p -fractional Hardy–Schrödinger–Kirchhoff systems with critical nonlinearities

: This paper deals with the existence of nontrivial solutions for critical Hardy–Schrödinger–Kirchhoff systems driven by the fractional p -Laplacian operator. Existence is derived as an application of the mountain pass theorem and the Ekeland variational principle. The main features and novelty of the paper are the presence of the Hardy terms as well as critical nonlinearities.


Introduction
In this paper, we study the existence of solutions for an elliptic system of Hardy-Schrödinger-Kirchhoff type, involving the fractional p-Laplacian as well as critical nonlinearities. More precisely, we first where 0 < s < 1 < p < ∞, sp < N, α > 1 and β > 1 with α + β = p * s and p * s = Np N−sp . The potential function V : ℝ N → ℝ + verifies V ∈ C(ℝ N ) and inf The nonlocal fractional operator L s p is defined along any φ ∈ C ∞ 0 (ℝ N ) by where B ε (x) denotes the ball in ℝ N of radius ε > 0 at the center x ∈ ℝ N and, throughout the paper, K : ℝ N \ {0} → ℝ + is a measurable function such that (a) there exists K 0 > 0 such that K(x) ≥ K 0 |x| −(N+ps) for any x ∈ ℝ N \ {0}, (b) mK ∈ L 1 (ℝ N ), where m(x) = min{|x| p , 1}, x ∈ ℝ N .
A typical example of K is given by K(x) = |x| −(N+sp) . In this case, the operator L s p simply reduces to the fractional p-Laplacian, denoted by (−∆) s p . In particular, (−∆) s p is consistent with the fractional Laplacian (−∆) s as p = 2, and it is well known that (−∆) s p reduces to the standard p-Laplacian as s ↑ 1 in the limit sense of Bourgain-Brezis-Mironescu, as shown in [4].
For critical equations in ℝ N we refer the reader to [2,7,11,14,29] and references therein for the study of scalar problems with critical nonlinearities.
The main solution space of (S) is W = W Then W = (W, ‖ ⋅ ‖) is a separable reflexive real Banach space, see [17,33] for more details.
Because of the presence of the Hardy terms in (S), we assume that the system is non-degenerate. We recall that the degenerate case for (S) corresponds to M(0) = 0. Hence, throughout the paper, we suppose that the Kirchhoff function M : ℝ + 0 → ℝ + 0 is continuous and satisfies Usually, the existence of solutions of fractional Kirchhoff problems is derived, when M is also nondecreasing in ℝ + 0 . For more comments we refer, e.g., to [18,31,33]. However, (M1)-(M2) do not force M to be monotone as the example M(t) = (1 + t) k + (1 + t) −1 for t ≥ 0, with 0 < k < 1, shows. For details we refer to [1,32].
The parameter σ is real and for the Hardy terms in (S) it is important to recall the fractional Hardy-Sobolev inequality. By [25, Theorems 1 and 2], we know that then, several papers have been devoted to stationary fractional Kirchhoff problems involving critical nonlinearities in the degenerate case. For further comments we refer to [1,7,33] [18,24] and for degenerate problems in [1,7,26,33,35]. For stationary Hardy-Kirchhoff fractional problems, with critical nonlinearities, even in the scalar case, very few contributions are known. We refer to [7,15,16] and the references therein. The main novelty of our paper is to treat (S) in the setting of fractional p-Laplacian involving critical nonlinearities and Hardy terms. The results are new even in the case M ≡ 1.

holds.
A very natural appealing open problem is to prove existence of nontrivial solutions for (S), when M(0) = 0 and M(t) > 0 for all t > 0. However, Theorem 1.1 was recently established in [27, Theorem 1.1], without the Hardy terms, that is, in the case σ = 0, but in the degenerate case. Because of the lack of compactness, due to the presence of the Hardy terms, Theorem 1.1 is more delicate to prove than in [27] and a tricky step in the proof is necessary to overcome this new difficulty. Theorem 1.1 extends to entire solutions the existence results recently obtained for fractional systems, with critical nonlinear terms, but in bounded domains, in [9,10,12,13,20,22,28], and generalizes to the fractional Hardy-Schrödinger-Kirchhoff case the systems driven by the p-Laplacian operator studied in [23]. However, in the systems treated in [17] the fractional p-Laplacian operator is replaced by two possibly different fractional Laplacian operators and H is not required to satisfy the Ambrosetti-Rabinowitz growth condition as assumed in (H). Finally, Theorem 1.1 extends in a broad sense [34,Theorem 1.1].
In what follows, we shall study system (S) under the solely assumption (M1) on the Kirchhoff function M. We first prove the next addition to Theorem 1.1.
The assumption M(0) = a, together with monotonicity of M, was assumed in [18,31] in the scalar case, as well as in numerous papers.
A very interesting open problem is to construct a nontrivial solution (u σ,λ , v σ,λ ) of (S) when μa ≤ pM(0) and  [15] for further comments.
In the sublinear case, that is, when q ∈ (1, p), we continue to assume on M solely (M1) but, following [27], we take H of the special separated form H(x, u, v) = h(x)f (u, v). Hence, we deal with the following new system in ℝ N : and there exist C > 0 and q ∈ (1, p) such that and f u , f v denote the partial derivatives of f with respect to the first and second variable, Concerning the function h in (S ), we assume from now on that h verifies In order to cover the more interesting case when γ > 0 in (S ), we need a further assumption on h. Fix σ < aH p and set We are now able to state the existence result for (S ).
where C and q are introduced in (f1) and C p * Clearly, condition (h) simply requires that h is nontrivial and (1.9) that the norm of h in L p * s p * s −q (ℝ N ) is sufficiently small. Theorem 1.3 was recently established in a weaker form in [27, Theorem 1.2] when σ = 0, that is, without the Hardy terms. Again, the nontrivial presence of the Hardy terms makes Theorem 1.3 more difficult to handle than in [27]. Furthermore, Theorem 1.3 generalizes the existence results obtained in [9,10,12,13,17,20,22,28]  However, as far as we know, Theorems 1.1-1.3 are new even when M ≡ 1 and p = 2. The paper is structured in the following way. In Section 2, we present some preliminary results, which are useful for the next main sections. In Section 3, we establish the key compactness theorems, particularly helpful to apply the mountain pass lemma at a special mountain pass level and to prove Theorems 1.1 and 1.2, that is, the existence of a nontrivial solution for (S). Finally, Section 4 is devoted to the proof of Theorem 1.3 via the Ekeland variational principle.

Variational framework
In this section we briefly recall the relevant definitions and notations related to real uniformly convex Banach space W and for a complete treatment, we refer to [2,17,32,33].
Combining the results of [ where C ν depends on ν, N, s, K 0 and p.
The next result can be proved similarly to the arguments used for [8, Lemma 2.2].
Let us present a technical lemma, which will play a crucial role in the study of compactness property of functional I. This result was proved in the scalar case in [30, Lemma 3.2] when K(x) = |x| −N−ps . For the sake of completeness, we report here the proof.
Proof. Let us define ω n : We want to prove that Define ω ε n : Brought to you by | University of Sussex Library Authenticated Download Date | 7/3/18 8:12 AM Since in particular u n → u a.e. in ℝ N , clearly ω ε n → 0 a.e. in ℝ 2N , so that lim n→∞ ∬ ℝ N ω ε n (x, y) dx dy = 0 by the dominated convergence theorem. Then, by (2.2), Consequently, since ε > 0 is arbitrary, the claim (2.1) holds true and it implies that A similar argument shows that This concludes the proof.

Proof of Theorems 1.1 and 1.2
In this section, we first assume, without further mentioning, that the assumptions required in Theorem 1.1 are satisfied. We say that the couple (u, v) ∈ W is a (weak) solution of problem (S) if Clearly, the entire (weak) solutions of (S) are exactly the critical points of the Euler-Lagrange functional I : W → ℝ associated with (S), given for all (u, v) ∈ W by which is well defined and of class C 1 (W) by (H) and the continuity of M.