Multiple solutions for a Kirchhoff-type equation with general nonlinearity

This paper is devoted to the study of the following autonomous Kirchhoff-type equation $$-M\left(\int_{\mathbb{R}^N}|\nabla{u}|^2\right)\Delta{u}= f(u),~~~~u\in H^1(\mathbb{R}^N),$$ where $M$ is a continuous non-degenerate function and $N\geq2$. Under suitable additional conditions on $M$ and general Berestycki-Lions type assumptions on the nonlinearity $f$, we establish several existence results of multiple solutions by variational methods, which are also naturally interpreted from a non-variational point of view.


Introduction and main results
In this paper, we consider the following autonomous nonlinear elliptic problem with a general subcritical nonlinearity: where N ≥ 2, M : R + → R + and f : R → R are continuous functions that satisfy some assumptions which will be specified later on.
In the case where M is not identically equal to a positive constant, the class of Problem (KT ) is called of Kirchhoff type because it comes from an important application in Physic and Engineering. Indeed, if we let M (t) = a + bt with a, b > 0 and replace R N and f (u) by a bounded domain Ω ⊂ R N and f (x, u) respectively in (KT ), then we get the following Kirchhoff problem: assuming the homogeneous Dirichlet boundary condition, which is related to the stationary analogue of the equation presented by G. Kirchhoff in [10]. Besides, (KT ) is also called a nonlocal problem in this case because of the appearance of the term M R N |∇u| 2 ∆u which implies that (KT ) is no longer a pointwise identity. And this phenomenon provokes some mathematical difficulties which make the study of Problem (KT ) particularly interesting.
On the other hand, when M is identically equal to a positive constant, for example M (t) ≡ 1, there has been a considerable amount of research on this kind of problems during the past years. The interest comes, essentially, from two reasons: one is the fact that such problems arise naturally in various branches of Mathematical Physics, indeed the solutions of (KT ) in the case where M (t) ≡ 1 can be seen as solitary waves (stationary states) in nonlinear equations of the Klein-Gordon or Schrödinger type, and the other is the lack of compactness, a challenging obstacle to the use of the variational methods in a standard way.
In the celebrated papers [4,5,6], the authors studied the case where M (t) ≡ 1, namely the following autonomous nonlinear scalar field problem − ∆u = f (u) in R N , under the following assumptions on the nonlinearity f : (f 0 ) f ∈ C(R, R) is continuous and odd.
(f 3 ) There exists ζ > 0 such that F (ζ) > 0, where F (t) := t 0 f (τ )dτ . With the aid of variational methods and critical points theory, by studying certain constrained problems, Berestycki-Lions and Berestycki-Gallouet-Kavian established the existence results of a ground state, namely a nontrivial solution which minimizes the action functional among all the nontrivial solutions, and infinitely many bound state solutions of (SF ) in [5,6] for N ≥ 3 and in [4] for N = 2 respectively.
As we can see, there is a difference in the assumption (f 1 ) between the cases N ≥ 3 and N = 2. We remark here that, in the proofs given by [4] for the case N = 2, the existence of a limit lim t→0 f (t)/t ∈ (−∞, 0) is used in an essential way to show the Palais-Smale compactness condition for the corresponding functional under suitable constraints. It is hard to generalize (1.2) to the general inequality (1.1) in that argument.
Later on, in a recent paper [8], Hirata, Ikoma and Tanaka revisited Problem (SF ) in the case N ≥ 2 assuming (f 0 ), (f 2 ), (f 3 ) and and managed to find radial solutions through the unconstrained functional In [8], following the approach introduced by Jeanjean in [9], Hirata, Ikoma and Tanaka considered the auxiliary functionalĨ : In this way, they were able to find a Palais-Smale sequence (θ j , u j ) +∞ j=1 in the augmented space R × H 1 r (R N ) such that θ j → 0 and u j "almost" satisfies the Pohozaev identity associated to (SF ). With the aid of this extra information, it was proved that Problem (SF ) possesses a positive least energy solution and infinitely many (possibly sign changing) radially symmetric solutions.
Our main goal of the present paper is to try to provide some multiplicity results for Problem (KT ) under the very general assumptions (f 0 ), (f 2 ), (f 3 ) and (f ′ 1 ) on f and some suitable conditions on M by variational methods.
In terms of (f 0 ), (f ′ 1 ) and (f 2 ), we conclude that the corresponding functional J of (KT ) given by It is easy to see that J is invariant under rotations of R N . Then, is a natural constraint to look for critical points of J, namely critical points of the functional restricted to H 1 r (R N ) are true critical points in H 1 (R N ). Therefore, from now on, we will directly define J on H 1 r (R N ).
Before stating our assumptions on M and the main results of this paper, we would like to mention the closely related works of Azzollini, d'Avenia and Pomponio [3] and Lu [11]. To the best of our knowledge, it seems that only articles [3] and [11] have considered the multiplicity of solutions for such problem under the very general assumptions on f .
In [3], under the same very general assumptions on f as above, Azzollini, d'Avenia and Pomponio considered a suitable perturbation of I, namely where I is given by (1.3), q > 0 is a positive parameter, R : H 1 (R N ) → R and N ≥ 3. The authors supposed that R = Σ k i=1 R i and, for each i = 1, · · · , k, the functional R i satisfies certain suitable assumptions and the following condition: (R2) There exists δ i > 0 such that, for any u ∈ H 1 (R N ), we have By a suitable combination of the method described in [8] and a certain truncation argument, they established an abstract theorem which claims the existence of (at least) n distinct critical points of I q for every n ∈ N and q ∈ (0, q n ), where q n > 0 is a suitable positive constant depending on n. As an application, in the case where N ≥ 3 and M (t) = a + bt with a, b > 0, they treated Problem (KT ) and obtained finitely many distinct radial solutions for sufficiently small b > 0. For another application to the nonlinear Schrodinger-Maxwell system, we refer the reader to [3]. We note that the truncation argument explored in [3] is important to the proof of the abstract existence result. Actually, the truncation argument is not only used to construct a suitable modified functional of I q , which satisfies the symmetric mountain pass geometry, but also, together with the method described in [8], plays a vital role in obtaining (at least) n distinct particular Palais-Smale sequences which are bounded for every n ∈ N and q ∈ (0, q n ). Thus, it is interesting to ask the question whether, at least for Problem (KT ) in the case where M (t) = a + bt with a, b > 0 and N ≥ 3, it is possible to prove the multiple result by some suitable arguments, e.g. variational methods, but without using a truncation technique as in [3].
In the more recent paper [11], by means of a scaling argument based on an idea of Azzollini [1,2] and a new description of the critical values, we investigated the following Kirchhoff Problem where a ≥ 0, b > 0 and N ≥ 1. When N ≥ 2, under some suitable conditions on the values of the nonnegative parameters a and b if necessary and the assumptions (f 0 ), (f 2 ), (f 3 ) and (f ′ 1 ) on f , certain multiplicity results for (K) were obtained as partial results in that paper. In particular, we obtained infinitely many distinct radial solutions in [11] for any a ≥ 0 and b > 0 fixed when N = 2, 3. We note here that [11] not only answers the question we raised above in the affirmative from the non-variational point of view, but also extends the result of Azzollini, d'Avenia and Pomponio in [3] concerning the existence of multiple solutions to (K).
As pointed out in [11], it is natural to know whether, at least for the nondegenerate case a > 0, one can still obtain the multiplicity results for (K) via variational methods. So far, this question has a positive answer for the case N ≥ 4 by the early work [3] of Azzollini, d'Avenia and Pomponio. However, this question is still open for the cases N = 2, 3, where, in fact, Problem (K) possesses infinitely many distinct radial solutions that .
Motivated by the articles [3,8,11] and the questions we raised above, by making some suitable assumptions on M , we shall show the existence of infinitely many distinct radial solutions for Problem (KT ) as our first result of this paper. For this purpose, we make the hypotheses on the function M as follows: Now, our first result of the present paper can be stated as follows. Remark 1.2 In our proof of Theorem 1.1, a truncation argument similar to that in [3] would and should be avoided; since, if not, in general it seems to be difficult or even impossible to get infinitely many distinct solutions. This can be seen as another reason why we try to find solutions of Problem (KT ) through the nonmodified functional J directly.  [7]. In that paper, under certain suitable conditions on M which are stronger than , the existence result of a least energy solutions to (KT ) was also established for N ≥ 3. Our Theorem 1.1 here can be viewed as a natural extension of [7].
Next, when N ≥ 3, for a suitable class of non-degenerate functions M which may not satisfy the hypothesis (M 3 ), we establish the following weaker multiplicity result, which claims the existence of finitely many radial solutions to (KT ).
. Besides, suppose that either (M 2 ) or the following is satisfied: Then, for any n ∈ N, there exists a positive constant q n > 0 such that Problem (KT ) has at least n distinct (possibly sign-changing) radial solutions for any q ∈ (0, q n ).

Remark 1.4
All the solutions that we obtain in Theorem 1.2 are characterized by the symmetric mountain pass minimax argument in H 1 r (R N ).

Remark 1.5
The fact that, under the assumptions of Theorem 1.2, we obtain only finitely many nontrivial solutions for sufficiently small q > 0 is not surprising and we can hardly expect more. Actually, the function M (t) = m 0 +qt 2 N −2 with m 0 , q > 0 satisfies (M 2 ). However, in this case, Theorem A.1 in the paper [7] by Figueiredo, Ikoma and Júnior showed the nonexistence of nontrivial solution for large enough q > 0. In addition, by repeating certain arguments explored in [11] for the proof of Theorem 1.2, Item (ii) in that paper, we can only show that more and more distinct solutions of (KT ) exist as q → 0 + . It seems to be difficult or even impossible to get infinitely many distinct solutions of (KT ) for sufficiently small but fixed q > 0. Thus, the conclusion of Theorem 1.2 seems to be the best possible result we could hope for when M does indeed not satisfy the hypothesis (M 3 ).
, then a straightforward computation shows that such M satisfies assumption (M ′ 2 ). However, in this case, (R2) is not satisfied due to the fact that, for any δ > 0 and u ∈ H 1 (R N ) \ {0}, we have Thus, our Theorem 1.2 can not be obtained by applying the abstract result given by [3] directly and the arguments there are also not valid here.
As a consequence of Theorems 1.1 and 1.2, we have the following result: . Then the following statements hold. (i) If N = 2, 3, Problem (K) has infinitely many distinct radially symmetric solutions for any b > 0, which are characterized by the symmetric mountain pass minimax argument in H 1 r (R N ). (ii) If N ≥ 4, for any n ∈ N, there exists a positive constant b n > 0 such that Problem (K) has at least n distinct radially symmetric solutions for any b ∈ (0, b n ). Moreover, all the solutions are characterized by the symmetric mountain pass minimax argument in H 1 r (R N ).
Remark 1.7 As we can see in Sections 3 and 4, the proofs of Theorems 1.1 and 1.2 are all based on a certain variational method described in [8] but without a truncation argument similar to that in [3]. Since Corollary 1.1 follows directly from Theorems 1.1 and 1.2, we thus answer the first question we raised above in the affirmative again from the variational point of view and address the second problem we raised above in the remaining cases N = 2, 3. As a by-product, in the case a > 0 and N ≥ 4, we provide another variational proof of the multiple result of (K) through the non-modified functional J, which is different from that in [3].
The remaining part of this paper is organized as follows. In Section 2, an auxiliary problem is constructed in the spirit of [8] and the corresponding conclusions are shown at the same time. With the aid of the method described in [8] and the conclusions in Section 2, the proofs of Theorems 1.1 and 1.2 are completed in Sections 3 and 4 respectively. Lastly, in Section 5, the non-variational proofs are presented which actually provide us a better understanding of the multiplicity results.

The auxiliary problem and its result
In this section, we shall construct an auxiliary problem in the spirit of [8], which will play an important role in the proofs of the main results of this paper. To be more precise, it will be proved that there is a sequence of positive critical values {e n } +∞ n=1 corresponding to the auxiliary problem which is divergent to infinity. This fact allows us to prove the multiplicity results for our original problem (KT ) based on the level sets argument.
(vi) The functions h and h satisfy (f 2 ).
(iii) There exists δ > 0 such that (vi) The functions H and H satisfy Now, we can construct the auxiliary problem as follow: where N ≥ 2, m 0 > 0 given by (M 1 ), ω > 0 and h ∈ C(R, R) defined as above. It is not difficult to see that the corresponding functional of (A) given by is well-defined on H 1 r (R N ) and of class C 1 . Moreover, as stated in the next lemma, K has the geometry of the Symmetric Mountain Pass theorem and satisfies the Palais-Smale compactness condition. In what follows, we set D n := {σ = (σ 1 , · · · , σ n ) ∈ R n | |σ| ≤ 1} and S n−1 := ∂D n .

Lemma 2.4
The functional K satisfies the following properties.
Due to Item (ii) of Lemma 2.4, for every n ∈ N, we can define a family of mappings Γ n by Γ n := γ ∈ C(D n , H 1 r (R N )) | γ is odd and γ(σ) = γ 0n (σ) on σ ∈ S n−1 , (2.1) which is nonempty since (i) For every n ∈ N, e n is a critical value of K and e n ≥ ρ > 0.
Remark 2.1 All of the conclusions stated in this section and their proofs can be found in [8]. For ease of exposition and completeness of this paper, it is better to outline the necessary conclusions that we need.

Proof of Theorem 1.1
In this section, we shall give the detailed proof of Theorem 1.1. Before going further, we would like to point out that assumption (M 3 ) is almost necessary when it comes to obtaining infinitely many distinct solutions to (KT ) in the case N ≥ 3, see Remark 1.5 in Section 1. On the other hand, as we can see below, actually assumption (M 3 ) is important to verifying the symmetric mountain pass geometry of J for every n ∈ N and, together with assumptions (M 1 ) and (M 2 ), is also sufficient to establish the existence result of infinite many distinct solutions to (KT ). Proof. In terms of Item (i) of Lemma 2.2 and (M 1 ), we have

Symmetric mountain pass geometry of J
which implies that Item (i) of Lemma 2.4 is applied to J. For every n ∈ N, arguing as in Theorem 10 of [6], an odd and continuous map π n : S n−1 → H 1 r (R N ) is defined such that 0 / ∈ π n S n−1 and R N F (π n (σ)) ≥ 1, for all σ ∈ S n−1 .
It is easy to see that, for every n ∈ N, there exists α n > 0 such that ∇π n (σ) 2 2 ≤ α n , for all σ ∈ S n−1 . For every n ∈ N and any σ ∈ S n−1 , setting β t n (σ)(x) := π n (σ)(t −1 x), we have When N = 2, it is clear that g n (t n ) < 0 for sufficiently large t n > 0. When N ≥ 3, in terms of (M 3 ), there also exists a sufficiently large t n > 0 such that g n (t n ) = t N where Γ n is given by (2.1). In view of (3.1) and Theorem 2.1, we have that d n ≥ e n ≥ ρ > 0 and d n → +∞ as n → +∞.
It is easy to see that the proof of Theorem 1.1 is completed if we can prove that, for every n ∈ N, the value d n defined above is a critical value of J.
For every n ∈ N, by Ekeland's principle, we can find a Palais-Smale sequence However, merely under the condition (3.2), it seems difficult to show the existence of a strongly convergent subsequence and even the boundedness of {u j } +∞ j=1 in H 1 r (R N ). Inspired by [8], by introducing an auxiliary functional, we find a Palais-Smale sequence that "almost" satisfies the Pohožaev identity associated to (KT ), which makes it possible for us to overcome these difficulties.
In the following subsection, based on the key idea above, we will show that d n is indeed a critical value of J for every n ∈ N.
Proof. For the sake of clarity, we divide the proof into three claims. We shall prove the boundedness of { ∇u j Thus, in association with the Item (i) of Lemma 3.3, we conclude the boundedness of { ∇u j By some simple calculations, we have which imply the boundedness of {v j } in H 1 r (R N ) with the aid of Claim 1 and (3.4). Without loss of generality, up to a subsequence, we may assume that v j ⇀ v 0 in H 1 r (R N ). Set ε j := ∂ u Φ(θ j , u j ) (H 1 r (R N )) −1 , with the help of (3.5) and Item (i) of Lemma 2.1, some calculations show that Then, by (3.4), Claim 1, Item (ii) of Lemma 2.3 and Items (i) and (iii) of Lemma 3.3, we have On the other hand, let ϕ ∈ H 1 r (R N ) be a function with compact support and, for every j ∈ N, set ψ j (·) := ϕ(t j ·). With the aid of the fact that v j ⇀ v 0 in H 1 r (R N ), Items (i) and (iii) of Lemma 3.3, Claim 1 and (3.4), we have Thus, there holds However, from (f ′ 1 ), it follows that 0 is an isolated zero point of f . In association with the fact that H 1 r (R N ) ⊂ C(R N \ {0}) and v 0 (x) → 0 as |x| → +∞, e.g. see [5], we have v 0 ≡ 0, which is a contradiction. On the other hand, a straightforward computation yields where

Proof of Theorem 1.2
In this section, the proof of Theorem 1.2 shall be completed. It is worth pointing out that, due to the loss of assumption (M 3 ), finding suitable candidate critical values of J becomes the major difficulty that we need to overcome in the proof of Theorem 1.2. Fortunately, as we can see below, we are able to get through this obstacle by Item (ii) of Theorem 2.1 and the non-negativeness of function λ. As the core of this section, the process of finding suitable candidate critical values will be shown in detail. For convenience, we rewrite the corresponding functional of (KT ) as where Λ(t) := t 0 λ(τ )dτ and I 0 ∈ C 1 (H 1 r (R N ), R) given by Apparently, there holds J q (u) ≥ I 0 (u) ≥ K(u) for all u ∈ H 1 r (R N ).
Redefining γ 0n if necessary, by Lemma 3.1, we have that I 0 also satisfies Items (i) and (ii) of Lemma 2.4 in Section 2. It is easy to see that, for such γ 0n , there exist α n , β n > 0 such that for all σ ∈ S n−1 .
Let q ∈ (0, α n β −1 n ], then for all σ ∈ S n−1 . Therefor, we can define a candidate critical value c q n of J q by where Γ n is given by (2.1). Obviously, for any 0 < q ≤ q ′ ≤ α n β −1 n , We claim that, for every m ∈ N, there exist {n k } m k=1 ⊂ N and q m > 0 such that, for q ∈ (0, q m ], the minimax values {c q n k } m k=1 of J q given by are all well-defined and satisfy 0 < ρ ≤ c q n1 < · · · < c q n k < · · · < c q nm < +∞.
Actually, let n 1 = 1, q n1 := α n1 β −1 n1 and q ∈ (0, q n1 ], it is easy to see that c q n1 is well defined and 0 < ρ ≤ e n1 ≤ c q n1 ≤ c q n 1 n1 < +∞. In view of Item (ii) of Theorem 2.1, there exists n 2 ∈ N such that c q n 1 n1 < e n2 . For such n 2 ∈ N, let q n2 := min{α n2 β −1 n2 , q n1 } and q ∈ (0, q n2 ], we have that {c q n k } 2 k=1 are well-defined and satisfy Thus, for every fixed m ∈ N, the desired sequence {n k } m k=1 ⊂ N can be obtained by an iterative procedure, and the desired positive number q m can also be found by letting q m := min 1≤k≤m {α n k β −1 n k }. It is not difficult to see that, under the assumptions of Theorem 1.2, the arguments explored in Subsection 3.2 are also valid here. This fact means that the candidate critical values of J q we define above are indeed critical values of J q . Therefor, the proof of Theorem 1.2 is finished.

Non-variational proofs of the multiplicity results
In this last section, inspired by [1,11], we shall present different new proofs of the multiplicity results which are non-variational, simple and fundamental. As we can see below, this gives us natural interpretations of the results we proved in previous sections.
Before going into the details of the non-variational proofs, some preliminary results are needed. Firstly, we have the following proposition which concerns the multiplicity result for (SF ) under the very general assumptions (f 0 ), (f 2 ), (f 3 ) and (f ′ 1 ) on f , see [6,8]. Now, under the assumptions of Theorem 1.1, the existence result of infinitely many distinct solutions to (KT ) can be proved in a convenient way.
Similarly, under the assumptions of Theorem 1.2, the existence result of finitely many distinct solutions to (KT ) can also be proved from the non-variational point of view. The detailed proof is provided here for reader's convenience.
For every fixed n ∈ N, let q ∈ (0, q n ), where Obviously, h v i , 1 √ 2m0 < 1 for every i ∈ {1, · · · , n}. On the other hand, (M 1 ) yields that h(v i , t) → +∞ as t → +∞ for every i ∈ {1, · · · , n}. Thus, there exists a positive sequence {t i } n i=1 such that h(v i , t i ) = 1 for every i ∈ {1, · · · , n}. In terms of (5.1), we also have that t 2−N i ∇v i 2 2 = t 2−N j ∇v j 2 2 for every i, j ∈ {1, · · · , n} and i = j. For such {t i } n i=1 , set u i (·) := v i (t i ·), i = 1, 2, · · · , n. Now, it is easy to see that {u i } n i=1 are the desired solutions. Remark 5.1 In some sense, J can be seen as a suitable perturbation of I. Additionally, Proposition 5.2 provides a clear and vital relation between the solutions of (KT ) and that of (SF ). Thus, in terms of Proposition 5.1, it is natural and well-founded to ask the existence of multiple solutions to (KT ).
Remark 5.2 As we can see in this section, the assumptions on M are mainly used to ensure the existence of t > 0 such that h(v, t) = 1. In this procedure, we observe that, when N ≥ 3, the behavior of function M (t) − (1 − 2/N ) M (t)t at infinity is actually not used, which, in contrast, plays a important role in the variational proofs, see Claim 1 and its proof in Subsection 3.2. This significant difference seems to imply that, in the variational arguments, the boundedness of { ∇u j 2 2 } +∞ j=1 could be established under some weaker assumptions on M or in a more natural way.