ON POSITIVE SOLUTIONS OF INTEGRAL EQUATIONS WITH THE WEIGHTED BESSEL POTENTIALS

. This paper is devoted to exploring the properties of positive solutions for a class of nonlinear integral equation(s) involving the Bessel potentials, which are equivalent to certain partial diﬀerential equations under appropriate integrability conditions. With the help of regularity lifting theorem, we obtain an integrability interval of positive solutions and then extend the integrability interval to the whole [1 , ∞ ) by the properties of the Bessel kernels and some delicate analysis techniques. Meanwhile, the radial symmetry and the sharp exponential decay of positive solutions are also obtained. Furthermore, as an application, we establish the uniqueness theorem of the corresponding partial diﬀerential equations.


1.
Introduction. Let α be a real number satisfying 0 < α < n. The Bessel potential of a positive function f ∈ L p (R n ) (1 ≤ p ≤ ∞) is dt t is the Bessel kernel.
In this paper, we will consider the following non-linear integral equation involving the weighted Bessel kernel functions and where 0 < β < α < n and p, q ≥ 1.
In recent years, there has been tremendous interest in studying integral equation(s), which are equivalent to some partial differential equation(s) in R n and also provide a special skill to investigate the global properties of corresponding differential equation(s), such as integrability and asymptotic behavior. First of all, we recount some closely related investigations and backgrounds.
As β = 0, the equation (1) can be reduced to the following form: By the method of moving planes in an integral form, Ma and Chen [22], combining with Sobolev embedding inequality, showed that all the positive solutions of (3) are radially symmetric and monotone decreasing about some point, and Han and Lu [8], by the regularity lift theorem, proved that all positive solutions are uniformly bounded in R n with u ∈ L q0 (R n ) and q 0 > max{n(p − 1)/α, p}. In addition, we know that, under certain integrability conditions, equation (3) is equivalent to the following differential equation: which has been extensively studied, as the parameters take special values. For example, when α = 2 and p = 3, Coffman [4] got the uniqueness of positive solution of (4) in R 3 , and Mcleod and Serrin [24] extended the corresponding result to R n ; when α = 2 and p = (n + 2)/(n − 2), the uniqueness theorem of (4) in bounded or unbouded annular region was given by Kwong in [11]. On the other hand, a natural extension of (3) is the following integral system: Under certain appropriate decay conditions on the solutions (u, v) at infinity, system (5) is equivalent to the following partial differential equations: When α = 2, system (6) is related to two coupled nonlinear Schrölder equations, which model the Bose-Einstein condensate. Lin and Wei [20] gave the existence and nonexistence theorem of ground state solutions of nonlinear Schrödinger equations.
Recently, for α ∈ (0, n), Lei [14] showed that all of positive solutions to system (5) or (6) are radially symmetric and decreasing about x 0 ∈ R n , under the conditions u ∈ L [n(p+q−1)]/µ (R n ) and µ ∈ (0, α]; moreover, the optimal integrable intervals and sharp decay rate were also given in [14]. Another integral system similar to (5) is the following Riesz potential type one: which is closely related to the stationary Schriönger system as α = 2. System (7) and its various weighted forms have recently attracted a great deal of attentions. The radial symmetry, optimal integrable intervals and sharp asymptotic behaviors of positive solutions to (7) were investigated successively. We refer the readers to [1,2,3,9,10,12,13,15,16,17,23,26,27,25,28] and the references therein for more information about the Riesz potential type integral equations. Inspired by the works mentioned above, this paper aims to explore the global properties, such as integrability and asymptotic behaviors etc., of positive solutions to (1) and (2). Our main results can be formulated as follows: • Symmetry and monotonicity: u(x) is radially symmetric and monotone decreasing about the origin. • Sharp decay rate: Here we use the notation u(x) g α (x) to denote that there exist two constants C 1 and C 2 such that for |x| large enough , is a pair of positive solutions of weighted integral system (2). Then • Symmetry and monotonicity: u(x), v(x) is radially symmetric and monotone decreasing about the origin. • Sharp decay rate for |x| large enough: System (2) is an extended model of (1) and (5). When α = 2, system (2), under certain integrability conditions, is also equivalent to the following system of differential equations: The system (12) arises in mathematical models from various physical phenomena, such as the incoherent solitons in nonlinear optics and the multispecies Bose-Einstein condensates in hyperfine spin states. The interested readers can consult [5,7,6], among numerous references, for more information. Set w(x) u(x) + v(x) and γ = p + q. Noting that Therefore there exists a bounded function R(x) such that system (2) can be rewritten as where R(x) is bounded function defined by Due to (13) and , by the same arguments as in the proof of Theorem 1.1, we can obtain the following results.
and for |x| sufficiently large, Therefore, to obtain Theorem 1.2, it suffices to show that (u(x), v(x)) is radially symmetric and monotone decreasing about the origin and, for sufficiently large |x|, Finally, based on the explanations above, as an application of Theorem 1.2, we will establish the following uniqueness theorem of (12).
, is a pair of positive solutions of the system (12). Then This paper is organized as follows. In Section 2, we will use the regularity lifting theorem, delicate analysis techniques, the properties of Bessel potentials and the interpolation theorem to obtain the best integrable interval of (1). In Section 3, we shall prove that all of positive solutions for (1) are radially symmetry by moving plane method. The sharp decay rate of (1) will be formulated by iteration in Section 4. The important properties of integral system (2) will be discussed in Section 5. Finally, the uniqueness theorem of system (12) will be shown.
Throughout the rest of this paper, we always use the letter C, sometimes with certain parameters, to denote positive constants that may vary at each occurrence but are independent of the essential variables.

2.
Integrability. This section is concerned with the proof of the integrability of positive solution u(x) of system (1) in Theorem 1.1, which will be divided into three steps. In Step 1, we will use the regularity lifting theorem to deduce the following integrability of u(x): In Step 2, with the help of some delicate analysis techniques, we will obtain the L ∞ estimation of u(x). In the last step, we will apply the properties of Bessel potential and L p interpolation theorem to extend the integrability interval to the whole [1, ∞].
Step 1. In order to show (18), we firstly introduce some necessary notions. For A > 0, set and and write is a solution of the following equation: We claim that Firstly, we recall the following important estimate (see [29, (2.63)] for the detail), where I α is the classical Riesz potential operator. Therefore, by the weighted Hardy-Littlewood-Sobolev inequality, we have Next, we show that F ∈ L ∞ (R n ). By (23) and the definition of w(x) , we get It suffices to show that R(x) < ∞, which will be divided into the following two cases: This together with (22) implies the claim holds. Now we turn to T A (f ). By the weighted Hardy-Littlewood-Sobolev inequality and Hölder's inequality, we conclude that, for s > n/(n − α) Step 2. In this step, we will show that u ∈ L ∞ (R n ). By (19)-(22), it suffices to verify that Write where and in what follows B r (x) denote the ball on R n centered at x with radius r.
We first estimate and the Hölder inequality leads to For x ∈ B 2A (0), by Hölder's inequality and Young's inequality, we conclude that This together with (26) implies the boundedness of D 1 (x).
This combing (18) with Hölder's inequality leads to which completes the proof of the first part in Theorem 1.1. 3. Radial symmetry. In this section, we will apply the method of moving plane in integral forms introduced by Chen et al in [2,3] to obtain the radial symmetry of positive solutions of system (1) in Theorem 1.1. For convenience, given λ ∈ R, set Σ λ {x = (x 1 , · · · , x n ) ∈ R n : x 1 ≥ λ}, x λ (2λ − x 1 , x 2 , · · · , x n ), and u λ (x) u(x λ ).
For any solutions u of system (1.1), it is easy to verify that The proof is made up of two steps. In step 1, to obtain the radial symmetry, we compare the value of u(x) with u λ (x) and show that for sufficiently negative λ < 0, there holds In Step 2, we continuously move the plane x 1 = λ along x 1 direction from near negative infinity to the right as long as (33) holds. By moving this plane in this way, we finally show that the plane will stop at the origin. Now we turn our attention to Step 1.
Step 1: Noting that as x, y ∈ Σ λ , it is easy to verify that |x λ − y| ≥ |x − y| and |y λ | ≥ |y|. By the monotonicity of g α (x) and (18), we know that A 1 (x) ≤ 0 and . Invoking the weighted Hardy-Littlewood-Sobolev inequality and Höler inequality leads to . On the other hand, for u ∈ L s (R n ), it is easy to see that u λ p−1 L s (Σ λ ) is sufficiently small, as λ → −∞. Therefore, we can choose N > 0 large enough, such that for which implies that Σ u λ = {x ∈ Σ λ : u(x) < u λ (x)} must be zero. This verifies (33).
Step 2: We continuously move x 1 = λ to the right as long as (33) holds. Indeed, suppose that at x 1 = λ 0 < 0, we have, for any x ∈ Σ λ0 \ {0}, Next, we will show that the plane can be moved further to the right. Precisely, there exists an such that By (32), we know that u(x) ≥ u λ (x) in the interior of Σ λ0 . Let Σ u λ0 = {x ∈ Σ λ0 : u(x) ≤ u λ0 (x)} . From the analysis mentioned above, it is easy to verify that the Σ u λ0 is a zero measure set in R n and lim λ→λ0 Σ u λ ⊂ Σ λ0 . This together with (34) and the integrability conditions u ∈ L s (R n ) ensures that one can choose small enough such that for any λ ∈ [λ 0 , λ 0 + ) and Σ u λ = {x ∈ Σ |u(x) < u λ (x)} must be zero. Finally, we show that the plane can't stop before hitting the origin. On the contrary, if the plane stops at x 1 = λ 0 < 0, then u(x) must be symmetric about the plane x 1 = λ 0 , i.e., On the other hand, noting that |x λ0 | > |x| for any x ∈ Σ λ0 , by (32), we have which obviously contradicts with (35). Since the direction is arbitrary, we deduce that u is radially symmetric about the origin and decreasing. This completes the proof of the second part of Theorem 1.2.
Proof. By (31) and the integrability of u, it is easy to verify that for |x| ≥ 2, Therefore, for sufficiently large |x|, which completes proof of Proposition4.1. Proof. By (31), we know that u p (y)/|y| β ∈ L 1 (R n ). Then On the other hand, since u(x) is radially symmetric and decreasing about the origin, we have This together with (38) implies that (37) holds. 2 Proposition 4.3. As |x| is sufficiently large, there exists C > 0 such that Proof. For η ∈ (0, 1/2), let { ζ k : k = 0, 1, 2, · · ·} be a sequence as follows: For every fixed R > 0, we can write Noticing that for y ∈ B R (0), we have which together with (31) implies that Next, we turn to II(x). By (37), we deduce that there exists R 1 large enough such that Therefore, Observing that |y−x| ≥ ζ 1 |x| as y ∈ {y | [R n \B R (0)]\B ζ1|x| (x)}, and for sufficiently large |x|, g α (x) ≤ C exp(−c|x|) ≤ 1, combining with that g α is radial and decreasing, we get It follows from (39)-(42) that Similarly to (39) and (43), for u([1 − ζ k ]x), we have and (45) Since g α is a radial and deceasing function, by the definition of {ζ k }, we have Therefore, inserting (45) from k = 1 to k = m into (43), we deduce that, for any given integer m > 1, Since g α (x) is continuous function on R n \ {0} and u(x) is a bounded function in R n , invoking that lim m→∞ ζ m = η, we get which combining with (31) implies that Therefore, by the continuity of g α (x) on R n \ {0}, we conclude that This completes the proof of Proposition 4.3.
Summing up the conclusions of Propositions 4.1 and 4.3, we get (9) and completes the proof of Theorem 1.1.

5.
Proof of Theorem 1.2. This section is devoted to proving Theorem 1.2. By (1.2) and (12), it suffices to obtain the symmetry and sharp decay of system (2). Therefore, we firstly discuss the radial symmetry and monotonicity of positive solutions of system (2). For simplicity, we use same notations as section 3. For any pair of solutions (u, v) of system (2), it is easy to check that and The proof of symmetry and monotonicity for system (2) is similar to equation (1), which is also made up of two steps. In step 1, we will compare the value of u(x) with u λ (x) and show that for sufficiently negative λ < 0, there holds In Step 2, we continuously move the plane x 1 = λ along x 1 direction from near negative infinity to the right as long as (48) holds. By moving this plane in this way, we finally show that the plane will stop at the origin. Now we turn to Step 1.
Step 2. We continuously move x 1 = λ to the right as long as (48) holds. Indeed, suppose that at x 1 = λ 0 < 0, we have, for any x ∈ Σ λ0 \ {0} Next, we will show that the plane can be moved further to the right. Precisely, there exists an depending on n, α, β and the solution (u, v) itself such that Under the assumption that v(x) ≡ v λ0 on Σ λ0 , by (46) and the non-negativity of From the analysis mentioned above, it is easy to check that the Σ u λ0 is a zero measure set in R n . Similarly, we also have m{ Σ v λ0 } = 0. This together with (52) and the integrability conditions u, v ∈ L τ (R n ) ensures that one can choose small enough such that for all λ in [λ 0 , λ 0 + ) C(α, n, p) u λ p−1 , C(α, n, q) v λ q−1 L τ (Σ λ ) u p L τ (Σ λ ) ≤  So (53) and (54) hold. Thus we also have u λ (x) − u(x) L τ (Σ u λ ) = 0 and v λ (x) − v(x) L τ (Σ v λ ) = 0, which implies that the measures of Σ u λ and Σ v λ must be zero. This verifies (56). Finally, we show that the plane can't stop before hitting the origin. On the contrary, if the plane stops at x 1 = λ 0 < 0, then u(x) and v(x) must be symmetric about the plane x 1 = λ 0 , i.e., u(x) = u λ0 (x) and v(x) = v λ0 (x), ∀x ∈ Σ λ0 .

6.
Uniqueness. This section is devoted to the proof of (14). Note that the system (10) is equivalent to the system (2) for α = 2. It follows from Theorem 1.