In this paper, we consider the following fractional -equation with a gradient term:
We first prove the uniqueness and monotonicity of positive solutions in a bounded domain. Then by estimating the singular integrals which define the fractional -laplacian along a sequence of approximate maximum points, we obtain monotonicity of positive solutions in the whole space via the sliding method. In order to solve the difficulties caused by the gradient term, we introduce some new techniques which may also be applied to investigate the qualitative properties of solutions for many problems with gradient terms. Our results are extensions of Berestycki and Nirenberg [Monotonicity, symmetry and antisymmetry of solutions of semilinear elliptic equations, J. Geom. Phys. 5 (1988), 237–275] and Wu and Chen [The sliding methods for the fractional p-Laplacian, Adv. Math. 361 (2020), 106933].
In studying differential equations it is often of interest to know if the solutions have symmetry, or perhaps monotonicity, in some direction. Monotonicity and symmetry for solutions to the -Laplace equation in bounded domains were first obtained in [1,2]. In , Gidas et al. obtained monotonicity and symmetry for positive solutions to the Dirichlet boundary value problem by using the maximum principle and the method of moving planes. Subsequently, Berestycki and Nirenberg  investigated the symmetry and monotonicity to the solutions to second-order elliptic equation with gradient term
During the last few decades, equations involving the fractional Laplacian and the fractional -Laplacian have been extensively studied. Chen et al.  developed a direct method of moving planes for the fractional semilinear equation and proved the symmetry and monotonicity of the solution. Dipierro et al.  obtained symmetry and monotonicity of bounded solutions for fractional equation in unbounded domain with the epigraph property. We also refer readers to [4,8] for some typical applications of the sliding method. Very recently, Liu , Wu and Chen [10,11] introduced a direct sliding method for the fractional Laplacian and the fractional -Laplacian; Wang  obtained uniqueness and monotonicity of solutions to the fractional equation with a gradient term in a bounded domain and upper half-space; Dai et al.  investigated nonlinear equations involving pseudo-relativistic Schrödinger operators; Wu  developed a sliding method for the higher-order fractional Laplacians.
Motivated by the aforementioned papers, the goal of this paper is to extend the results in  to the fractional -equation. That is, we study the monotonicity and uniqueness of solutions for the following nonlinear nonlocal equation with a gradient term
where , and denotes the gradient of , fractional -Laplacian is given by
where P.V. represents the Cauchy principal value. The fractional -Laplacian we consider in this paper is actually a special case of the following fully nonlinear nonlocal operator, introduced in , with , ,
In order for the integral (1.3) to make sense, we require that
On the one hand, we extend the case in  to the fractional -Laplacian case , and extend the bounded domain to . On the other hand, the nonlinear term we will deal with contains the nonlinear term or . The first difficulty is caused by the gradient term. And the second difficulty is that compared with the fractional Laplacian; the fractional -Laplacian shows more complexity due to its nonlinearity.
In order to apply the sliding method, we give the exterior conditions on . Let and assume that
( ) for any three points and lying on a segment parallel to the axis, , with , we have
For bounded domain, we prove
Suppose that satisfies ( ) and is a solution of problem
where is a bounded domain which is convex in direction. Assume that is continuous in all variables, Lipschitz continuous in and nondecreasing in . Then is strictly monotone increasing with respect to in , i.e., for any ,
Furthermore, the solution of (1.6) is unique.
For the finite cylinder , where and is a bounded domain in with smooth boundary, the results of Theorem 1.1 still hold.
The conditions assumed in Theorems 1.1 and 1 of  are different, and there is no relation between them. Dai et al.  studied the positive solution and obtained that was strictly increasing in the left half of (i.e., ) in direction with by the method of moving planes, but the solution obtained can be negative and is strictly increasing with respect to in the whole domain by the sliding method.
(Antisymmetry) Assume that the conditions of Theorem 1.1 are satisfied and in addition that is odd in on . If is odd in , then is odd, i.e., antisymmetric in :
This follows from the fact that is a solution satisfying the same conditions, and so equals .
We say that is in for , if there exists a constant such that
For the whole space, we prove
Suppose that is a solution to
Assume that is bounded, continuous in all variables and satisfies
Suppose that there exists such that
Then u is strictly monotone increasing with respect to , and furthermore, it depends on only.
Theorem 1.2 is closely related to the well-known De Giorgi conjecture . As an example, we may think of , which yields the fractional Allen-Cahn equation, that is, a widely studied model in phase transitions in media with long-range particle interactions . The conclusion of Theorem 1.2 is also valid for , and there is no corresponding result in .
In order to solve the difficulty that the nonlinear terms at the right-hand side of (1.6) and (1.7) contain the gradient term, in bounded domains when deriving the contradiction for the minimum point of the function (see Section 2 for definition), we use the technique of finding the minimum value of the function for the variables and at the same time. This is different from the previous sliding process which only finds the minimum value of the variable for the fixed . In the whole space, we estimate the singular integrals defining the fractional -Laplacian along a sequence of approximate maximum, and and the sequence of approximate maximum are estimated at the same time.
For more related articles on symmetry and nonexistence results of local and nonlocal equations, we also refer readers to  for Laplace equations with a gradient term, [20,21, 22,23,24, 25,26] for fractional equations,  for weighted fractional equation, [28,29] for fractional equations with a gradient term, [30,31,32] for fully nonlinear equations with a gradient term, [33, 34,35,36] for fractional -Laplace equation,  for parabolic -Laplace equation and  for -Laplace equation. For results related to the existence of solutions for fractional -Laplacian problems, we refer to  and references therein.
The paper is organized as follows. In Section 2, Theorem 1.1 is proved via the sliding method. In Section 3, we derive monotonicity for the fractional -equation with a gradient term in .
2 Monotonicity and uniqueness of solutions in bounded domains
For simplicity, we list some notations used frequently. For denote . Set
Proof of Theorem 1.1
When , reduces to fractional Laplacian , see  for the proof of Theorem 1.1. And the following proof will omit the case of . When , the main difference between the fractional -Laplacian and the fractional Laplacian is that the former is a nonlinear operator, hence we need to use instead of and this makes the proof different. Now we will give a detailed proof of Theorem 1.1 in the case of .
For , it is defined on the set , which is obtained from by sliding it downward a distance parallel to the axis, where Set
We mainly divide the following two steps to prove that is strictly increasing in the direction, i.e.,
Step 1. For sufficiently close to , i.e., is narrow, we claim that there exists small enough such that
If (2.13) is false, we set
From the condition , can be obtained for some . Noting that , we arrive at . So . Since is a minimizing point, we have , i.e., . Since satisfies the same equation (1.6) in as does in , and is nondecreasing in , so we have
where is a function satisfying
On the other hand, note that is a strictly increasing function in , and By the definition of the fractional -Laplacian, we have
To estimate noting that is the minimum point of in , it follows
By the monotonicity of , we derive
Next we prove that . Similar to , we need the following lemma.
 For , there exists a constant such that
for arbitrary .
Noting that in and , it implies
By Lemma 2.1, we have
From (1.4) in the condition , there exists a point such that
Since function is continuous in , there exists a small and such that
where denotes the width of in the direction and is bounded. This contradicts (2.15).
Hence, (2.13) is true for sufficiently close to .
We will prove
Otherwise, assume we will show that the domain be slided upward a little bit more and we still have
which contradicts the definition of .
Since by the condition and it follows
If there exists a point such that , then is the minimum point, similar to (2.20), we have
which contradicts to
The minimum can be obtained for some where by condition . We carve out of a closed set such that is narrow. According to (2.23),
From the continuity of in , we have for small
From , it follows
So and . Since and small , we obtain that is a narrow domain. Similar to (2.15), we have
Similar to (2.20), by narrow domain , we have
This is a contradiction. Hence we derive (2.21), which contradicts to the definition of . So Therefore, we have shown that
Let us prove (2.12). Since
if there exists a point for some such that then is the minimum point and similar to (2.20), we have
Therefore, we arrive at (2.12).
Now we prove uniqueness. If is another solution satisfying the same conditions, the same argument as before but replace with . Similar to (2.25), we have in for any . Hence, . Interchanging the roles of and , we find the opposite inequality. Therefore, .
This completes the proof of Theorem 1.1.□
3 Monotonicity of solutions in
In the section, we prove Theorem 1.2 by the sliding method based on the following lemma.
Proof of Theorem 1.2
Denote . For any , define
From (1.9), there exists a constant such that
For any , no matter where is, we have either
Outline of the proof: We will use the sliding method to prove the monotonicity of and divide the proof into three Steps.
This provides the starting point for the sliding method. Then in Step 2, we decrease continuously as long as (3.28) holds to its limiting position. Define
We will show that . In Step 3, we deduce that the solution must be strictly monotone increasing in and only depends on .
Now we will show the details in the following three steps.
Step 1. We will show that
If not, then
so for some there exists a sequence , such that
Denote . Since as it implies that there exists such that
where is a constant, taking such that . Set
From (3.31), there exists a sequence , with such that
Since for any and hence
It follows that there exists a point such that
Since , no matter where is, one of the points and is in the domain where is nonincreasing in by (1.11). Since , from (3.26), both and are close to 1, while from (3.27), both and are close to . So we apply the monotonicity of to derive that
From (3.33), we have . It follows that as . Let , we have
where is a positive constant which can be different from line to line. Denote
Since is bounded, one derive that is uniformly Hölder continuous, by the Arzelà-Ascoli theorem, up to extraction of a subsequence, we obtain
Since is bounded, from (1.9), we have
for any . This is impossible because is bounded. This verifies (3.30).
Step 2. Note that (3.30) provides a starting point, from which we can carry out the sliding. We decrease and show that for any , it yields
We prove that . Otherwise, we have . To derive a contradiction, we prove that can be decreased a little bit while inequality (3.37) is still valid.
We first prove that
If not, then
and there exists a sequence
Let , where is in (3.32). Then there exists a sequence such that
Since for we have and . Then there exists such that
On one hand, similar to the argument in Step 1, we have
One has Letting we obtain
It yields that
Hence, for any
Next we prove that, there exists an such that
First, (3.38) implies immediately that there exists an such that
So we only need to prove that
If not, then