Radial symmetry for a generalized nonlinear fractional p-Laplacian problem

School of Mathematics and Computer Science, Shanxi Normal University, Linfen, Shanxi 041004, China lebron_hww@163.com; zhanglih149@126.com; wgt2512@163.com Department of Mathematics, Texas A&M University, Kingsville, TX 78363-8202, USA ravi.agarwal@tamuk.edu Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, Jeddah 21589, Saudi Arabia


Introduction
Fractional-order differential equations are very suitable for describing materials and processes with memory and heritability, and their description of complex systems has the advantages of simple modeling, clear physical meaning of parameters and accurate description. Examples include a fractional differential model for the free dynamic response of viscoelastic single degree of freedom systems [18], a new noninteger model for convective straight fins with temperature-dependent thermal conductivity associated with Caputo-Fabrizio fractional derivative [20] and so on.
In this paper, we are concerned with a generalized nonlinear fractional p-Laplacian equation with negative power where Here 0 < s < 1, 2 < p < ∞, PV means the Cauchy principal value and F is a continuous function. For the purpose of making the integral meaningful, we need that The operator (−∆) s F ,p introduced in this paper includes some special cases. When F(·) is an identity map, (−∆) s F ,p becomes the fractional p-Laplacian (−∆) s p . Based on this, (−∆) s p will become fractional Laplacian(−∆) s if p = 2, this is well known. In order to surmount the nonlocality of fractional Laplacian, Caffarelli and Silvestre [4] introduced the extension method that reduced this nonlocal problem into a local one in higher dimensions. This method is briefly described below. Given a function g : R n → R, let the extension G : R n × [0, ∞) → R n that meets the following condition: They concluded that The extension method mentioned above has been utilized to discuss equations involving fractional Laplacian; see [3,5,15]. Another way to overcome the nonlocality is the integral equations method. Applications of this method can be found in [7,12,13,23,25,26,36]. However, there are still some operators that cannot be solved by the above methods; see [5]. To overcome the difficulty, a direct method of moving planes is introduced in [11]. Gradually, it is used to tackle a series of problems involving kinds of nonlinear operators. For example, the relevant properties of solutions for nonlinear elliptic equations are obtained, besides, it has been highly applied in studying the properties of fractional Laplacian equations and systems; see [8,14,24,28,31,32]. Furthermore, there are some excellent results by using this method to study the radial symmetry and monotonicity of http://www.journals.vu.lt/nonlinear-analysis the solutions of fractional p-Laplacian equations and systems; see [9,16,22,27,[33][34][35]. For fully nonlinear nonlocal operators, for example, this direct method has been further developed by Chen and Li in [10].
In [17], Davila, Wang and Wei proved sharp Hölder continuity and an estimate of rupture sets for sequences of solutions of the following nonlinear problem with negative exponent: ∆u = 1 u p in Ω, p > 1. The above problem arises in modeling an electrostatic microelectromechanical system (MEMS) device. The solution of the singularity in the equation, namely u ≈ 0 in some region, represents the rupture in the device in the physical model.
In [19], Jiang and Ni studied the singular elliptic equation where Ω ⊂ R n , n 2, is a bounded smooth domain and α > 1. When n = 2 and α = 3, the above equation is used to model steady states of van der Waals force driven thin films of viscous fluids. They also considered the physical problem when total volume of the fluid is prescribed. Singular elliptic equations modeling steady states of van der Waals force driven thin films have been mathematically rigorously studied with no flux Neumann boundary condition. They gave a complete description of all continuous radially symmetric solutions. In particular, they constructed both nontrivial smooth solutions and singular solutions.
In [6], Ma and Cai studied the following nonlinear fractional Laplacian equation with negative powers by using the direct method of moving planes: The above results of all encourage us to further study a generalized nonlinear fractional p-Laplacian equation with negative powers by the direct method of moving planes. As far as authors know, up to now, this is a new attempt to study a class of equations (1) combining a generalized fractional p-Laplacian with negative powers. Interestingly, this method can also be analogously applied to a generalized Hénon-type nonlinear fractional p-Laplacian equation with negative power where σ < 0 and γ > 0 are constants.
The paper is structured as follows. In Section 2, we mainly present some lemmas used in the following part. In Section 3, we study radial symmetry and monotonicity of two generalized fractional p-Laplacian equations with negative powers by applying the direct method of moving planes.

Auxiliary lemmas
Throughout the next section, we assume that there exists a constant L > 0 such that Then If φ(x) = 0 at some point x ∈ ℵ, then φ(x) = 0 holds for almost all points x in R n . When ℵ is unbounded area, we need further assume that lim |x|→∞ φ(x) 0, then the same conclusion still holds.
Proof. Suppose (4) is not true, then there is an x 0 such that φ(x 0 ) = min ℵ φ < 0. According to the second inequality in (3), This is a direct contradiction to the first inequality in (3), hence (4) holds. When φ(x 0 ) = 0 at some point x 0 ∈ ℵ, then On the other hand, from (3) we have so, the integral result is 0. Because u is nonnegative, one could get that φ(x) = 0 almost everywhere in R n . This completes the lemma.
then the same conclusions still hold. where To estimate J 1 , we notice the fact on the basis of strict monotonicity of Q, we have Therefore, To evaluate J 2 , by using the mean value theorem we obtain Combining (6), (7) and (8), one can deduce This inequality contradicts the first condition in (5), thus d κ (x) 0. If d κ (x) = 0 at some point x ∈ ℵ, equivalently, x is a minimum of d κ in ℵ, so, J 2 = 0. Now, according to the first inequality in (5), we get J 1 0, which means Considering the monotonicity of Q, for almost all y ∈ Σ, http://www.journals.vu.lt/nonlinear-analysis Consequently, d κ (y) = 0 almost everywhere in Σ. Besides, in the light of the antisymmetry of d κ , we receive d κ (y) = 0 almost everywhere in R n . If ℵ is unbounded area, under this circumstance, in view of assumption lim |x|→∞ d κ (x) 0, suppose that d κ (x) 0, x ∈ Σ, is false, then a negative minimum of d κ is obtained at some point x ∈ Σ. Being similar to the above argument, one can find a contradiction. The proof is completed.
Lemma 3 [Narrow region principle]. Let ℵ be bounded narrow area in Σ such that it is contained in {x|κ − δ < x 1 < κ} with small δ. Presume that c(x) is bounded from below in ℵ and and there exists y 0 ∈ Σ satisfying d κ (y 0 ) > 0, then when δ is sufficiently small, one can get Proof. Suppose the contrary, then for any δ > 0, there exists an x δ ∈ ℵ δ such that Then for δ k = 1/k, k = 1, 2, . . . , there exists x δ k and ℵ δ k , let us call them x k and ℵ k such that By inequality (6) we deduce where Similarly to (8), we can get H 2 0, from [33] we can get Combining (9) with (10), one can get (1))δ x k < 0. This contradicts with the equation, hence the proof is completed.

Lemma 4 [Decay at infinity].
Let ℵ be unbound area in Σ, and let φ ∈ l s p ∩ C 1,1 loc (ℵ) be a solution of Then there exists a positive constant R 0 (depending on c(x) and independent of φ(x) and Proof. By inequality (6) we get where M is a constant. For each fixed κ, when |x 0 | κ, Then we have this contradicts with the condition of c(x). This completes the proof.
3 The generalized fractional p-Laplacian equations with negative powers where sp/(γ+1) < t < 1 and > 0 are constants. Then φ(x) must be radially symmetric and monotone increasing about some point in R n .
Step 2. In this step, we will move P κ to the limiting position, this moving process could give the symmetry about the positive solution φ(x). (11) means that there is an initial point to move P κ , we could move P κ so long as (11) holds. Let If (12) does not hold, then by Lemma 2 we get therefore, there exists a small δ > 0 and a constant b δ , which satisfy According to the continuity of d κ about κ, there exists 0 < < δ such that In This, combining with (13), indicates This violates the definition of κ 0 . Thus, (12) holds. The proof is completed.
Next, we discuss the radial symmetry of a generalized nonlinear Hénon-type fractional p-Laplacian equation with negative power.
Step 1. We demonstrate that holds when κ is sufficiently negative. Near the singular point 0 κ of φ(x) and φ κ (x), we manifest that Σ − κ has no intersection with B (0 κ ) for certain small ε > 0. In fact, we consider that when κ is sufficiently negative, obviously, 0 κ is a sufficiently negative point, we have Since c(x) ∼ o(1/|x| sp ), by applying Lemma 4 we could work out that there is an R 0 > 0, when d κ (x) gets the negative minimum at x * in Σ κ , then the following relation is true: That is to say, (14) holds when κ sufficiently negative.
We prove that Suppose κ 0 < 0, we can use Lemmas 3 and 4 to state that P κ can be moved further right, this contradicts with κ 0 . By condition of u(x), That is, there exist ε, h 0 > 0 such that From (15) the situation where the negative minimum of d κ0 (x) is obtained in B c R0 (0) does not exist. We also show that it cannot be gained in the internal of B R0 (0). That is, when κ is close enough to κ 0 , When κ 0 < 0, by Lemma 2 one can get There is a positive constant j 0 , which satisfies d κ0 (x) j 0 , x ∈ (Σ κ0−δ ∩ B R0 (0)) \ 0 κ0 .