Existence and nonexistence of positive solutions to an integral system involving Wolff potential

In this paper, we are concerned with the sufficient and necessary 
conditions for the existence and nonexistence of the positive 
solutions of the following system involving Wolff type potential: 
\begin{eqnarray} 
& u(x) =c_{1}(x)W_{\beta,\gamma}(v^{q})(x), 
 \\ 
 &v(x) 
=c_{2}(x)W_{\alpha,\tau}(u^{p})(x). 
\end{eqnarray} 
Here $x\in R^{n}$, $1 0$, 
$0< \beta\gamma$, $\alpha \tau < n $, and the functions $c_{1}(x),c_{2}(x)$ 
are double bounded. This system is helpful to well understand some 
nonlinear PDEs and other nonlinear problems. Different from the 
case of $\alpha=\beta,\gamma=\tau$, it is more difficult to handle 
the critical condition. Fortunately, by applying the special 
iteration scheme and some critical asymptotic analysis, we 
establish the sharp criteria for existence and nonexistence of 
positive solutions to system (0.1). Then, we use the method of 
moving planes to prove the symmetry and monotonicity for the 
positive solutions of (0.1) when $c_{1}(x)\equiv c_{2}(x)\equiv1$ 
in the case 
\begin{eqnarray} 
\frac{\gamma-1}{p+\gamma-1}+\frac{\tau-1}{q+\tau-1}=\frac{n-\alpha\tau}{2n-\alpha\tau+\beta\gamma} 
+\frac{n-\beta\gamma}{2n-\beta\gamma+\alpha\tau}. 
\end{eqnarray}

(Communicated by Wenxiong Chen) Abstract. In this paper, we are concerned with the sufficient and necessary conditions for the existence and nonexistence of the positive solutions of the following system involving Wolff type potential: Here x ∈ R n , 1 < γ, τ ≤ 2, α, β > 0, 0 < βγ, ατ < n, and the functions c 1 (x), c 2 (x) are double bounded. This system is helpful to well understand some nonlinear PDEs and other nonlinear problems. Different from the case of α = β, γ = τ , it is more difficult to handle the critical condition. Fortunately, by applying the special iteration scheme and some critical asymptotic analysis, we establish the sharp criteria for existence and nonexistence of positive solutions to system (0.1). Then, we use the method of moving planes to prove the symmetry and monotonicity for the positive solutions of (0.1) when c 1 (x) ≡ c 2 (x) ≡ 1 in the case 1. Introduction. The Wolff potential is defined for any non-negative Borel measure µ: where 1 < γ < ∞, β > 0, βγ < n, and B t (x) is the ball of radius t centered at point x.
There have been a series of studies on the relations between the Wolff potentials and the nonlinear PDEs (for example, see (cf. [10] and [17]). In [17], an integral estimate between the Wolff potential and Riesz potential was established as follows: where q > p − 1 and is the Riesz potential. The result (1) plays a key role in the proof of Section 3 of this paper.
The Wolff potentials are helpful to well understand the nonlinear PDEs (cf. [10]- [13] and [17]) and other nonlinear problems (cf. [1] and [16]). For example, W 1,γ (w) can be used to estimate the solutions u of γ-Laplace equation If inf R n u = 0, then there exist positive constants C 1 and C 2 such that (cf. [10]) In this paper, we first consider the existence and nonexistence of positive solutions of the following system involving Wolff type Here x ∈ R n , 1 < γ, τ ≤ 2, α, β > 0, 0 < βγ, ατ < n, and the functions c 1 (x), c 2 (x) are double bounded. Namely, there exist positive constants c and C such that When α = β, γ = τ , system (3) becomes: The nonexistence result, known as Liouville type theorems, are fundamental in PDE theory and applications. Recently, by some new iteration technique and critical asymptotic analysis, Lei and Li [14] established the sharp criteria for nonexistence and nonexistence of positive solutions to system (5). If The critical case is For (6), Chen and Li [5] proved that the solutions u and v are radial symmetry and decreasing about some point x 0 . Furthermore, Ma, Chen and Li [16] used the regularity lifting lemmas to derive the optimal integrability and the Lipschitz continuity. Based on these results, Lei [15] obtained the decay rates of the integrable solutions when |x| → ∞.
Obviously, system (6) is a natural generalization of the Hardy-Littlewood-Sobolev type integral system. Namely, when γ = 2, and β = α 2 in (6), we get (8) is associated with the study of the sharp constant of the well-known classical Hardy-Littlewood-Sobolev (HLS) inequality, |x − y| n−α dxdy ≤ C s,α,n f r g s for any f ∈ L r (R n ) and g ∈ L s (R n ). Here 1 < r, s < ∞ and 0 < α < n such that 1 r + 1 s + n−α n = 2. It was shown in [6] that the integral system (8) is equivalent to the system of PDE: Several authors studied the special cases about (3) and (5). For example, Huang, Hong, Li [9] investigated system of integral equations involving Wolff potential on a bounded domain such as They showed that if u and v are constants on ∂Ω, then Ω is a ball. Furthermore, u and v are radially symmetric and monotone decreasing about the center of the ball. The inverse is also true by a similar procedure of their proof. Intuitively, it would be interesting to study the system (3). Based on the results above, in this paper we first establish sufficient and necessary conditions for the existence of the positive solutions to (3). Then, by using the method of moving planes, we show that the solutions of system (3) are radially symmetric and decreasing about some x 0 ∈ R n .
When γ = τ = 2, β = s 2 , α = t 2 , system (3) becomes the following integral system involving Riesz potential Therefore, we can easily obtain the following corollary. Corollary 1. Let 0 < s, t < n. The HLS system (11) has positive solutions u, v for some double bounded functions c 1 (x), c 2 (x), if and only if pq > 1 and According to the Theorem 3 in [6], the property can be extended to the following semilinear Lane-Emden the type system Corollary 2. Let k 1 and k 2 be positive integers less than n 2 , and assume p, q > 1. There exist positive u, v of the system (12) for some double bounded functions According to Theorem 1.1, we assume hereafter Since c 1 (x) and c 2 (x) are not constants, the solutions of (3) have no radial structure. Then we consider another system for c 1 (x) ≡ c 2 (x) ≡ 1, that is: The critical case is Remark 1. As βγ = ατ , (10) and (14) becomes and respectively. Clearly, (15) is weaker than (16). But as βγ = ατ , we can't be sure that (10) is weaker than (14).
For (13), using the method of moving planes, we get Theorem 1.2. Let p, q > 1, and assume (u, v) is a pair of positive solutions of (13).
Then u and v are radially symmetric and monotonic about some points in R n .

Remark 2. When
and (14) becomes Huang, Li, and Wang [8] proved the symmetry and monotonicity of its solution in such case.
2. Existence and nonexistence for system (3). In this section, we prove Theorem 1.1 by applying the special iteration scheme and some critical asymptotic analysis (cf. [14]).
Proof of Theorem 1.1.
Step 1. Existence. Similar to the argument in the proof of Theorem 3.3 in [14], We can find three pairs solutions. Let where θ 1 , θ 2 > 0 will be determined later.
When |x| ≤ R for some R > 0, then u is proportional to W β,γ (v q )(x). So we only consider suitably large |x|. Clearly, Case I. Take the slow rates .
First, write H := Bt(x) v q (y)dy. By Hölder inequality, Therefore, exchanging the order of variables yields Thus, (24) Without loss of generality, we suppose Integrating on B R 4 , we get We claim that the exponent of R is zero. In fact, (26) implies pq(n−βγ) The claim is proved.
and letting R → ∞, we also have R n v q dy = 0. It is impossible. Thus, we complete our proof.
3. Radial symmetry and monotonicity of solutions. In this section, we use the method of moving planes in integral forms which was established by Chen et al. [6] to prove that the positive solutions u and v are radially symmetric and decreasing about some point in R n . Claim 1. There holds s > max{1, n(γ−1) n−βγ } and r > max{1, n(τ −1) n−ατ }. In fact, by the relationship of p, q in (14), it is easy to see On the other hand, by (10), it is easy to discover that Claim 2. There holds q r = γ−1 s + βγ n and p s = τ −1 r + ατ n . In fact, Similarly, we can get the second equality by a simple calculation.
Proof of Theorem 1.2. Let be the boundary of Σ λ , the hyper plane we will move with. Let x λ = (2λ − x 1 , x 2 , . . . , x n ) be the reflection point of x about the plane T λ and write u λ (x) = u(x λ ).
Denotes D t (x) the intersection of the ball B t (x) with its mirror image B t (x λ ), and Ω t (x) = B t (x)\D t (x).
Step 1. We first show that for λ sufficiently negative, we have To this end, we introduced the sets We will show that Σ u λ and Σ v λ must have measure zero for λ sufficiently negative. We have for Applying the Mean Value Theorem to the above, we derive where ξ t (x) is valued between

y)dy and
Bt(x) v q λ (y)dy and is obviously positive and less than the sum of the two. By the definition of Ω t (x) and Σ v λ and again by the Mean Value Theorem, we have where as usual f (x) + = max{0, f (x)}. Applying Hölder inequality to (30) and taking account of (31), we deduce Here we have used the inequality and the equation Since u ∈ L s (R n ), v ∈ L r (R n ) by the Hardy-Littlewood-Sobolev inequality, we derive from (32) that The last two inequalities above was deduced by the integral estimate between the Wolff potential and Riesz potential, for s γ−1 > 1. Then by Claim 1 and Claim 2, we can use the Hardy-Littlewood-Sobolev inequality and the Hölder inequality on (33) to obtain Combining (34) and (35), we arrive at Since u ∈ L s (R n ) and v ∈ L r (R n ), for sufficiently negative λ, we have Then it follows from (36) that u − u λ L s (Σ u λ ) = 0, and hence Σ u λ must be zero. Then from (35), Σ v λ must also be measure zero. This verifies (29).
Step 2. Inequality (29) provides a starting point to move the plane T λ . Now we start from this neighborhood of x 1 = −∞ and move the plane to the right as long as symmetric about the limiting plane. More precisely, define λ 0 = sup{µ | (29) holds for any λ ≤ µ}.
One can see that λ 0 < +∞ by using the similar argument in Step 1 and starting the plane T λ near x 1 = +∞.
Again the integrability conditions u ∈ L s (R n ) and v ∈ L r (R n ) ensure that one can choose sufficiently small, so that for all λ in [λ 0 , λ 0 + ), Now by (36), we have u − u λ L s (Σ u λ ) = 0, and hence Σ u λ must be zero. Similarly, Σ v λ must also be measure zero. This verifies (39) and hence (37).
Since x 1 direction can be chosen arbitrarily, we deduce that (u, v) must be radially symmetric and monotone decreasing about some point in R n . This completes the proof of Theorem 1.2.