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

1 Introduction and main results

Consider the following quasilinear equation with nonlocal nonlinearity

where N ≥ 3 , μ ∈ ( 0 , N ) , κ > 0 is a parameter, λ is a positive constant, f satisfies some appropriate assumptions, and F ( t ) = ∫ 0 t f ( s ) d s . (1.1) is formally the variational formulation of the following functional

A function u ∈ X is called a weak solution of (1.1) if

for all φ ∈ C 0 ∞ ( R N ) . 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

which is related to the standing waves ψ ( x , t ) = u ( x ) e − i E t of the following Schrödinger equation

where i denotes the imaginary unit, λ = μ − E , and g ( t ) = h ( ∣ t ∣ 2 ) t . 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 [2,12,16,23,26–28,34]).

Different from the semilinear case (1.1), due to the presence of the quasilinear term, one lacks an appropriate working space to deal with J . In fact, there is no natural function space in which J 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 ∣ − μ * F ( u ) ) f ( u ) has caused many analytical difficulties and made the study of this problem more interesting. Moreover, the critical exponent of (1.1) is 4 N − 2 μ N − 2 rather than 2 2 * (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., ∫ R N ∣ u ∣ 2 = m > 0 , 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 , μ ∈ ( 0 , N + 2 2 ) , and f ( u ) = ∣ u ∣ p − 2 u with p ∈ ( 2 , 4 N − 4 μ N − 2 ) , the second author, Zhang and Zhao [46], first prove the existence of positive, negative, and high-energy solutions via perturbation method. Subsequently, when N ≥ 3 , μ ∈ ( 0 , N ) , and f ( u ) = ∣ u ∣ p − 2 u with p ∈ 4 N − 2 μ N , 4 N − 2 μ N − 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, 0 < μ < 2 , and p N < p < 4 ( N − μ ) N − 2 with p N = max { 4 , 3 N − 2 μ + 2 N − 2 } . Under appropriate assumptions on potential functions, when N ≥ 3 , μ ∈ ( 0 , N ) , and f ( u ) = ∣ u ∣ p − 2 u with p ∈ [ 4 N − 2 μ N , 4 N − 2 μ N − 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.

If f ( u ) = ∣ u ∣ p − 2 u , the existing results to (1.1) in references mainly focus on p ∈ [ 4 N − 2 μ N − 2 , 4 N − 2 μ N − 2 ) . It is worth pointing out that the possible existence interval of solutions for this problem is p ∈ [ 2 N − μ N , 4 N − 2 μ N − 2 ] , since 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, p ∈ [ 2 N − μ N , 4 N − 2 μ N − 2 ] is similar to p ∈ [ 2 , 2 2 ∗ ] for the nonlocal case (1.1). Recently, Liu et al. [16] established the existence of infinitely many solutions for (1.1), including the case f ( u ) = ∣ u ∣ p − 2 u with p ∈ ( 2 , 2 2 ∗ ) . So a natural question is whether the existence of infinitely many solutions can be obtained to (1.1) when f ( u ) = ∣ u ∣ p − 2 u with p ∈ ( 2 N − μ N , 4 N − 2 μ N − 2 ) 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-Lions-type 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

1. f ( t ) ∈ C ( R , R ) .

2. For p 1 = 2 N − μ N , lim t → 0 f ( t ) t p 1 − 1 = 0 .

3. For p 2 = 4 N − 2 μ N − 2 , lim t → ∞ f ( t ) t p 2 − 1 = 0 .

4. F ( t ) ≢ 0 , i.e., there exists t 0 ∈ R , t 0 ≠ 0 such that F ( t 0 ) ≠ 0 .

5. F is odd or even.

Second, we are concerned with infinitely many normalized solutions for the following constraint problem

In addition, f satisfies the following hypothesis

1. There exists p 3 > 2 such that lim t → 0 f ( t ) t p 3 − 1 = 0 .

2. For p 4 = 3 N − μ + 2 N , lim t → + ∞ f ( t ) t p 4 − 1 = 0 .

3. For p 5 = 2 N − μ + 2 N , lim t → 0 ∣ F ( t ) ∣ ∣ t ∣ p 5 = + ∞ .

Our second main result is stated as follows:

In what follows, we use the notation

• H 1 ( R N ) is the usual Sobolev space endowed with the inner product and norm

  ( u , v ) = ∫ R N ( ∇ u ∇ v + u v ) d x ; ‖ u ‖ 2 = ∫ R N ( ∣ ∇ u ∣ 2 + ∣ u ∣ 2 ) d x .

• H r 1 ( R N ) denotes the space of radially symmetric Sobolev functions.

• L s ( R N ) , 1 ≤ s ≤ + ∞ , denotes a Lebesgue space; the norm in L s ( R N ) is denoted by ‖ u ‖ s .

• Let S be the best Sobolev constant

  S ‖ u ‖ 2 * 2 ≤ ‖ ∇ u ‖ 2 2 , ∀ u ∈ H 1 ( R N ) .

• The weak convergence is denoted by ⇀ , and the strong convergence by → .

• The embeddings H r 1 ( R N ) ↪ L s ( R N ) are continuous for s ∈ [ 2 , 2 * ] , i.e., there exist constants C s such that

  ‖ u ‖ s ≤ C s ‖ u ‖ , ∀ u ∈ H r 1 ( R N ) .

  The embeddings H r 1 ( R N ) ↪ L s ( R N ) are compact for s ∈ ( 2 , 2 * ) .

• We omit the symbol d x in the integrals over R N when no confusion can arise. C and C i denote various positive constants.

• Let E be a Banach space and ℐ : E → R be a functional of C 1 class. For c ∈ R , we recall ℐ satisfied ( C ) c condition, if any sequence { w n } ⊂ E with

  ℐ ( w n ) → c , ( 1 + ‖ w n ‖ ) ‖ I ′ ( w n ) ‖ → 0 ,

  as n → ∞ , then { w n } has a strongly convergent subsequence in E .

2 Unconstrained problem