The Solvability of Fractional Elliptic Equation with the Hardy Potential

(− Δ)u � C(N, s)P.V.􏽚 R u(x) − u(y) |x − y|N+2s dy, (2) where P.V. stands for the Cauchy principal value and constant C(N, s) is a constant. Recently, much attention has been devoted to the study of fractional Laplacian equations. One of the reasons comes from the fact that the fractional Laplacian arises in various areas and different applications, such as phase transitions, finance, stratified materials, flame propagation, ultrarelativistic limits of quantum mechanics, and water waves. For more details, see [1–6] and references therein. For fractional elliptic problems with the Hardy potential, Abdellaoui et al. [7] obtained the existence and summability of solutions to a class of nonlocal elliptic problem:

For fractional elliptic problems with the Hardy potential, Abdellaoui et al. [7] obtained the existence and summability of solutions to a class of nonlocal elliptic problem: in Ω, with f ∈ L m (Ω) and 0 < λ < Λ N,s . ey mainly considered the summability of solutions to (3) with f(x, u) � f(x) and the existence and regularity of solutions to (3) with f(x, u) � h(x)/u σ . Mi et al. [8] obtained the combined influence of the Hardy potential and lower order terms on the existence and regularity of solutions to the problem: Barrios et al. [9] discussed the existence and multiplicity of solutions to the following fractional elliptic equation: and α λ ∈ (0, (N − 2s/2)) is a parameter depending on λ. ey shown that problem (5) has at least one solution if 1 < p < p(λ, s) and problem (5) has no solution if p > p(λ, s).
Recently, Shang et al. [10] studied the existence and multiplicity of positive solutions to the following problem: where s ∈ (0, 1), N > 2s, 0 < p < 2 * s − 1, and 0 < μ < Λ N,s . Some other results of fractional elliptic equations with the Hardy potential, see [7,9,[11][12][13][14] and references therein. e local version of quasilinear problem related to problem (8) has been considered by Boccardo et al. [15]. ey analyzed the existence of nontrivial solutions to the following problem: where Ω ⊂ R N is a smooth bounded domain, 1 < p < N, r > p, g: Ω × R ⟶ R is a Carathéodory function, and there exist constants c 1 > 0 and q ∈ (1, p) such that g(s) ≤ c 1 s q− 1 for any s > 0. Motivated by the above works, the aim of this paper is to study the existence of solutions to problem (1) by the method of subsuper solutions and taking into advantage the combined effect of concave and convex nonlinearity.
Remark 1. In order to prove the above theorem, we study directly to the pseudodifferential operator, without the harmonic extension to an extra dimension by transforming the nonlocal problem into a local problem due to Caffarelli and Silvestre [16].

Remark 2.
To establish the upper bound for r (see (9)), we consider a radial solution w � A|x| ((2s− N)/2)+β with constant A > 0 to the problem: We obtain In order to have homogeneity, we have [17,18]. Hence, there is a unique element α λ such that c α λ � λ. us, we have α λ > β, that is, which implies that erefore, we can construct a supersolution to problem (1) for r < r(λ, s), just modifying the w found above. us, r(λ, s) is the threshold for the existence to problem (1). Now, we consider the nonexistence of solution to problem (1).

(21)
We have to prove that δ is smaller than the minimum of Φ(C 1 ). erefore, we have Moreover, by (10) and (13), we have for any τ > 1, us, for any τ > 1, . Hence, problem (1) has no solution at least δ > c 1 λ . erefore, the result of the above theorem is more general than [9].
Remark 4. We consider the function for 0 < α < θ < 1. We easily deduce that conditions (10) and (11) are fulfilled and M 0 � ∞. On the contrary, Similarly, in this case, problem (1) has no solution provided (27) , for some τ 0 > 1, the results of eorem 1 will be true for any τ > τ 0 . e paper is organized as follows. In Section 2, we present some definitions and preliminary tools, which will be used in the Proof of eorems 1 and 2. e Proof of eorems 1 and 2 are given in Section 3 and Section 4, respectively.

Preliminaries and Function Setting
In this section, we recall some known results for reader's convenience.
Denote the space equipped with the norm

Complexity 3
Let Ω be an open subset of R N . Given u ∈ L s and φ in the Schwartz class, the distribution (− Δ) s u in D ′ (Ω) is defined as We give some useful facts for the fractional Sobolev space. Let s ∈ (0, 1), and define the fractional Sobolev space:
We have to use the classical Sobolev theorem.
Theorem 3 (see [20], eorem 6.5). Let s ∈ (0, 1), then there exists a positive constant S � S (N, s), such that, for any measurable and compactly supported function u: R N ⟶ R, we have where 2 * s is the so-called Sobolev critical exponent.
In this paper, we consider the existences of energy solution to problem (1) with the critical and subcritical cases.
Definition 2. We say that u ∈ X s 0 (Ω) is an energy solution to problem (1), if for any φ ∈ X s 0 (Ω), We also need to consider the weak solution to problem (1).

Definition 3.
We say that u ∈ L 1 (Ω) is a weak solution to problem (1), if u ≥ 0 a.e. in Ω, u � 0 in R N \ Ω, and for all φ ∈ C 2s+β (Ω) ∩ C s (Ω), β > 0, in the weak sense, we say that u is a supersolution to problem (1). If u satisfies in the weak sense, we say that u is a subsolution to problem (1). Now, we recall the comparison lemma.
Lemma 1 (see [9]). Let u ∈ H s (R N ) and v ∈ H s (R N ) be solutions, respectively, to For the supercritical case, we need a prior regularity result, see [9], Lemma 2.2.

The Existence Result
We are now ready to prove eorem 1 by employing the idea contained in [9,15], whose proof will be split into several steps.
Step 2: supersolution for subcritical and critical case: 2 < r ≤ 2 * s . We look for a supersolution of the form w(x) ≔ A|x| − β with A ≥ 0 and β > 0 as real parameters and verify Since r ≤ 2 * s , we obtain By (49), we deduce that for the appropriate choice of A.

Complexity 5
For u, there is nothing to prove. Suppose the result is true up to order j. en, in Ω, So w j is well-defined by (54) and Lemma 2. By the induction hypothesis, for x ∈ Ω, and w j+1 − w j � 0, in R N \ Ω. en, by Lemma 1, we obtain w j+1 ≥ w j in Ω.

Complexity
On the contrary, by (11), the function F is nondecreasing. erefore, By the comparison principle we deduce that w 1 ≤ u in Ω.
In particular, for all x ∈ Ω, w j is a nondecreasing sequence which is bounded. erefore, w j monotone converges in L 1 (R N ) to a weak nonnegative solution u δ to (1) for 2 * s < r < r(λ, s). erefore, for δ small enough, we have built a minimal solution in both subcritical and supercritical case. Let that is, we show that M > 0.
We complete the Proof of eorem 1.

Nonexistence Result
In this section, we consider the nonexistence of solution to problem (1) in H s 0 .

Data Availability
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Conflicts of Interest
e authors declare that they have no conflicts of interest. Complexity 7