Combined effects of Choquard and singular nonlinearities in fractional Kirchhoff problems

1 Introduction

In this paper, we study the following Choquard-Kirchhoff problem involving singular nonlinearity:

where N≥2 , 1<p<N/s with s∈(0,1) , 1<q<pμ,s* and

with a>0,b≥0 , θ∈[1,2q) and

Here μ ∈ 0 , N , p μ , s ∗ = p 2 ⋅ 2 N − μ N − p s is the upper critical exponent in the sense of Hardy-Littlewood-Sobolev inequality, λ,μ>0 are two parameters,0<β<1 , and − Δ p s is the fractional p− Laplacian which, up to a normalization constant, is defined for any x∈ℝN as

for any φ∈C0∞(ℝN). Here Bε(x) denotes the ball in ℝN centered at x with radius ε>0. For further details about the fractional Laplacian and its applications, we refer to [31]. Throughout our paper, the following assumptions will be satisfied:

• f:ℝN→ℝ+ such that f∈Lq(ℝN), where q = p s ∗ p s ∗ − 1 + β with p s ∗ = N p N − p s is the fractional critical Sobolev exponent;

• g ∈ L 2 μ ∗ p s ∗ p s ∗ − q 2 μ ∗ ( R N ) , where 2μ*=2N2N−μ.

When s = 1 and b = 0, the equation (1.1) covers the Choquard-Pekar equation which has come forth in quantum physics of a polaron at rest [39] and which describes the modeling of an electron ensnared in its own hole [26], see also [32]. Equation (1.1) is also related with the fractional Kirchoff model which was first proposed by Fiscella and Valdinoci [11]. Indeed, the study of Kirchhoff–type problems, which arise in various models of physical and biological systems, have received more and more attention in recent years. Precisely, Kirchhoff in [24] extended the classical D’Alembert wave equation by considering the effects of the changes in the length of the strings during the vibrations and established a model given by the equation

where M(∫0L|∂u∂x|2dx)=(p0h+E2L∫0L|∂u∂x|2dx) and ρ, p₀, h, E, L are constants. For recent results about fractional Kirchhoff problems, we refer to [10, 19, 21, 23, 22, 20, 28, 29, 30, 49, 50] and the references cited there.

In recent years, much attention has been focused on the existence and properties of nontrivial solutions for fractional Choquard equation involving fractional p-Laplacian, see for example [47, 37, 41, 51]. In [37], Mukherjee and Sreenadh studied the following subcritical Choquard system involving fractional p-Laplacian and perturbations

The authors proved that the system admites at least two solutions by means of Nehari manifold and minimax methods. Pucci, Xiang and Zhang in [41] discussed the following Schrödinger-Choquard-Kirchhoff type fractional p-Laplacian equations with upper critical exponents

M ( ∥ u ∥ s p ) [ ( − Δ ) p s u + V ( x ) | u | p − 2 u ] = λ f ( x , u ) + ∫ R N | u | p μ , s ∗ | x − y | μ d y | u | p μ , s ∗ − 2 u i n R N ,

where p μ , s ∗ = p N − p μ / 2 / N − p s is the critical exponent in the sense of Hardy–Littlewood–Sobolev inequality and

The authors established the existence and asymptotic behavior of solutions for above problem in the cases of f satisfies superlinear and sublinear nonlinearities, respectively. The main techniques used in the paper are the mountain pass lemma and Ekeland’s variational principle. Recently, Yang et al. [51] considered the following problem

where Ω is a bounded domain in ℝN with Lipschitz boundary, 1<p<∞,0<s<1,0<μ<N,N>sp,0≤a≤sp,p≤r≤pa* , and p ≤ 2 q ≤ p μ , s ∗ with p a ∗ = p N − a N − p s . The authors analyzed the minimizer of energy functional associated to the problem (1.2) on positive Nehari and sign-changing Nehari sets, and obtained the existence of positive and sign-changing solutions for the problem (1.2). Further discussions on Choquard equation can be found in the survey papers [33, 36] and the references cited there.

On the other hand, critical and subcritical fractional problems involving singular nonlinearity have received more and more attention. In [2], Barrios etal. considered the existence of solutions for the problem

where γ>0,p>1 and M∈{0,1} . The existence of solutions were obtained by using the approximation method by Boccardo and Orsina. Canino etal. [6] extended the above problem to the fractional p-Laplacian and considered the following problem

where γ>0 . The authors considered two cases: γ∈(0,1) and γ>1 . The approximation method by Boccardo and Orsina was used to get the existence of solutions. Moreover, the uniqueness and symmetry of solutions were also investigated. In [12], Fiscella and Mishra studied the following Kirchhoff problem involving singular and critical nonlinearities

where 0 > q > 1 , L u is defined as in (1.1) with p = 2 and θ ∈ [ 1 , 2 s ∗ / 2 ) , λ < 0 , and g is a sign-changing function. The authors analyzed the fibering map and gave the compactness property of the energy functional corresponding to the problem (1.3). The authors required b small enough in order to get some key estimates of the energy functional on the Nehari manifold. When λ is small enough, the authors obtained the existence of two positive solutions by using Nehari manifold method. For fractional Kirchhoff problems with singular nonlinearity, we also refer the interested readers to [44]. Very recently, Goel and Sreenadh [16] used the similar method as in [12] to discuss the following critical Choquard-Kirchhoff type problem

where K ( u ) = − a + ϵ p ( ∫ Ω | ∇ u | 2 d x ) θ − 1 Δ u with a>0,p>N−2(N≥3) and θ ∈ [ 1 , 2 μ ∗ ) . Here 0<μ<N,1<q≤2 and λ is a positive parameter. The authors established the existence of two positive solutions for (1.4). They applied minimization argument on the Nehari sub-manifolds to obtain the first solution. To get the second solution, the authors divided the proof into two cases: μ<min{4,N} and μ≥min{4,N} . Furthermore, do Ó, Giacomoni and Mishra [9] studied fractional Kirchhoff system with critical and concave-convex nonlinearities.

In present paper, we are interested in the multiplicity of solutions for fractional Kirchhoff equations with Choquard and singular nonlinearities. Since the energy functional associated to (1.1) in general is not differentiable on Ds,p(ℝN) , the usual critical point theory is not available. Inspired by [9, 45, 46], we shall use the Nehari manifold approach to get the existence of two solutions for (1.1). Clearly, equation (1.1) is different from the problems considered in the literature, since (1.1) deals with fractional p-Kirchhoff equations with Choquard type and singular nonlinearities. Thus, our equation and result are new. Definitely, we encounter some difficulties in analyzing the fiberling map and discussing the existence of local minimizes on Nehari manifold. Our discussions are more elaborate than the papers in the literature.

To introduce the main result of this paper more precisely, we first give the definition of weak solutions.

Let

and

Here S>0 denotes the best constant of embedding from Ds,p(ℝN) to L p s ∗ ( R N ) and Cg(N,μ)>0 will be given by (2.2).

Our result is the following theorem.

The rest of our paper is organized as follows. In Section 2, we recall some definitions and preliminaries which will be used in our discussion. In Section 3, the properties of fibering maps are analyzed. Furthermore, a compactness result is also given. In Section 4, two nontrivial and nonnegative solutions are obtained by applying the Nehari manifold approach.

2 Preliminaries

In this section, we recall some basic results on fractional Sobolev spaces and the Hardy-Littlehood-Sobolev inequalty. For the details, we refer to [8, 42, 43, 38].

Firstly, we denote by Ds,p(ℝN) the usual fractional Sobolev space

endowed with the norm

Note that, by Theorem 2.1, we know

is finite if g|u|q∈L2μ*(ℝN) for some 2μ*>1 defined as

2 μ ∗ = 2 N 2 N − μ .

Hence, by the fractional Sobolev embedding theorem, if u∈Ds,p(ℝN) this occurs provided that 1 > 2 μ ∗ q > p s ∗ and g ∈ L p s ∗ 2 μ ∗ p s ∗ − 2 μ ∗ q ( R N ) . Thus, q has to satisfy

Hence, p μ , s ∗ is said to be the upper critical exponent in the sense of the Hardy–Littlewood–Sobolev inequality.

By Theorem 2.1 and g ∈ L p s ∗ 2 μ ∗ p s ∗ − 2 μ ∗ q ( R N ) , we have

where C(N,μ)>0 is a suitable constant. Here and from now on, we shortly denote by ∥⋅∥ν for the norm of Lebesgue space Lν(ℝN) . Further, by the fractional Sobolev inequality, we get

where C g ( N , μ ) = C ( N , μ ) ∥ g ∥ p s ∗ 2 μ ∗ p s ∗ − 2 μ ∗ q 2 and S>0 is given by

3 Fibering map analysis

First, we define the energy functional Iλ:Ds,p(ℝN)→ℝ corresponding to equation (1.1) as

Here u+=max{u,0} . For each u∈Ds,p(ℝN) , we define the fibering map Φλ,u:ℝ+→ℝ as

Φλ,u(t)=Iλ(tu)

for all t>0 . A simple calculation gives that

and

Define the Nehari manifold

Then u ∈ N λ if and only if Φ λ , u ′ ( 1 ) = 0 . Thus, it is natural to divide N λ into three parts as

and