Radial Symmetry for Weak Positive Solutions of Fractional Laplacian with a Singular Nonlinearity

Abstract: This paper is concerned with the radial symmetry weak positive solutions for a class of singular fractional Laplacian. The main results in the paper demonstrate the existence and multiplicity of radial symmetry weak positive solutions by Schwarz spherical rearrangement, constrained minimization, and Ekeland’s variational principle. It is worth pointing out that our results extend the previous works of T. Mukherjee and K. Sreenadh to a setting in which the testing functions need not have a compact support. Moreover, we weakened one of the conditions used in their papers. Our results improve on existing studies on radial symmetry solutions of nonlocal boundary value problems.


Introduction
In this paper, we focus on radial symmetry positive solutions to a singular elliptic problem involving a nonlocal operator: the fractional powers of the Laplacian in a bounded sphere domain in B R (0) ⊆ R N (N ≥ 3).Nonlinear equations with fractional powers of the Laplacian are actively studied.The fractions of the Laplacian are the infinitesimal generators of Lévy stable diffusion processes and appear in anomalous diffusion in plasmas, population dynamics, American options in finances, and geophysical fluid dynamics.For more details, we refer the reader to [1,2].To circumvent the nonlocal nature of the fractional Laplacian operator, Caffarelli, Salsa, and Silvestre [3,4] introduced the s-harmonic extension, which turns the nonlocal problem into a local one in higher dimensions.In our paper, we are interested in the existence of radial symmetry weak positive solutions that satisfy the singular fractional Laplacian boundary value problem, where B R (0) ⊂ R N (N ≥ 3) is the ball centered at 0 with radius R, N > 2s(0 < s < 1), 0 < γ < 1 < β < 2 * s − 1 = N+2s N−2s , λ > 0 is a real parameter, and the nonnegative real function a : Ω −→ R is integrable.
Let Ω ⊂ R N (N ≥ 3) be a bounded domain with a smooth boundary.For 0 < s < 1, the fractional Laplacian (−∆) s u is defined as (−∆) s u(x) := C N, s P.V. Γ(1−s) is a normalization constant.The fractional Sobolev space H s (Ω) is as follows: endowed with the natural norm .
As in the classical case, we denote by H s 0 (Ω) the closure of C ∞ 0 (Ω) with respect to the norm • H s .The inequalities about a Sobolev space and the embedding of the spaces H s 0 (Ω) into the Lebesgue spaces L p have been exhaustively researched in [6,9,10].Moreover, there is another norm .
In recent years, elliptic problems with a singular nonlinearity have attracted many researchers who study partial differential equations.Firstly, for the local operator (s = 1), the pioneering work by Crandall, Rabinowitz, and Tartar [11] starts the following singular Laplacian Equation (2).
where Ω is a bounded domain with a smooth boundary, 0 , λ > 0 is a real parameter, and the real function a : Ω −→ R is integrable in domain Ω.In [11], the authors proved that Equation (2) has a unique class solution u ∈ C 1 (Ω) ∩ C 2 (Ω) when λ = 0, a(x) = 1.By means of the Ekeland's variational principle, Sun [12] proved that Equation (2) has at least two weak positive solutions in H 1 0 (Ω) when the parameter λ is sufficiently small and β < 2 * − 1.By using the geometry of the Nehari manifold and the concentration-compactness method, our previous work [13] achieved results similar to [12].Furthermore, our results improved on the existing research on the power exponent β = 2 * − 1. Local Equation (1) and some other versions of it have been extensively studied over the past decades; for further information, one can refer to [11][12][13][14][15][16][17][18][19][20] and references therein.
In the nonlocal setting (0 < s < 1), the existence of weak solutions and various properties of solutions have been considered for the fractional Laplacian with a singular nonlinearity, Equation (1), by many authors in recent years.In [21], the author stated that u ∈ H s 0 (Ω) is a weak solution of Equation ( 1) with λ = 0 if the identity holds.By using the sub-supersolution method, the author proved the existence and uniqueness of a weak positive solution of Equation ( 1) with λ = 0.In [22], using variational methods, the authors proved that Equation (1) has at least two distinct weak positive solutions u, v ∈ L ∞ (Ω) when β < 2 * s − 1, among other conditions.Before stating the main results contained in this paper, we need to clarify the concept of weak positive solutions.We say that the function u ∈ H s 0 (B R (0)) is a weak solution of Equation (1) if u satisfies Equation (1) weakly.More precisely, we are looking for a function u from B R (0) to R such that u ∈ H s 0 (B R (0)), u(x) > 0 a.e. in B R (0), and where The greatest difficulty in this problem is that the vanish boundary value is such that the nonlinearity singular is at the boundary ∂B R (0).Therefore, the essence of this problem is determining which class of the testing function φ makes Equation (3) hold.It is worth emphasizing that since 0 < 1 − γ < 1, the natural associated functional is not Frechet-differentiable.So, the fractional singular elliptic Equation (1) cannot be studied by directly using critical point theory.In recent years, the study of elliptic problems with a singular nonlinearity has attracted many researchers of partial differential equations ( [23][24][25] and the references therein).In [24], the authors studied the existence, regularity, and multiplicity of weak solutions for fractional p-Laplacian equations with singular nonlinearities via fibering maps.The authors studied the existence and regularity of weak solutions to Equation (1). A. Capella, J. Davila, L. Dupaigne, and Y. Sire [25] provided new results with respect to the existence and regularity of radial extremal solutions for some nonlocal problems with smooth nonlinearity by following the s-harmonic extension approach, as in [3].Recently, W.X. Chen and C.M. Li [26] established radial symmetry and monotonicity for positive solutions to the fractional p-Laplacian by moving planes.
As far as we know, there are no published results with respect to the existence and multiplicity of radial symmetry weak solutions to Equation (1) in the sense of Equation (3).As we know, the moving plane method is one of the most effective strategies to establish radial symmetry for weak solutions of the classic Laplacian equations.However, in our case, because of the singular nature of our problem, we have to manage more difficulties.One way to overcome these difficulties is by using the variational principle combined with the Schwarz spherical rearrangement.
The structure of the paper is as follows.In Section 2, we give some preliminaries and basic facts, and we formulate our main results.In Section 3, we use variational methods, Nehari manifold, and Schwarz spherical rearrangement to prove our main results.

Preliminaries and Main Results
In this section, we present some collected preliminary facts for future reference.Before proceeding, we need some definitions for the spaces, results, and notations.Throughout this paper, we make use of the following notations.C, C 0 , C 1 , C 2 , • • • denote (possibly different) positive constants.We denote B R (0) as B. Let |Ω| be the measure of domain Ω and let • q be the L q (Ω)(1 By using the embedding theorems in [4], we derive H s 0 (Ω) so that it is compactly and continuously embedded in L p (Ω) when 1 ≤ p < 2 * s = 2N N−2s (N > 2s), and the embedding is continuous but not compact if p = 2 * s .We can let (see, for instance, [6,27,28]) Now, we can state the main results of our paper.
Then, there exists a real λ * such that for all λ ∈ (0, λ * ), Equation (1) has at least two radial symmetry weak positive solutions u λ , v λ ∈ H s 0 (B) in the sense of Equation (3).
Therefore, one cannot replace H s 0 (B) in (3.1) by C ∞ 0 (B).We point out that there is a requirement for the testing functions φ in Definition 2.1 in [24] and Definition 2.1 in [22], i.e., φ ∈ C ∞ 0 (B).This indicates that our results include and extend their previous conclusions.
Remark 2. It is worth pointing out that we only assume the coefficient a(x) ≥ 0. This greatly relaxes the condition for a(x) in [22].Here, the authors require that a(x) has a uniform positive lower bound and that there exists a positive constant θ > 0 such that a(x) ≥ θ for all x ∈ Ω.Hence, the results reported in this paper are new in the area of singular fractional elliptic problems.

Existence and Multiplicity of Weak Positive Solution of Equation (1)
We are now in a position to give the proof of Theorem (4).To start, let us define the Nehari manifold, Notice that u ∈ Λ if u is a weak positive solution of Equation ( 1).The fact suggests that we apply the following splitting for Λ.
To obtain Theorem 1, we divide the proof into several preliminary lemmas.
Proof.Since u ∈ Λ + , using the definition of Λ and Λ + , we obtain From the H ölder inequality, we derive the existence of the constant C > 0 such that Thus, we obtain Therefore, the result of Lemma 1 follows by letting 2 > 1 − γ.This completes the proof of Lemma 1.
Lemma 2. The functional I λ is coercive and bounded below on Λ + .
Proof.Let u ∈ Λ + .Combining the definition of Λ + and Equation ( 5), we have for some positive constants C 1 and C 2 .This implies that I λ is coercive and bounded below on Λ + .This completes the proof of Lemma 2.
Lemma 3. The minimal value m λ = inf Proof.By using the Hölder inequality and Equation (4), we get Applying the inequality in Equation ( 5), we deduce where U r = {u ∈ Λ + | u ≤ r}.Then, we can choose a small enough λ 1 > 0 such that for any fixed λ ∈ (0, λ 1 ), it follows that Furthermore, for any fixed v > 0, simple calculations show that we conclude that if t > 0 is sufficiently small, then I λ (tv) < 0 for any fixed v > 0. This implies that m λ = inf Λ + I λ < 0. This completes the proof of Lemma 3.
For the reader's convenience, we are ready to describe Lemma 4 (below) on the embedding properties of H s 0 (Ω).We refer to [6,9] and their references for its proof.
Lemma 4. Let s ∈ (0, 1) and p ∈ [1, +∞) such that sp < N. Let Ω ⊂ R N be an extension domain for W s,p (Ω).Then, there exists a positive constant C(N, s, p, Ω) such that, for any u ∈ W s,p (Ω), for any q ∈ [p, p * s = N p N−sp ]; that is, the space W s,p (Ω) is continuously embedded in L q (Ω) for any q ∈ [p, p * s ].If, in addition, Ω is bounded, then the space W s,p (Ω) is compactly embedded in L q (Ω) for any q ∈ [p, p * s ].
To state the next results, we need the next Lemma on Schwarz symmetrization and rearrangement, presented without proofs.One can refer to [29][30][31].Assume u is a real function defined in R N .The distribution function of u is defined as Then, µ u is non-increasing and right-continuous.The decreasing rearrangement of u is given by is defined as the Schwarz symmetrization of u.The function u has the following basic properties.
Lemma 5. Assume u, v are integral functions in R N , and let g : R N → R be non-decreasing nonnegative functions.Then, we conclude that (1) Lemma 6.For all λ ∈ (0, λ 1 ), there exists a function u λ ∈ Λ + which is radially symmetric about the origin such that Proof.The proof is inspired by [14].Let {u n } ⊆ U r be a minimizing sequence such that and .
Consequently, we obtain Using the Brezis-Lieb Lemma, we derive and Recall Equation ( 7) and m λ < 0; thus, u n ≤ r 0 < r for some positive constant r independent of n.So, from Equation ( 11) and u λ ∈ U r , we have u n − u λ ∈ U r while n is large enough.By using Equation ( 8) again, we deduce Combining the above arguments with Equations ( 9)-( 11), we have Since u n u λ weakly in H s 0 (B), by Lemma 4, we infer that u n → u λ strongly in L 1+β (B), thus, In order to apply the Schwarz symmetrization rearrangement of Lemma 5, we should extend u λ to a function defined in R N .In fact, by using the extension theorems, u λ can be extended to a function in H s (R N ) by defining it as zero outside of R N \ B. Since the functions t 1+β and t 1−γ are non-decreasing, by using Lemma 5, we have Consequently, we deduce that Therefore, the radial symmetry function u λ := ( u λ ) ∈ Λ + is also a minimizer of I λ in Λ + .This completes the proof of Lemma 6.

Existence of radial symmetry weak positive solution u
Proof.For any φ ∈ H s 0 (B) with φ ≥ 0 and t > 0 small enough, since u λ is a minimizer, we have Dividing by t > 0 and letting t → 0 therefore shows This means that u λ ∈ H s 0 (B) is a weak subsolution (−∆) s u ≥ 0, in B. In the following, we prove that u λ > 0, in B. We need the following strong maximum principle for the nonlocal operator (−∆) s (Theorem 4.1 in [32]).For the convenience of the reader, we report the main result of Theorem 4.1 in [32].If u ∈ H s 0 (Ω) satisfies, in a weak sense, that (−∆) s u ≥ 0 in Ω and This completes the proof of Lemma 6.
From Lemma 3 in [12], we have the following Lemma 8 immediately below.

Proof. Let
Using the continuity of g, g(0) > 0, and u λ ∈ Λ + , we deduce that there exists T > 0 such that g(t) > 0 for all t ∈ [0, T].On the other hand, applying Lemma 8, for each t > 0 there exists t > 0 such that t (u λ + tϕ) ∈ Λ + .Therefore, t → 1 as t → 0, and for each t ∈ [0, T], we obtain Proof.The novelty of Equation ( 1) lies not only in the non-differentiability of the corresponding functional I λ (u) but also in the singularity of Equation (1).There seem to be difficulties to get that the minimizer u λ is a weak solution of Equation ( 1) directly from critical point theory.Inspired by Y.J.Sun [12], using direct and detailed computations, we still proved that minimizer u λ is a weak solution of Equation (1).
Recall that in Lemma 9, we infer that for any ϕ ∈ H s 0 (B), ϕ ≥ 0, and t ∈ [0, T], there is Dividing t > 0 and letting t → 0 + implies that and the conclusion follows.The proof of this lemma is completed.
Existence of a weak positive H s 0 (B)− solution v λ .
Proof.We claim Λ 0 = {0}.Suppose, by contradiction, that there exists an h ∈ Λ 0 with h = 0. From the definitions of Λ 0 and Λ, it follows that Therefore, we find that On the other hand, by using Equation ( 5) and fractional Sobolev inequality, we infer that where the constant C > 0 is independent of λ.Since lim λ→0 1+γ λ(β+γ) = +∞, it means that there exists λ 2 > 0 small enough to satisfy and, consequently, Using the Sobolev inequalities and continuous compact embedding, we have u n → u in L 1+β (B) and u ∈ Λ − ∪ Λ 0 .Recalling the definition of Λ − once more, we infer that thus, u = 0, i.e., u ∈ Λ − .This completes the proof of Lemma 11.
1, the sequence {u n } is bounded in H s 0 (B).Thus, we can claim that there exists a subsequence of {u n } (still denoted by {u n }) such that u n u λ weakly in H s 0 (B), strongly in L p (B)(1 ≤ p < 2N N−2s ), and pointwise a.e. in B. According to H ölder's inequality, as n