Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities

: In this article, we study the following quasilinear equation with nonlocal nonlinearity

N 0 When = κ 0, (1.1) is reduced to the semilinear Choquard equation The existence and qualitative properties of equation (1.1) have been widely studied in the last decades.We refer the interested readers to, e.g., [1,5,9,13,14,21,22,31,32,38,41] as well as to references therein.When > κ 0 and the nonlinearity behaves like a local one, (1.1) is reduced to the well-known modified nonlinear Schrödinger equations Yue Jia: School of Mathematics and Computer Science, Yunnan Minzu University, Kunming, 650500, P. R.China  ( ) which is related to the standing waves ( ) ( ) = − ψ x t u x e , iEt of the following Schrödinger equation where i denotes the imaginary unit, = − λ μ E, and ( ) (| | ) = g t h t t 2 .The quasilinear Schrödinger equation (1.1) is derived as a model of several physical phenomena.For example, it is used for the superfluid film equation in plasma physics by Kurihara [18].It also appears in the theory of Heisenberg ferromagnetism and magnons (see [17,35]), in dissipative quantum mechanics, and in the condensed matter theory [30].As far as we know, the study related to variational methods to equation (1.1) can go back to [11].After this work, a series of subsequent studies have been carried concerning with the existence, multiplicity, and qualitative properties of solutions to (1.1) (see for example [2,12,16,23,[26][27][28]34]).Different from the semilinear case (1.1), due to the presence of the quasilinear term, one lacks an appropriate working space to deal with .In fact, there is no natural function space in which is well defined and possesses compactness properties.Consequently, the standard critical point theory cannot be applied directly.In order to overcome this difficulty, several approaches have been successfully developed in the last decades, such as the constraint minimization [25], the Nehari manifold method [24], the perturbation method [29], and the nonsmooth critical point theory [3,19].
Compared with the nonlocal case (1.1), the convolutional term (| | ( )) ( ) x F u f u * μ has caused many analytical difficulties and made the study of this problem more interesting.Moreover, the critical exponent of (1.1) is 2 rather than 22* (see [4]).
General speaking, the study of standing waves of nonlinear Schrödinger equations in the literature has been pursed in two main directions: the L 2 -norm of the solution is not confirmed and the L 2 -norm of the solution is prescribed, i.e., , we customarily call them as unconstrained problem and constrained problem, respectively.We also point out that this difference opened two different challenging research fields.It seems that the first contribution to equation (1.1) is due to [46].Under coercive assumptions on V , when ≥ N 3, ( ) , and 2 , the second author, Zhang and Zhao [46], first prove the existence of positive, negative, and high-energy solutions via perturbation method.Sub- 2 , Zhang and Wu [47] established the existence, multiplicity, and concentration of positive solutions for the following problem by a dual approach where > ε 0 is a parameter, < < μ 0 2, and .Under appropriate assumptions on potential functions, when ≥ N 3, ( ) ∈ μ N 0, , and 2 , Chen and Wu [4] established the existence of positive solutions.We also refer the interested readers to [20,39,40] and the references therein.
or not?On the other hand, to the best of our knowledge, there is no work considering the existence of normalized solutions for (1.1).Very recently, under assumption of Berestycki-Lions-type nonlinearity, Cingolani et al. [7] prove the existence of infinitely many solutions for both unconstrained and constrained problems to semilinear Choquard equation (1.1).So another natural question is whether this result can be generalized to quasilinear case or not?
Motivated by the aforementioned results especially by [16] and [7], in this study, we are concerned with the existence and multiplicity of solutions to the quasilinear Choquard equation (1.1) with Berestycki-Lionstype nonlinearity.First, we search for solutions of (1.1) with a prescribed frequency λ and free mass, i.e., the so- called unconstrained problem.To state our main results, we make the following assumptions on f Our first main result is stated as follows: Theorem 1.1.Assume that ( f 1 )-( f 5 ) hold.For > λ 0 fixed, there exist infinitely many radial solutions of the quasilinear Choquard equation (1.1).Moreover, we have n Second, we are concerned with infinitely many normalized solutions for the following constraint problem In addition, f satisfies the following hypothesis Remark 1.1.Without loss of generality, we assume that < p p 3 4 .
Our second main result is stated as follows: In what follows, we use the notation is the usual Sobolev space endowed with the inner product and norm denotes the space of radially symmetric Sobolev functions.
, denotes a Lebesgue space; the norm in ( ) • The weak convergence is denoted by ⇀ , and the strong convergence by → .
w n has a strongly convergent subsequence in E.

Variational setting and Palais-Smale-Pohozaev condition
Without loss of generalities, throughout the remainder of this article, we assume = κ 1. Because, unless = N 1, is not defined for all u in the space ( ) H N 1 (see [34]), it is difficult to apply variational methods to the functional .To overcome this difficulty, we employ an argument developed in [25], which helps us to transform the quasilinear problem (1.1) into a semilinear problem.More precisely, we make a change of variables ( ) , where g is defined by , on 0, , and , on , 0 .
, for all ∈ t .(9) There exists a positive constant C such that for all ∈ t and ≤ r 1.
After making the change of variables, we consider the functional where Note that the critical points of ( ) v are the weak solutions of the following equation [10] ( ) . Moreover, Proof.Using Lemma 2.2 and ( f 1 )-( f 3 ), we note that is continuous in , and hence, ( . Moreover, a direct calculation deduces that (2.1.3)holds.
□ .Moreover, it satisfies the Pohozaev identity Therefore, we also introduce the Pohozaev functional We consider the action of 2 on n , ∈ n We note that, under the assumption ( f 5 ) and g is odd, ( ) v and ( ) v are even under this action, i.e., ( ) Finally, we denote by ( ) ( ) the projection on the second component.For every ∈ c , we set As already observed, if ( f 1 )-( f 3 ) hold, then ( ) = v 0 for each ∈ v K c .We note also that, assuming that ( f 5 ) hold, K c is invariant under the following 2 -action, i.e., Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  5 Under our assumptions on f , it seems difficult to verify the standard Palais-Smale condition for the functional .Inspired by [8,15,33], we introduce the Palais-Smale-Pohozaev condition, which is a weaker compactness condition than the standard Palais-Smale one.Using this new condition, we will show that K c is compact when > c 0. Step 1: . By (2.1.4)and (2.1.6),we have which implies that there exists > C 0 such that (2.1.7)By Lemma 2.1 and the Sobolev embedding theorem, there holds Step 2: { } v n strongly converge.By Step 1, up to a subsequence if necessary, we may assume For any . Thus, for small > σ 0, by Lemma 2.1 and the Hölder inequality, there exists (2.1.10)which, together with (2.1.9),implies that On the other hand, Similarly, there holds .

By Lemma 2.6 of [45], we have
Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  7 , and this completes the proof.□

Deformation theory
Following [15], we define and introduce the augmented functional → M : By a direct calculation, we obtain We introduce a metric on M by for any ( ) ( ) . We also denote the dual norm on ( ) .
We denote a natural distance in M as follows The following two propositions can be found in [7] and [15].
is non-increasing along ͠ η , and in particular ( ( is non-increasing along η, and in particular ( ( )) ( ) Proof.We introduce the following notation By a direct calculation, we observe that .

Construction of multidimensional odd paths
In this section, we will introduce a sequence of minimax values = a n , 1, 2, 3,….
n These minimax values play important roles to find multiple solutions for the unconstrained Problem (1.1).We divide the proof into two steps as follows.
For ∈ n N *, let and we introduce the set of paths Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  9

N
The following two lemmas can be found in [7].independent of h such that By scaling, we have Lemma 2.5.For < i j, we have , and thus, a n is well defined.
Proof.Essentially, the proof can be obtained by Proposition 6 in [7] through some minor modifications.Because we deal with quasilinear problems, here we give the details for completeness.
Step 1: Construction of an odd path in ( ) . Here, we assume ( )≡ χ R x , 0; 0 i .On the one hand, setting > s 0 0 and ( ) . On the other hand, It is easy to see that (i) By (i)-(iv), we can derive for ≫ R 1 Step 2: Construction of an odd path in ( ) ε Here, we assume : .

N
By a simple proof, we can derive is odd and continuous.Since and g are, respectively, continuous on ( ) and ( ) , for > ε 0 small enough, we have we obtain for large ≫ θ 1 In this subsection, we will define the new minimax families that allow us to find multiple solutions.We borrow an idea from [36] where genus theory is effectively developed in general.
Definition 2.2.Let X be a Banach space.For a closed symmetric set { } ⊂ ⧹ A X 0 with ∉ A 0 , we recall And the basic properties of the genus can be seen in [36].Our function is considered in the following 2 -action: , Θ Γ , 0 is closed, symmetric in 0 and genus n , and the following statements hold.
n Proof.Since the ( ) PSP c condition holds for > c 0 by Proposition 2.1, we can develop deformation theory given in Proposition 2.3.We can also observe that the minimax classes Λ n are stable under the deformation (see [15]  and [6] for details).□ The proof of Theorem 1.1 Proof.Theorem 1.1 follows from Propositions 2.4 and 2.5.□ 3 Constrained problem

Variational setting
For the sake of simplicity, we denote λ by e λ .Since we are looking for normalized solutions to constrained problems, we consider the following functional which is not well defined in the normal Sobolev space . After making a change of variables ( ) = − v g u 1 , we reconsider the following energy functional , for all , .
The following results will play important roles in our argument (see [43] for details).
, then ( ( )) e g v , λ solves equation (1.1).Now, we introduce the Pohozaev functional as follows For every ∈ c , we set Consider the action of 2 on ( ) given by Then, , are invariant under this action, i.e., ( ) n n n which implies that { } λ n is bounded from below.By (3.2.5), we deduce that ‖ ( )‖ → g v m n 2 2 as → +∞ n .

Denote
In this subsection, we give some key estimates of the asymptotic behavior of ( ) given in Proposition 2.4.Recall that for small > ε 0, for large ≫ θ 1, Next, we will prove the monotonicity and positivity of ( ) . The monotonicity of ( ) a λ n with respect to λ is obvious, so we omit its proof here.In order to prove we first claim that there exist small > ρ 0 and > α 0 such that In fact, by Lemma 2.1, for sufficiently small ρ and for any

2* 2
It follows Hardy-Littlewood-Sobolev inequality and the Sobolev theorem that , the claim follows if ρ is small enough.Thus, for any By the arbitrariness of γ, ( ) > a λ 0 1 .

□
For > σ 0 and ≥ A 0, let where Next, we will prove that ( ) satisfies the assumptions of Mountain pass theorems.
Lemma 3.2.Functional σ A , satisfies the Mountain pass geometry, i.e., (1) There exist > α ρ , 0 Proof.As in the proof of (3.3.1),we have then (1) follows if ρ is small enough.For ≥ t 0 and ≠ w 0, we can derive . By a direct calculation, there holds which implies that there exists > C 0 such that jointly with (2.1.8),implies that { } Step 2: { } w n has strongly convergent subsequence.By Step 1, up to a subsequence, we may assume that  .To complete the proof of this lemma, it suffices to prove → , 0 as → ∞ n .Here, we only prove Next, we only prove that → 0 n12 because others are similar.By Lemma 2.2 and the Hölder inequality, we obtain  [36], there exists such that Using the proof of Lemma 2.5 in [44] and Proposition 3.11 in [37], we obtain where Proposition 3.4.We have the following properties: Proof.Without loss of generality, we may assume = σ 1.For fixed A, we denote ), for all k, we can choose some Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  19 .Up to a subsequence, if ( ) φ v k is bounded above, one can obtain a contradiction.Note that By claim 2 of Lemma 2.2 in [44], we have Consequently, which leads to ‖ ‖ ≥ > v α 0 k .Moreover, without loss of generality, we may assume On the other hand, since Then, 1 .
Next, we will prove as → + σ 0 .Otherwise, there exists constant > C 0 such that where we denote v σn by v n for simplicity.Using Claim 2 of Lemma 2.2 in [44] again, we have which implies that there exists . On the other hand, as a contradiction.This completes the proof of (ii).□ The following proposition plays an important role in our construction of minimax level, which will be used in the next subsection., and we have  By Lemma 2.1-( 13), for > l 0, we have , we have where For any By the arbitrariness of γ, we have In view of Proposition 3.4(i),

Constructions of negative minimax level
Following [7], we define the Pohozaev's mountain as follows As in the proof of (3.3.1),we have {( ) } ( ) In particular, > −∞ B m and Thus (i) follows.
(ii) For ≫ t 1, we have We observe that ( ) and hence, By a suitable rescaling, there exists > t 0 which implies (ii) and (iii).By (ii), we have ( ) to be 2 -equivariant, i.e., ( ) It is easy to see that  Then, ⧹ ∈ − A Z Λ n i m .
Proof.The proof is essentially given in [15] and [6].□ Moreover, fix ∈ n N * and by Proposition 3.8, we can obtain following proposition.
Proposition 3.9.Assume that ( f 1 ) and ( f 4 )-( f 7 ) hold.We have that are the critical values of m .Moreover, (i) for some ∈ q *, then we have + q 1 different nonzero critical values.
Proof.Essentially, the proof is similar to Proposition 3.3 in [15].Here, we give some details of (ii) for completeness.By the fundamental properties of genus [36], there exists a closed neighborhood N of ( )

If 2 . 2 , 2 , 22 .
It is worth pointing out that the possible existence interval of solutions for this problem is[ by Hardy-Littlewood-Sobolev inequality, the associated functional enjoys smooth properties in a given space when we use the perturbation method or dual method.From this view of point, 22 for the nonlocal case (1.1).Recently, Liu et al.[16] established the existence of infinitely many solutions for (1.1), including the case ( So a natural question is whether the existence of infinitely many solutions can be obtained to (1.1) when (

for any > m 0 ,
Problem (1.1) has infinitely many radial solutions.

nn 11 2. 4
that the path ∈ ∼ γ Γ n .□ Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  Proof of Theorem 1.1 Palais-Smale-Pohozaev condition and deformation theory Definition 3.1.For ∈ c , we say that ( -Smale-Pohozaev sequence for m at level c (shortly a ( ) PSP c sequence) if

1 Proposition 3 . 2 .
(Deformation lemma) Let < c 0 and be a neighborhood of K c m with respect to the standard distance of ( ) × r N 1

□
By Lemmas 3.2-3.3and the Mountain pass theorem argument to Lemma 3.3 deduces that { } v k is bounded in ( ) H r N 1

23 Definition 3 . 3 .
Next, we will define a family of minimax values as follows.Multiple solutions for the quasilinear Choquard equation with Berestycki-Lions-type nonlinearities  For > m 0 and ∈ n N *, we define