Infinitely many solutions for fractional Kirchhoff – Sobolev – Hardy critical problems

We investigate a class of critical stationary Kirchhoff fractional p-Laplacian problems in presence of a Hardy potential. By using a suitable version of the symmetric mountain-pass lemma due to Kajikiya, we obtain the existence of a sequence of infinitely many arbitrarily small solutions converging to zero.


Introduction
In this paper we consider the following fractional problem where 0 < s < 1 < p < ∞, 0 ≤ α < sp < N, 1 < q < p, p * s (α) = p(N−α) N−sp ≤ p * s (0) = p * s is the critical Hardy-Sobolev exponent, γ and λ are real parameters, w is a positive weight whose assumption will be introduced in the sequel and Ω ⊆ R N is a general open set.Naturally, the condition u = 0 in R N \ Ω disappears when Ω = R N .
Here (−∆) s p denotes the fractional p-Laplace operator which, up to normalization factors, may be defined by the Riesz potential as , where B ε (x) = {y ∈ R N : |x − y| < ε}.See [11,23] and the references therein for further details on the fractional Sobolev space W s,p (Ω) and some recent results on the fractional p-Laplacian.
Problem (1.1) is fairly delicate due to the intrinsic lack of compactness, which arise from the Hardy term and the nonlinearity with critical exponent p * s (α).For this reason, we strongly need that the Kirchhoff coefficient M is non-degenerate, namely M(t) > 0 for any t ≥ 0. Hence, along the paper, we suppose that the Kirchhoff function M : R + 0 → R + 0 is continuous and satisfies (M 2 ) there exists θ ∈ [1, p * s (α)/p), such that M(t)t ≤ θM (t) for all t ∈ R + 0 , where M (t) = t 0 M(τ)dτ.
Concerning the positive weight w, we assume that (w) w(x)|x| qα p * s (α) ∈ L r (R N ), with r = p * s (α) p * s (α)−q .Condition (w) is necessary, since it guaranties that the embedding Z(Ω) → L q (Ω, w) is compact, even when Ω is the entire space R N .Indeed, the natural solution space for problem (1.1) is the fractional density space Z(Ω), that is the closure of C ∞ 0 (Ω) with respect to the norm [ • ] s,p , given by Thus, by arguing similarly to Lemma 4.1 of [15], we have that the embedding Z(Ω) → L q (Ω, w) is compact with where the weighted norm is set by denotes the best fractional critical Hardy-Sobolev constant, given by Of course number H α is well-defined and strictly positive for any α ∈ [0, ps], since Lemma 2.1 of [15].We observe that when α = 0 then H 0 coincides with the critical Sobolev constant, while when α = sp then H sp is the true critical Hardy constant.In order to simplify the notation, throughout the paper we denote the true fractional Hardy constant and the true fractional Hardy norm with H = H sp and • H = • H sp , in (1.3) when α = sp.When s = 1 and p = 2, our problem (1.1) is related to the celebrated Kirchhoff equation proposed by Kirchhoff [21] in 1883 as a nonlinear generalization of D'Alembert's wave equation for free vibrations of elastic strings.This model describes a vibrating string, taking into account the changes in the length of the string during vibrations.In the equation (1.4) u = u(x, t) is the transverse string displacement at the space coordinate x and time t, L is the length of the string, h is the area of the cross section, E is Young's modulus of the material, ρ is the mass density, and P 0 is the initial tension.The early studies devoted to the Kirchhoff model were given by Bernstein [6], Lions [22] and Pohozaev [26].
In the nonlocal setting, Fiscella and Valdinoci [17] proposed a stationary Kirchhoff variational model in smooth bounded domains of R N , which takes into account the nonlocal aspect of the tension arising from nonlocal measurements of the fractional length of the string, given by Caffarelli et al. in [8].In other words, the problem studied in [17] is the fractional version of the Kirchhoff equation (1.4).Starting from [17], a great attention has been devoted to the study of fractional Kirchhoff problems; see for example [1-3, 9, 13-16, 24, 27].
The true local version of problem (1.1), namely when M ≡ 1 and s = 1, given by has been widely studied in [10,12,18,19].In these works, the authors proved the existence of infinitely many solutions of (1.5), when the parameter λ is controlled by a suitable threshold depending on the following Sobolev-Hardy constant .
In order to overcome the lack of compactness, due to the presence of two Hardy potentials in (1.5), they exploit a concentration compactness principle, applied to the combined norm |x| α dx.Because of the bi-nonlocal nature of the problem (1.1), the same approach of [10,12,18,19] can not work in our case.Indeed, due to the presence of a Kirchhoff coefficient M, for which the equation in (1.1) is no longer a pointwise identity, we have difficulties in considering a combined norm.Since Ω could be unbounded, we can not apply a concentration compactness argument because of the nonlocal nature of (−∆) p s , as well explained in Section 2.3 of [25].For these reasons, we use a tricky analysis of the energy functional which allows us to handle the two Hardy potentials in (1.1); see Sections 2 and 3.
Thus, we get the next multiplicity result for (1.1), which involves the main geometrical parameter κ σ = κ(σ) defined by for any σ ∈ (pθ, p * s (α)).A parameter similar to (1.6) already appeared in [9].Clearly κ σ ≤ a, since θ ≥ 1 and pθ ≤ σ.When θ = 1 in (M 2 ), we observe that parameter κ σ = a does not depend by the choice of σ.As shown in Section 2 of [9], the situation θ = 1 holds true in other cases, besides the obvious one M ≡ a.Now, we are ready to state the main result of the present paper.
The proof of Theorem 1.1 is obtained by applying suitable variational methods and consists of several steps.In Section 2 we study the compactness property of the Euler-Lagrange functional associated with (1.1).After that, in Section 3, we introduce a truncated functional which allows us to apply the symmetric mountain pass lemma in [20].Finally, we prove that the critical points of the truncated functional are indeed solutions of the original problem (1.1).
According to the variational nature, (weak) solutions of (1.1) correspond to critical points of the Euler-Lagrange functional J γ,λ : where Now, we discuss the compactness property for the functional J γ,λ , given by the Palais-Smale condition.We recall that We say that J γ,λ satisfies the Palais-Smale condition at level c if any Palais-Smale sequence {u n } n at level c admits a convergent subsequence in Z(Ω).

The truncated functional
In this section we prove that problem (1.1) admits a sequence of solutions which goes to zero.Firstly, we recall the definition of genus and some its fundamental properties; see [29] for more details.
Let E be a Banach space and A a subset of E. We say that A is symmetric if u ∈ A implies that −u ∈ A. For a closed symmetric set A which does not contain the origin, we define the genus µ(A) of A as the smallest integer k such that there exists an odd continuous mapping from A to R k \ {0}.If there does not exist such a k, we put µ(A) = ∞.Moreover, we set µ(∅) = 0.
Let us denote by Σ k the family of closed symmetric subsets A of E such that 0 / ∈ A and µ(A) ≥ k.Then we have the following result.(ii) If there is an odd homeomorphism from A onto B, then µ(A) = µ(B).
(iv) The n-dimensional sphere S n has a genus of n + 1 by the Borsuk-Ulam Theorem.
(v) If A is compact, then µ(A) < ∞ and there exist δ > 0 and a closed and symmetric neighborhood Now, we state the following variant of symmetric mountain pass lemma due to Kajikija [20].Lemma 3.2.Let E be an infinite-dimensional Banach space and let I ∈ C 1 (E, R) be a functional satisfying the conditions below: (h 1 ) I(u) is even, bounded from below, I(0) = 0 and I(u) satisfies the local Palais-Smale condition; that is, for some c * > 0, in the case when every sequence {u n } n in E satisfying I(u n ) → c < c * and I (u n ) → 0 in E * has a convergent subsequence; (h 2 ) For each n ∈ N, there exists an A n ∈ Σ n such that sup u∈A n I(u) < 0.
Then either (i) or (ii) below holds.
(i) There exists a sequence {u n } n such that I (u n ) = 0, I(u n ) < 0 and {u n } n converges to zero.
(ii) There exist two sequences {u n } n and {v n } n such that I (u Remark 3.3.It is worth to point out that in [20] the functional I verifies the Palais-Smale condition in global.Anyway, a careful analysis of the proof of Theorem 1 in [20], allows us to deduce that the result in [20] holds again if I satisfies the local Palais-Smale condition with the critical levels below zero. Let us note that the functional J γ,λ is not bounded from below in Z(Ω).Indeed, assumption (M 1 ) implies that M(t) > 0 for any t ∈ R + 0 and consequently by (M 2 ) we have M(t) M (t) ≤ θ t .Thus, integrating on [1, t], with t > 1, we get From this, by using (1.2) and (1.3), for any u ∈ Z(Ω) we have Now, fix γ ∈ (−∞, aH) and λ > 0 and let us consider the function and define Taking in mind the fact that Q γ,λ (t) ≤ 0 for t near zero, since q < p < p * s (α), and Q γ,λ * (R 1 ) > 0, we can infer that Q ∞).Thus, we consider the truncated functional It immediately follows that J γ,λ (u) → ∞ as [u] s,p → ∞, by (M 1 ), since γ ∈ (−∞, aH) and q < p.Hence, J γ,λ is coercive and bounded from below.Now, we prove a local Palais-Smale result for the truncated functional J γ,λ .
Taking into account that Z(Ω) is reflexive and separable (see Appendix A in [28]), we can find a sequence {ϕ n } n ⊂ Z(Ω) such that Z(Ω) = span{ϕ n : n ∈ N}.For any n ∈ N we can set X n = span{ϕ n } and Y n = ⊕ n i=1 X i .Lemma 3.5.For any γ ∈ (−∞, aH), λ > 0 and k ∈ N, there exists ε with M * = max τ∈[0,R 0 ] M(τ) < ∞, by continuity of M. Now, let be a positive constant such that Then, for any u ∈ Y k such that [u] s,p = , by (3.5) we get where the last inequality follows from (3.6).Hence we can find a constant ε By using (ii) and (iv) of Proposition 3.1 we have the thesis.
For any c ∈ R and any k ∈ N, let us define the set Lemma 3.6.For any γ ∈ (−∞, aH), λ > 0 and k ∈ N, we have that c k < 0.
Therefore, recalling that J γ,λ is bounded from below, we get Proof.Fix γ ∈ (−∞, aH) and λ > 0. It is easy to see that c k ≤ c k+1 for all k ∈ N. By Lemma 3.6 it follows that c k < 0, so we can assume that c k → c ≤ 0. Since J γ,λ satisfies the Palais-Smale condition at level c k by Lemma 3.4, we can argue as in [29] to see that all c k are critical value of J γ,λ .Now, we prove that c = 0. We argue by contradiction, and we suppose that c < 0. In view of Lemma 3.4, we know that K c is compact, so, by applying part (v) of Proposition 3.1 we can deduce that µ(K c) = k 0 < ∞ and there exists δ > 0 such that µ(K c) = µ(N δ (K c)) = k 0 .By Theorem 3.4 of [5], there exists ε ∈ (0, c) and an odd homeomorphism η : Z(Ω) → Z(Ω) such that η( J c+ε γ,λ \ N δ (K c)) ⊂ J c−ε γ,λ .Now, taking into account that c k is increasing and c k → c, we can find k ∈ N such that c k > c − ε and c k+k 0 ≤ c.Take A ∈ Σ k+k 0 such that sup u∈A J γ,λ (u) < c + ε.By using part (iii) of Proposition 3.1, we obtain However, in view of (3.7) and (3.9) we can see that which gives a contradiction in virtue of (3.10).Therefore, c = 0 and c k → 0.
p |x| p dx and to the critical norm Ω |u| p * (α)

Proposition 3 . 1 .
Let A and B be closed symmetric subsets of E which do not contain the origin.Then we have (i) If there exists an odd continuous mapping from A to B, then µ(A) ≤ µ(B).