Multiple concentrating solutions for a fractional (p, q)-Choquard equation

1 Introduction

In this paper, we are concerned with the multiplicity and concentration behavior of solutions for the following fractional (p, q)-Choquard problem:

where ɛ > 0 is a small parameter, 0 < s < 1, 1 < p < q < N s and 0 < μ < sp. We assume that the potential V ∈ C ( R N , R ) satisfies the following conditions:

1. there exists V ₀ > 0 such that V 0 = inf x ∈ R N V ( x ) ,

2. there exists an open bounded set Λ ⊂ R N such that

   V 0 < min ∂ Λ V    and    0 ∈ M = { x ∈ Λ : V ( x ) = V 0 } ,

   and that the nonlinearity f ∈ C ( R , R ) fulfills the following hypotheses:

3. lim | t | → 0 | f ( t ) | | t | q − 1 = 0 ,

4. there exists σ ∈ q , q ( N − μ ) N − s q such that

   lim | t | → ∞ | f ( t ) | | t | σ − 1 = 0 ,

5. for ϑ = 2q we have 0 < ϑ F ( t ) ≤ 2 f ( t ) t for all t > 0, where F ( t ) = ∫ 0 t f ( τ ) d τ ,

6. the function t ∈ ( 0 , ∞ ) ↦ f ( t ) t q − 1 is increasing.

Since we look for positive solutions to (1.1), we suppose that f(t) = 0 for t ≤ 0.

For r ∈ {p, q}, the operator ( − Δ ) r s denotes the fractional r-Laplacian operator defined, up to a normalization constant, by setting

for every u ∈ C c ∞ ( R N ) . We stress that fractional and nonlocal operators are currently studied in the literature due to their application in several contexts such as, for instance, optimization, finance, crystal dislocations, phase transitions, conservation laws, quasi geostrophic flows, ultrarelativistic limits of quantum mechanics, material science, and water waves; see Refs. [1], [2] for more details. We notice that the fractional (p, q)-Laplacian operator ( − Δ ) p s + ( − Δ ) q s in (1.1) is nonhomogeneous in the sense that does not exist any υ ∈ R such that

As s → 1, the fractional (p, q)-Laplacian operator reduces to the (p, q)-Laplacian operator −Δ _p − Δ _q , which appears in the study of reaction-diffusion problems arising in biophysics, plasma physics, and chemical reaction design; see Ref. [3]. More precisely, the prototype for these problems can be written in the form

In this context, the function u in (1.2) denotes a concentration, div[D(u)∇u] represents the diffusion with diffusion coefficient D(u), and c(x, u) corresponds to the reaction term related to source and loss processes. On the other hand, the functional associated with the (p, q)-Laplacian operator falls in the realm of the following double-phase functional

where 0 ≤ a(x) ∈ L ^∞(Ω), which was introduced by Zhikov [4], [5] to describe the behavior of strongly anisotropic materials in the context of homogenization phenomena. We also recall that, from a regularity point of view, F p , q belongs to the class of nonuniformly elliptic functionals with nonstandard growth conditions of (p, q)-type, according to Marcellini’s terminology; see Ref. [6]. For more details on (p, q)-Laplacian problems, we refer the interested reader to Refs. [7]–[13] and the references therein.

When s ∈ (0, 1) and p = q, after rescaling and up to a multiplicative constant, Equation (1.1) boils down to the following nonlinear fractional p-Choquard equation

for which different existence and multiplicity results have been achieved by means of appropriate variational and topological methods; see Refs. [14], [15], [16], [17], [18], [19], [20]. In particular, in Ref. [14] the author examined the multiplicity and concentration of solutions for (1.3) with p = 2, V fulfilling (V ₁)–(V ₂) and f is a subcritical continuous nonlinearity, whereas in Ref. [15] the author provided a multiplicity result for (1.3) with p ∈ (1, ∞) and requiring that the potential V satisfies the following global condition due to Rabinowitz [21]:

We observe that as s → 1, (1.3) becomes the well-known quasilinear Choquard equation

which has been investigated in Refs. [22], [23], [24]. We remark that in Ref. [24] the authors obtained multiple concentrating solutions to (1.5) by assuming (V ₁)–(V ₂) and considering C ¹ subcritical nonlinearities. We note that when N = 3, p = 2, V = μ = ɛ = 1 and F ( t ) = t 2 2 , (1.5) is the so-called Choquard–Pekar equation

which goes back to the description of the quantum theory of a polaron at rest by Pekar in 1954 [25] and which reemerged in 1976 in the work of Choquard on the modeling of an electron trapped in its own hole, in a certain approximation to Hartree–Fock theory of one-component plasma [26]. Equation (1.6) was also proposed in 1996 by Penrose as a model of self-gravitating matter and is known in that context as the Schrödinger–Newton equation [27]. For more details on the Choquard equation, one can see Refs. [28], [29], [30], [31], [32], [33] and the references therein.

For what concerns fractional (p, q)-Laplacian problems, some intriguing existence and multiplicity results appeared in the last years; see Refs. [34], [35], [36], [37], [38], [39], [40], [41]. In particular, for (1.1) in the absence of the convolution term, that is

the authors in Refs. [37], [38] combined variational methods and the Ljusternik–Schnirelmann category theory to get multiple solutions whenever ɛ > 0 is sufficiently small, under conditions (1.4) in Ref. [37] and (V ₁)–(V ₂) in Ref. [38], respectively.

Strongly motivated by Refs. [14], [24], [38], in this paper we analyze the multiplicity and concentration behavior as ɛ → 0 of solutions to (1.1). The main result of this work is the following.

The proof of Theorem 1.1 is based on suitable variational and topological arguments. Inspired by Refs. [14], [24], [38], we adopt a del Pino–Felmer penalization approach [42] by modifying the nonlinearity f outside of the set Λ and introducing an auxiliary problem whose corresponding modified energy functional J ε satisfies all assumptions of the mountain-pass theorem [43]. Lastly, we check that, for ɛ > 0 small enough, the solutions of the modified problem are indeed solutions of the original one. In order to explore (1.1) via variational methods, we recall the following Hardy–Littlewood–Sobolev inequality which will be frequently used throughout the paper.

We note that if μ ∈ (0, N), F ( t ) = | t | σ 0 for some σ ₀ > 0 and u ∈ W s , p ( R N ) ∩ W s , q ( R N ) , then Theorem 1.2 implies that the term

is well-defined whenever F ( u ) ∈ L τ ( R N ) with τ ∈ (1, ∞) such that 2 τ + μ N = 2 , i.e., τ = 2 N 2 N − μ . Bearing in mind that W s , p ( R N ) ∩ W s , q ( R N ) is continuously embedded into L r ( R N ) for all r ∈ [ p , q s * ] , we have to require that τ σ 0 ∈ [ p , q s * ] , namely,

Consequently, if F fulfills | F ( t ) | ≤ C | t | σ 1 + | t | σ 2 with p 2 2 − μ N ≤ σ 1 ≤ σ 2 ≤ q s * 2 2 − μ N , then the quantity in (1.7) is finite. Since we deal with the subcritical case and we aim to examine positivity, regularity and concentration behavior of solutions to (1.1), we impose the restrictions 0 < μ < sp and q < σ < q ( N − μ ) N − s q . Then, we will be able to treat the convolution 1 | x | μ * F ( u ) as a bounded term and we will use assumptions (f ₃) and (f ₄) to implement the penalization method. We underline that the presence of the convolution term and the fractional (p, q)-Laplacian operator make our study much more complicated with respect to Refs. [14], [24], [38] and a more refined analysis will be carried out to accomplish compactness; see Lemma 3.5. To produce multiple solutions for the modified problem, we follow the strategy due to Benci and Cerami [45], which consists in precise comparisons between the category of some sublevel sets of J ε and the category of the set M. We point out that, because f is merely continuous, standard Nehari manifold arguments for C ¹ functionals are unusable and so we exploit the generalized Nehari manifold method proposed by Szulkin and Weth [46]. We could prove Theorem 1.1 without the additional condition V ∈ L ∞ ( R N ) ; see Remark 6.1. However, for the sake of simplicity, we assume here the boundedness of V. As far as we know, this is the first time that the penalization method and the Ljusternik–Schnirelmann category theory are combined to obtain multiple solutions to (1.1).

The paper is organized as follows. In Section 2 we fix the notations and recall some technical results. In Section 3 we introduce the modified problem and we study the corresponding modified energy functional. In Section 4 we focus on the limiting problem associated with (1.1). In Section 5 we provide a multiplicity result for the modified problem. The last section is devoted to the proof of Theorem 1.1.

2 Preliminary results

Let p ∈ [1, ∞] and A ⊂ R N be a measurable set. With A c = R N \ A we denote the complement of A. We will use | ⋅ | L p ( A ) to denote the L ^p (A)-norm. When A = R N , we simply write |⋅| _p . For a generic function u : R N → R , we set u ⁺ = max{u, 0} and u ⁻ = min{u, 0}.

Let s ∈ (0, 1), p ∈ (1, ∞) and N > sp. The space D s , p ( R N ) is given by the closure of C c ∞ ( R N ) with respect to

or equivalently

where p s * = N p N − s p is the fractional critical Sobolev exponent. The fractional Sobolev space W s , p ( R N ) is defined as

The space W s , p ( R N ) is endowed with the norm

We recall that W s , p ( R N ) is a separable reflexive Banach space (see Ref. [47]). For u , v ∈ W s , p ( R N ) , we put

The next embeddings are well-known.

The following vanishing Lions type-result will be frequently used along the paper.

Let s ∈ (0, 1) and p, q ∈ (1, ∞) be such that p < q. We introduce the space

equipped with the norm

Obviously, W is a separable reflexive Banach space. For each ɛ > 0, we consider the space

endowed with the norm

where

Let X be a real Banach space, X′ its topological dual and J ∈ C 1 ( X , R ) . We say that { u n } n ∈ N ⊂ X is a Palais–Smale sequence for J at the level c ∈ R (a (PS) _c sequence for short) whenever J(u _n ) → c and J′(u _n ) → 0 in X′. The functional J is said to satisfy the Palais–Smale condition at the level c ∈ R (the (PS) _c condition for short) if each (PS) _c sequence has a convergent subsequence in X.

3 The modified problem

We adapt the del Pino–Felmer penalization type approach [42] to our problem (1.1). Fix ℓ ₀ > 0 such that

and let a > 0 be the unique number such that

We define

and

where χ _A stands for the characteristic function of A ⊂ R N . Put G ( x , t ) = ∫ 0 t g ( x , τ ) d τ . In light of (f ₁)–(f ₄), we see that g : R N × R → R is a Carathéodory function satisfying the following conditions:

1. lim t → 0 g ( x , t ) t p − 1 = 0 uniformly with respect to x ∈ R N ,

2. g(x, t) ≤ f(t) for all x ∈ R N and t > 0,

3. (i) 0 < ϑG(x, t) ≤ 2g(x, t)t for all x ∈ Λ and t > 0,

4. ( i i ) 0 < q G ( x , t ) ≤ g ( x , t ) t ≤ V 0 ℓ 0 ( t p + t q ) for all x ∈ Λ ^c and t > 0,

5. for each fixed x ∈ R N , the functions t ↦ g ( x , t ) t q 2 − 1 and t ↦ G ( x , t ) t q 2 are increasing in (0, ∞).

Let us consider the auxiliary problem

Clearly, if u _ɛ is a solution to (3.1) such that u _ɛ (x) ≤ a for all x ∈ Λ ε c , where Λ ε = { x ∈ R N : ε x ∈ Λ } , then u _ɛ is also a solution to (1.1). The energy functional J ε : X ε → R associated with (3.1) is given by

From the growth assumptions on g, Theorems 1.2 and 2.1, we deduce that J ε ∈ C 1 ( X ε , R ) and

The next lemma ensures that J ε possesses a mountain pass geometry [43].

In light of Lemma 3.1 and a variant of the mountain pass theorem without the Palais–Smale condition (see Ref. [48]), we can find a Palais–Smale sequence { u n } n ∈ N ⊂ X ε for J ε at the mountain pass level c ε ′ defined by

where

Note that, thanks to (g ₄), c ε ′ can be characterized as

Because supp(u ₀) ⊂ Λ _ɛ , there exists κ > 0, independent of ɛ, ℓ ₀, and a, such that

Let us define

and we set