A fractional Kirchhoff problem involving a singular term and a critical nonlinearity

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. By combining variational methods with an appropriate truncation argument, we provide the existence of two solutions.


Introduction
This paper is devoted to the study of a class of Kirchhoff-type problems driven by a nonlocal fractional operator and involving a singular term and a critical nonlinearity. More precisely, we consider where Ω is an open bounded subset of ℝ N with continuous boundary, dimension N > 2s with parameter s ∈ (0, 1), 2 * s = 2N/(N − 2s) is the fractional critical Sobolev exponent, λ > 0 is a real parameter, θ ∈ (1, 2 * s /2), while γ ∈ (0, 1). Here (−∆) s is the fractional Laplace operator defined, up to normalization factors, by the Riesz potential as |y| N+2s dy, x ∈ ℝ N , along any φ ∈ C ∞ 0 (Ω); we refer to [11] and the recent monograph [22] for further details on the fractional Laplacian and the fractional Sobolev spaces H s (ℝ N ) and H s 0 (Ω). As is well explained in [11,22], problem (1.1) is the fractional version of the following nonlinear problem: where ∆ denotes the classical Laplace operator while, just for a general discussion, M(t) = t θ−1 for any t ∈ ℝ + 0 . In literature, problems like (1.1) and (1.2) are called of Kirchhoff type whenever the function M : ℝ + 0 → ℝ + 0 models the Kirchhoff prototype, given by In particular, when M(t) ≥ constant > 0 for any t ∈ ℝ + 0 , Kirchhoff problems are said to be non-degenerate and this happens for example if a > 0 in the model case (1.3). While if M(0) = 0 but M(t) > 0 for any t ∈ ℝ + , Kirchhoff problems are called degenerate. Of course, for (1.3) this occurs when a = 0.
This kind of nonlocal problems has been widely studied in recent years. We refer to [17][18][19][20][21] for different Kirchhoff problems with M like in (1.3), driven by the Laplace operator and involving a singular term of type u −γ . In [21], Liu and Sun study a Kirchhoff problem with a singular term and a Hardy potential by using the Nehari method. The same approach is used in [19] for a singular Kirchhoff problem with also a subcritical term. In [17], strongly assuming a > 0 in (1.3), Lei, Liao and Tang prove the existence of two solutions for a Kirchhoff problem like (1.2) by combining perturbation and variational methods. While in [18], Liao, Ke, Lei and Tang provide a uniqueness result for a singular Kirchhoff problem involving a negative critical nonlinearity by a minimization argument. By arguing similarly to [17], Liu, Tang, Liao and Wu [20] give the existence of two solutions for a critical Kirchhoff problem with a singular term of type |x| −β u −γ .
Problem (1.1) has been studied by Barrios, De Bonis, Medina and Peral [4] when θ = 1, namely without a Kirchhoff coefficient. They prove the existence of two solutions by applying the sub/supersolutions and Sattinger methods. In [8], Canino, Montoro, Sciunzi and Squassina generalize the results of [4,Section 3] to the delicate case of the p-fractional Laplace operator (−∆) s p . While in the last section of [1], Abdellaoui, Medina, Peral and Primo provide the existence of a solution for nonlinear fractional problems with a singularity like u −γ and a fractional Hardy term by perturbation methods. Concerning fractional Kirchhoff problems involving critical nonlinearities, we refer to [2,9,13,14,16,23] for existence results and to [5,12,24,25,29] for multiplicity results. In particular, in [9,13,14,23] different singular terms appear, but are given by the fractional Hardy potential. Inspired by the above works, we study a multiplicity result for problem (1.1). As far as we know, a fractional Kirchhoff problem involving a singular term of type u −γ has not been studied yet. We can state our result as follows. Theorem 1.1. Let s ∈ (0, 1), N > 2s, θ ∈ (1, 2 * s /2), γ ∈ (0, 1) and let Ω be an open bounded subset of ℝ N with ∂Ω continuous. Then there exists λ > 0 such that for any λ ∈ (0, λ) problem (1.1) has at least two different solutions.
The first solution of problem (1.1) is obtained by a suitable minimization argument, where we must pay attention to the nonlocal nature of the fractional Laplacian. Concerning the second solution, because of the presence of u −γ , we can not apply the usual critical point theory to problem (1.1). For this, we first study a perturbed problem obtained by truncating the singular term u −γ . Then by approximation we get our second solution of (1.1).
Finally, we observe that Theorem 1. The paper is organized as follows: In Section 2, we discuss the variational formulation of problem (1.1), and we introduce the perturbed problem. In Section 3, we prove the existence of the first solution of (1.1), and we give a possible generalization of this existence result at the end of the section. In Section 4, we prove the existence of a mountain pass solution for the perturbed problem. In Section 5, we prove Theorem 1.1.
Problem (1.1) has a variational structure, and the natural space where to find solutions is the homogeneous fractional Sobolev space H s 0 (Ω). In order to study (1.1) it is important to encode the "boundary condition" u = 0 in ℝ N \ Ω in the weak formulation, by considering also that the interaction between Ω and its complementary in ℝ N gives a positive contribution in the so-called Gagliardo norm, given as The functional space that takes into account this boundary condition will be denoted by X 0 and it is defined as We refer to [26] for a general definition of X 0 and its properties. We also would like to point out that when ∂Ω is continuous, by [15,Theorem 6] the space C ∞ 0 (Ω) is dense in X 0 , with respect to the norm (2.1). This last point will be used to overcome the singularity in problem (1.1).
In X 0 we can consider the norm which is equivalent to the usual one defined in (2.1) (see [26,Lemma 6]). We also recall that (X 0 , ‖ ⋅ ‖ X 0 ) is a Hilbert space, with the scalar product defined as From now on, in order to simplify the notation, we will denote ‖ ⋅ ‖ X 0 and ⟨ ⋅ , ⋅ ⟩ X 0 by ‖ ⋅ ‖ and ⟨ ⋅ , ⋅ ⟩, respectively, and ‖ ⋅ ‖ L q (Ω) by ‖ ⋅ ‖ q for any q ∈ [1, ∞].
In order to present the weak formulation of (1.1) and taking into account that we are looking for positive solutions, we will consider the following Kirchhoff problem: We say that u ∈ X 0 is a (weak) solution of problem (2.2) if u satisfies for any φ ∈ X 0 . Problem (2.2) has a variational structure and J λ : X 0 → ℝ, defined by is the underlying functional associated to (2.2). Because of the presence of a singular term in (2.2), the functional J λ is not differentiable on X 0 . Therefore, we can not apply directly the usual critical point theory to J λ in order to solve problem (2.2). However, it is possible to find a first solution of (2.2) by using a local minimization argument. In order to get the second solution of (2.2) we have to study an associated approximating problem. That is, for any n ∈ ℕ, we consider the following perturbed problem: For this, we say that u ∈ X 0 is a (weak) solution of problem (2.4) if u satisfies for any φ ∈ X 0 . In this case, solutions of (2.4) correspond to the critical points of the functional J n,λ : X 0 → ℝ, set as It is immediate to see that J n,λ is of class C 1 (X 0 ). We conclude this section by recalling the best constant of the fractional Sobolev embedding, which will be very useful to study the compactness property of the functional J n,λ . That is, we consider which is well defined and strictly positive, as shown in [10, Theorem 1.1].

A first solution for problem (1.1)
In this section, we prove the existence of a solution for problem (1.1) by a local minimization argument. For this, we first study the geometry of the functional J λ .

Theorem 3.2.
Let λ 0 be given as in Lemma 3.1. Then for any λ ∈ (0, Proof. Fix λ ∈ (0, λ 0 ] and let ρ be as given in Lemma 3.1. We first prove that there exists u 0 ∈ B ρ such that Since γ ∈ (0, 1), by the Hölder inequality, for any k ∈ ℕ we have Let w k = u k − u 0 ; by [7,Theorem 2] it holds true that s ≥ α > 0, and from this, since ρ ≤ 1, for k sufficiently large we have Thus, by (3.2), (3.4)-(3.6) and considering θ ≥ 1, it follows that, as k → ∞, Hence, u 0 is a local minimizer for J λ , with J λ (u 0 ) = m λ < 0, which implies that u 0 is nontrivial. Now, we prove that u 0 is a positive solution of (2.2). For any ψ ∈ X 0 , with ψ ≥ 0 a.e. in ℝ N , let us consider a t > 0 sufficiently small so that u 0 + tψ ∈ B ρ . Since u 0 is a local minimizer for J λ , we have From this, by dividing by t > 0 and passing to the limit as t → 0 + , it follows that We observe that in Ω as t → 0 + . Thus, by the Fatou lemma, we obtain Therefore, combining (3.7) and (3.8), we get Since u 0 is a local minimizer for J λ in B ρ , the functional I λ has a minimum at t = 0, that is, For any φ ∈ X 0 and any ε > 0, let us define ψ ε = u + 0 + εφ. Then by (3.9) we have We observe that, for a.e. x, y ∈ ℝ N , we obtain from which we immediately get From the last inequality it follows that N+2s dx dy. (3.13) Hence, denoting Ω ε = {x ∈ ℝ N : u + 0 (x) + εφ(x) ≤ 0} and combining (3.11) with (3.13), we get where the last equality follows from (3.10). Arguing similarly to (3.12), for a.e. x, y ∈ ℝ N we have N+2s , by the symmetry of the fractional kernel and (3.15), we get where we set U(x, y) = (u 0 (x) − u 0 (y))(φ(x) − φ(y)) |x − y| N+2s .
We end this section by observing that the result in Theorem 3.2 can be extended to more general Kirchhoff problems. That is, we can consider the problem where p * s = pN/(N − ps), with N > ps and p > 1, while the Kirchhoff coefficient M satisfies the following condition: (M) M : ℝ + 0 → ℝ + 0 is continuous and nondecreasing. There exist numbers a > 0 and ϑ such that for any t ∈ ℝ + 0 , The main operator (−∆) s p is the fractional p-Laplacian which may be defined, for any function φ ∈ C ∞ 0 (Ω), as where B ε (x) = {y ∈ ℝ N : |x − y| < ε}. Then, arguing as in the proof of Theorem 3.2 and observing that we have not used yet the assumption that ∂Ω is continuous, we can prove the following result.

A mountain pass solution for problem (2.4)
In this section, we prove the existence of a positive solution for the perturbed problem (2.4) by the mountain pass theorem. For this, throughout this section we assume n ∈ ℕ without further mentioning. Now, we first prove that the related functional J n,λ satisfies all the geometric features required by the mountain pass theorem.
Proof. Since γ ∈ (0, 1), by the subadditivity of t → t 1−γ , we have in Ω (4.1) for any u ∈ X 0 and any n ∈ ℕ. Thus, we have J n,λ (u) ≥ J λ (u) for any u ∈ X 0 and the first part of the lemma directly follows by Lemma 3.1.
We discuss now the compactness property for the functional J n,λ , given by the Palais-Smale condition. We recall that {u k } k ⊂ X 0 is a Palais-Smale sequence for J n,λ at level c ∈ ℝ if We say that J n,λ satisfies the Palais-Smale condition at level c if any Palais-Smale sequence {u k } k at level c admits a convergent subsequence in X 0 . Before proving this compactness condition, we introduce the following positive constants, which will help us for a better explanation: Proof. Let λ > 0 and let {u k } k be a Palais-Smale sequence in X 0 at level c ∈ ℝ, with c satisfying (4.4). We first prove the boundedness of {u k } k . By (4.2), there exists σ > 0 such that, as k → ∞, where the last two inequalities follow by (2.7), (4.1) and the Hölder inequality. Therefore, Thus, by inequality (3.12) we deduce that ‖u − k ‖ → 0 as k → ∞, which yields Hence, we can suppose that {u k } k is a sequence of nonnegative functions. By the boundedness of {u k } k and by using [26,Lemma 8] and [6,Theorem 4.9], there exist a subsequence, still denoted by {u k } k , and a function u ∈ X 0 such that in Ω, u k ≤ h a.e. in Ω, (4.5) as k → ∞, with h ∈ L p (Ω) for a fixed p ∈ [1, 2 * s ). If μ = 0, then immediately u k → 0 in X 0 as k → ∞. Hence, let us assume that μ > 0.
We now give a control from above for the functional J n,λ under a suitable situation. For this, we assume, without loss of generality, that 0 ∈ Ω. By [10], we know that the infimum in (2.7) is attained at the function (4.11) that is, it holds true that .
Let us fix r > 0 such that B 4r ⊂ Ω, where B 4r = {x ∈ ℝ N : |x| < 4r}, and let us introduce a cut-off function For any ε > 0, we set Then we can prove the following result.
By the elementary inequality with p = 2 * s /2, considering ε < r 1/q sufficiently small, with r given by (4.12), and since q < 1, we have with two positive constantsC and C 2 independent of ε. By combining (4.15) with (4.17), we get Thus, let us consider λ * > 0 such that and let us set where r and q are given in (4.12) and (4.16), respectively, while we still consider D 1 and D 2 as defined in (4.3). Then, by considering ε = λ ν 1 /q in (4.18), since (4.16) implies that ν 3 < 0, for any λ ∈ (0, λ 1 ) we have which concludes the proof.
We can now prove the existence result for (2.4) by applying the mountain pass theorem.
Proof. Let λ = min{λ 0 , λ 1 }, with λ 0 and λ 1 given in Lemmas 3.1 and 4.3, respectively. Let us consider λ ∈ (0, λ). By Lemma 4.1, the functional J n,λ verifies the mountain pass geometry. For this, we can set the critical mountain pass level as By Lemmas 4.1 and 4.3, we get Hence, by Lemma 4.2 the functional J n,λ satisfies the Palais-Smale condition at level c n,λ . Thus, the mountain pass theorem gives the existence of a critical point v n ∈ X 0 for J n,λ at level c n,λ . Since we obtain that v n is a nontrivial solution of (2.4). Furthermore, by (2.5) with test function φ = v − n and inequality (3.12), we can see that ‖v − n ‖ = 0, which implies that v n is nonnegative. By the maximum principle in [28, Proposition 2.17], we have that v n is a positive solution of (2.4), concluding the proof.

A second solution for problem (1.1)
In this last section, we prove the existence of a second solution for problem (1.1), as a limit of solutions of the perturbed problem (2.4). For this, here we need the assumption that ∂Ω is continuous in order to apply a density argument for the space X 0 .