DIRECT METHOD OF MOVING PLANES FOR LOGARITHMIC LAPLACIAN SYSTEM IN BOUNDED DOMAINS

. Chen, Li and Li [Adv. Math., 308(2017), pp. 404-437] developed a direct method of moving planes for the fractional Laplacian. In this paper, we extend their method to the logarithmic Laplacian. We consider both the logarithmic equation and the system. To carry out the method, we establish two kinds of narrow region principle for the equation and the system separately. Then using these narrow region principles, we give the radial symmetry results for the solutions to semi-linear logarithmic Laplacian equations and systems on the ball.

= C n P.V. B1(x) u(x) − u(y) |x − y| n dy − C n R n \B1(x) u(y) |x − y| n dy. C n is a normalization positive constant. Let For u ∈ C 1,1 loc (R n ) ∩ L 0 , the integral on the right side of (2) is well defined. In recent years, motivated by impressive applications in obstacle problem [27,25], optimization [13], anomalous diffusion phenomena [28,24], finance [12] and et.al., there has been tremendous interest in developing equations with nonlocal operators. In particular, the fractional Laplacian operator (−∆) α 2 can be defined by u(x) − u(y) |x − y| n+α dy, α ∈ (0, 2). ( And the prototype of fractional Laplacian equation is Nonlocal equations, such as (1) and (4), do not act by point-wise differentiation but a global integral with respect to a singular kernel, that causes the main difficulty in studying problems involving it. There are several ways to deal with the nonlocal difficulty. By constructing a Dirichlet to Neumann operator of a degenerate elliptic equation, Caffarelli and Silvestre [4] introduced an extension method to localize equations involving fractional Laplacian. This extension method has been applied to a wide kinds of fractional Laplacian equations [11,26,14,3,2]. Another way to deal with the nonlocal difficulty is considering the corresponding equivalent integral equation, tremendous results including symmetry properties for the fractional Laplacian equation have been obtained [9,8,7,20,21,23,19,29].
In the literature, many new kinds of maximum principles have been developed [18,15,16,30,6]. Those maximum principles together with the moving plane methods [17,1,5] make it possible to work directly on the nonlocal equation. By showing radially symmetry for solution of (4) with the direct method of moving plane, Chen et al. [6] proved that problem (4) with Ω = R n has no positive solution in L α ∩ C 1,1 loc (R n ) (L α :={u : R n → R| R n |u(x)| 1+|x| n+α dx < +∞}) in the subcritical case and the solution is radially symmetric in the critical case for all α ∈ (0, 2). In our previous work [22], we have generalized the direct method of moving planes to the system of fractional Laplacian.
Logarithmic Laplacian (2) could be seen as an extremal of the fractional Laplacian. In fact, F((−∆) L )(ζ) = 2 ln |ζ| and (see [10] for more information). Different from the fractional Laplacian, the kernel of the logarithmic Laplacian is |x| −n , which is non-integrable at the origin and also at infinity. It is natural to seek the properties of the equation involving the logarithmic Laplacian operator, Chen and Weth [10] has considered (5) and obtained the following Theorem 1.1 by the moving planes method with a maximum principle for narrow domain based on the Aleksandrov-Bakelman-Pucci (ABP) estimate. The motivation of this present paper is to generalize the direct method of moving planes [6] to the logarithmic Laplacian equation and system. We work directly on the equation and give a new narrow region principle for logarithmic Laplacian (see Proposition 1). Then as an application, we give a simple proof of Theorem 1.1.
Our main concern is the system case. A narrow region principle for problem (1) is given in Proposition 2, by using which we obtain the following symmetry result.
be a positive solution of system (1) with f (·, ·), g(·, ·) being locally Lipschitz continuous and satisfying Then u, v must be radially symmetric and monotone decreasing about the origin.
The paper is organized as follows. We devote Section 2 to the scalar equation (5), including the important narrow region principle for the logarithmic Laplacian equation. We prove Theorem 1.2 in Section 3. Note that in the following, c and C will be constants which can be different from line to line.

Symmetry result for the logarithmic Laplacian equation.
In what follows, we shall use the method of moving planes. Choose any direction to be the For any A ⊂ Σ λ , letÃ be the reflection of A with respect to T λ . According to the fact that for all f ∈ C 1,1 we know that for all x ∈ Σ λ ∩ Ω, Without the classical maximum principle, we shall introduce the following narrow region principle, which will play an important role in the proof of Theorem 1.1.
Combining (11) and (12), we get Plugging the above inequality into (10), we obtain Using the assumption that c(·) is bounded from below, i.e. there is a constant c 1 , such that c(x) ≥ c 1 . Now we have Therefore, there is l 0 > 0, such that for all l ∈ (0, l 0 ], we have C n A∪B 1 |x0−y λ | n dy > cC n − c 1 , and

BAIYU LIU
which leads to a contradiction with (9). Now we prove Theorem 1.1 by using the method of moving planes directly on the equation. We divide our proof into two steps.
We now move the plane T λ to the right as long as (17) holds to its limiting position. Define Step 2. λ 0 = 0.
Proof of Step 2. Suppose λ 0 < 0, we show that the plane T λ can be moved further right.
On the one hand, equation (7) tells us that One the other hand, by direct computation and using the fact that |x 0 − y λ0 | > |x 0 − y|, ∀y ∈ Σ λ0 and U λ0 ≥ ( ≡) 0 in Σ λ0 , we know that = C n P.V.
3. Symmetry result for the logarithmic Laplacian system. To deal with the logarithmic Laplacian system (1), we shall use the same sprite of direct method of moving planes as in Section 2, while we need a new version of the narrow region principle to treat the coupling of the system.
We use Σ λ , T λ and x λ in the same way as in Section 2. Define the reflected functions by u λ (x) = u(x λ ), v λ (x) = v(x λ ) and introduce functions It follows that for all −R < λ ≤ 0 and for all x ∈ Σ λ ∩ Ω, in which Similarly, there holds where
The proof of conclusion (b) is similar and we shall omit it.
With the help of the narrow region principle for system, we give the proof of Theorem 1.2. The proof is divided in two steps.
We now move the plane T λ to the right as long as (30) holds to its limiting position. Define Step 2. λ 0 = 0.
In this case, equation (23) becomes in which we have used (6). According to the fact that u is positive in Ω and zero outside Ω, it follows U λ0 (x) ≡ 0 in Σ λ0 . By a same computation as in (20), we have (−∆) L U λ0 (x * ) < 0, which contradicts with (34).
There are also two possible cases. One is y * ∈ Σ λ0 ∩ Ω and it will lead to a contradiction as in Case 1. The other one is y * ∈ T λ0 ∩ Ω. Hence, for sufficiently large k, y k ∈ Ω λ0,l0/2 and Proposition 2 implies that U λ k (y k ) < 2V λ k (y k ) < 0.
Since any direction can be chosen as x 1 , we have u and v are radially symmetric and monotone decreasing about the origin. The proof is completed.