view of (3.3) and the Hölder inequality again, we derive Thus, for every b > , there exists R b > such that Next, we prove that Note that, by (3.3), for any r ∈ [ , * ] and i = , , there holds Then from (3.7) and (3.8), we obtain From (3.9), it is obvious that I b is bounded from below and coercive on H (R N ) and The proof is complete.  (3.13) and (3.14) as n → ∞. Similarly, we obtain for some A > . De ne Then it follows from (3.15)-(3.16) that Note that from (2.1) and (3.16), we can also easily obtain  (3.20) and In view of (3.19)-(3.21), there holds which, together with (3.17)-(3.18), implies that On the other hand, since m ∈ L q * (R N ), thus for every ε > , there exists R * > such that |x|>R * |m| q * dx < ε q * . (3.24) Then from (3.24), we obtain since un and u are bounded in L p (R N ) for ≤ p < min{ , * }.
Note that from (3.11), we may also have as n → ∞. In view of (3.25) and (3.26), there holds It is now deduced from (3.21)-(3.23) and (3.27) that Thus, un → u in H (R N ) by (3.28). The proof is complete. H (R N ) for each < |m|q * ≤ Γ , where Γ is given by (3.2). Moreover, Proof. In view of the hypothesis (H ), there exists ω ∈ H (R N )\{ } satisfying When t > is su ciently small, there holds Thus, from ( . ) and Lemma 3.1, we obtain From the Ekeland variational principle [14], there exists a sequence {un} ⊂ Bρ ( ) satisfying It is easy to know that I b satis es the (PS) θm -condition in Bρ ( ) by Lemma 3.3. As a result, there exists u ( ) b ∈ Bρ ( ) satisfying un → u ( ) b as n → +∞. This implies that u ( ) b is a local minimizer on Bρ ( ) that satis es with ζ H > ρ such that I (ζ ) < . Since I b (ζ ) → I (ζ ) as b → + , then there exists b > such that for every < b < b , there holds I b (ζ ) < . It is easy to know that ζ H < R b by Lemma 3.2. The rest proof is analogous to that of Theorem 3.4, we omit it here.

The ltration of Nehari manifold
and the Sobolev and Hölder inequalities imply that since u q H < ρ q− u H . This implies that for there are two constants C , C satisfying < C < A(p) w Ω H < C such that Hence, we get the following result.
Proof. Let Then It is obvious that In view of ( . ), we obtain provided that Next, we consider the problem in the following two cases.

Case (I)
: Obviously, f (t) ≤ f (t) for t > . In view of ( . ), there holds Evidently, f ( ) = , lim t→ + f (t) = ∞ and lim t→∞ f (t) = . Moreover Observe that Thus, for each This shows that there exist two constants t + b and t − b that satisfy That is, t ± b w Ω ∈ M b . Besides, it is evident that These imply that t ± b w Ω ∈ M ± b .
From ( . ), ( . ) and ( . ), we derive (4.10) In view of ( . ) and ( . ), there are two constants t + b and t − b that satisfy such that That is, t ± b w Ω ∈ M b . A straightforward calculation shows that If Ω m(x)w q Ω dx < , it follows from (4.11) that .
for all v ∈ B( , σ), and The proof is complete.