On the integral systems with negative exponents

This paper is concerned with the integral system 
$$\left \{ 
\begin{array}{ll} 
&u(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{v^q(y)},\quad u>0~in~R^n,\\ 
&v(x)=\int_{R^n}\frac{|x-y|^\lambda dy}{u^p(y)},\quad v>0~in~R^n, 
\end{array} \right. 
$$ 
where $n \geq 1$, $p,q,\lambda \neq 0$. 
Such an integral system appears in the study of the conformal 
geometry. We obtain several necessary conditions for the existence of 
the $C^1$ positive entire solutions, particularly including the 
critical condition 
$$ 
\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}, 
$$ 
which is the necessary and sufficient condition for the invariant 
of the system and some energy functionals under the scaling 
transformation. The necessary condition 
$\frac{1}{p-1}+\frac{1}{q-1}=\frac{\lambda}{n}$ can be relaxed to 
another weaker one $\min\{p,q\}>\frac{n+\lambda}{\lambda}$ for the 
system with double bounded coefficients. In addition, we classify 
the radial solutions in the case of $p=q$ as the form 
$$ 
u(x)=v(x)=a(b^2+|x-x_0|^2)^{\frac{\lambda}{2}} 
$$ 
with $a,b>0$ and $x_0 \in R^n$. Finally, we also deduce some analogous 
necessary conditions of existence for the weighted system.

 u(x) = R n |x−y| λ dy v q (y) , u > 0 in R n , v(x) = R n |x−y| λ dy u p (y) , v > 0 in R n , where n ≥ 1, p, q, λ = 0. Such an integral system appears in the study of the conformal geometry. We obtain several necessary conditions for the existence of the C 1 positive entire solutions, particularly including the critical condition 1 which is the necessary and sufficient condition for the invariant of the system and some energy functionals under the scaling transformation. The necessary condition 1 p−1 + 1 q−1 = λ n can be relaxed to another weaker one min{p, q} > n+λ λ for the system with double bounded coefficients. In addition, we classify the radial solutions in the case of p = q as the form with a, b > 0 and x 0 ∈ R n . Finally, we also deduce some analogous necessary conditions of existence for the weighted system. 1. Introduction. In this paper, we study the existence and the asymptotic behavior for the integral system where n ≥ 1, p, q > 0 and λ = 0. When p = q and u ≡ v, (1.1) becomes This equation is related to the study of the conformal geometry and the nonlinear elliptic PDEs with the negative exponent (cf. [2], [9], [10], [12], [21], [22] and [25]).

YUTIAN LEI
A problem posed by Li [18] is whether or not does (1.2) admit any positive (regular) solutions for all n ≥ 1, λ > 0 and p > (2n + λ)/λ. Xu [26] gave a positive answer and obtained the following results.
For the system (1.4), there exists positive solution when p, q satisfy the critical condition (cf. [20] Furthermore, (1.4) has positive solution in L p+1 (R n ) × L q+1 (R n ) if and only if the critical condition (1.6) holds (cf. [15]). According to the results in [24], the positive solutions u, v are bounded and decay with the fast rates as long as (u, v) ∈ L p+1 (R n ) × L q+1 (R n ). In the subcritical case the Hardy-Littlewood-Sobolev conjecture states that (1.4) has no positive solution (cf. [1] and [5]). When α = 2 and n ≤ 4, this problem (Lane-Emden conjecture) was resolved (cf. [23]). The corresponding results on the discrete Hardy-Littlewood-Sobolev inequality can be seen in [8].
In this paper, we also consider the problems mentioned above for (1.1). The results listed as follows will be proved in section 2.
(Rt2) p, q must satisfy min{p, q} > (n + λ)/λ and Two interesting problems are posed naturally. One is whether (1.1) with λ > 0 has no positive solution in the case of max{p, q} ≤ (n + λ)/λ. The other is whether the necessary condition (1.7) is still the sufficient condition for the existence of positive solutions.
Eq. (1.7) can be viewed as the necessary and sufficient condition ensuring the conformal invariant. Namely, the system (1.1) and the energy functionals u −1 p−1 and v −1 q−1 are invariant under the scaling transformation (see section 3). Here we deduce (1.7) by using the integral estimates instead of the ideas in [26]. Certainly those ideas in [26] also work and the readers can refer to [13].
Clearly, if p = q in (1.7), then p is the critical exponent (2n + λ)/λ in (R1). When p = q, we will prove u ≡ v in section 4 by the same ideas in [4]. Thus, applying the result (R1), we can also classify the positive C 1 entire solutions as the form (1.3).
We expect to remove the assumption of radial property. However, it seems difficult to verify the radial property for the positive solutions of (1.1). For the system (1.4) with positive exponents, the radial symmetry of the positive solutions was proved by the method of moving planes as long as (u, v) ∈ L p+1 (R n ) × L q+1 (R n ) (cf. [6]). According to Theorem 1.1, the pair of positive solutions (u, v) of (1.1) belongs to L 1−p (R n ) × L 1−q (R n ) naturally. Although the Hardy-Littlewood-Sobolev inequality does not work for λ, p, q > 0 as in [6], we conjecture that u, v are still radially symmetric by the ideas in [19]. Theorem 1.1 shows that (1.7) is the necessary condition of the existence of positive C 1 entire solutions. For the following system the necessary condition (1.7) can be relaxed to another weaker one min{p, q} > (n + λ)/λ. Here c 1 (x) and c 2 (x) are double bounded functions. Namely, there exists C > 0 such that In section 5, we will prove the following results. (Rt5) When −n < λ < 0 and p, q > 0, for arbitrary double bounded functions c 1 (x) and c 2 (x), there does not exist solution of (1.8) satisfying u(x) |x| θ1 and v(x) |x| θ2 for all θ 1 , θ 2 ∈ R.
Here u(x) |x| θ means that there exists C > 0 such that Another interesting problem is whether (1.8) with λ > 0 has no positive solution for any double bounded c 1 (x), c 2 (x) in the case of min{p, q} ≤ (n + λ)/λ.
When λ, p, q < 0, this system is associated with the study of the extremal functions of the weighted Hardy-Littlewood-Sobolev inequality (cf. [20]). The optimal integrability intervals of positive solutions were established in [14]. Based on this result, the authors of [16] obtained the fast asymptotic rates of u, v when |x| → 0 and |x| → ∞ respectively. Afterwards, the authors of [17] discussed the symmetry property of the locally bounded solutions. In section 6, we will also consider the same problems and prove the following results.
Remark. After completing this work, the author learned that Jiankai Xu recently completed a preprint where he proves Theorem 1.2 without the assumption of radial constraint by the ideas in [19] and [26].
2. Necessary conditions of existence. Assume u, v are the positive C 1 entire solutions of (1.1). We will give the proof of Theorem 1.1.
Before verifying (Rt1), we need the following result.
When y ∈ R n \ B 2|x| (0), |x − y| ≤ |x| + |y| ≤ 3|y|/2. Therefore, by (2.1) and (2.6) Noting n + λ − pλ < 0 which is implied by (2.1), from the estimates of (2.4)- Since u is an entire solution, we also see from the result above Therefore, we can estimate the upper bound of u by the same process for v, and still obtain u(x) |x| λ when |x| → ∞. Thus, Proposition 2.1 is proved.
Proof of (Rt1). By Proposition 2.1, there exists R > 0 sufficiently large, such that Therefore, u −1 ∈ L s (R n ) as long as s > n/λ. Similarly, v has the same consequence. (Rt1) is proved.
Proof of (Rt2). From (2.1) and (2.8), it follows that By the definition of the improper integral, we have Therefore, integrating by parts we get Similarly, we also have (x · ∇v 1−q ) ∈ L 1 (R n ). By (1.1) and the Fubini theorem, On the other hand, for µ > 0, Differentiate it with respect to µ, and then let µ = 1. Thus, Multiplying u −p and integrating on R n , and using (2.12), we obtain Similar to the derivation of (2.10), we also have Inserting this result and (2.11) into (2.13) yields (2.14) Applying (1.1) and the Fubini theorem, we also have Substituting this result into (2.14), we get which leads to the critical condition (1.7). (Rt2) is proved.
Similarly, we also have Proposition 2.2 is proved.
In fact, if we set y =z, then the modulus of the Jacobi determinant is +1. Therefore, Similarly, v θ (x) = R n |x−z| λ u p (z) dz. By the same argument, we can also see the system is invariant under the reflection transformation.
In the following, we show that (1.7) is the necessary and sufficient condition of the fact that the system (1.1) and the energy are invariant under the scaling transformation.
For r > 0, set the scaling of u, v as follows: where σ 1 , σ 2 = 0 will be determined later to ensure that u r , v r still solve (1.1). Clearly, dz.
The necessity can also be proved by the same calculation. Finally, we should point out that (1.1) is not invariant under the inversion transformationū except λ 1 = λ 2 = λ and p = q = (2n + λ)/λ. If the system is replaced by the weighted one (1.9), then it is invariant under the inversion transformation with the newᾱ,β which are dual to the origin ones. In fact, However, the invariant of the weighted system is clearly absent under the translation transformation. 4. Classification of radial solutions when p=q. In this section, we will prove Theorem 1.2. Assume u, v solve (1.1). According to Theorem 1.1, (1.7) holds. In view of p = q, we have We can see later that this exponent ensures us to use the properties of the conformal invariant. Define For any a ∈ R n , let r > 0 satisfy Our main aim is to prove Proof of (4.3). At first, for a = 0, consider  Let e be an arbitrary unit vector. Take the Kelvin type transforms Noting p − 1 = 2n λ which is implied by (4.1), we deduce that Similarly,v has the same property. Claim 2. The pair (ū,v) solves (1.1).
In view of (4.5) and (1.1), we obtain by using q = (2n + λ)/λ that Similarly, we also havev These results show thatū andv are solutions of (1.1). Claims 1 and 2 imply thatū andv are also radially symmetric and decreasing about some point x * ∈ R n . By virtue of (4.6), we get x * = e 2 . Therefore, for any h ∈ R. The definition (4.5) ofū leads to Denote Λe by x. Since h and e are arbitrary, there holds By a translation, we see that (4.3) holds for any a ∈ R n . Similarly, we also get v(a + rx) = |x| λ v(a + rx |x| 2 ).
Combining this with (4.11), we obtain u(a + e) = t −kλ u(a + t 2k e). Therefore, letting k → ∞ in (4.12), we get It is impossible. This contradiction leads to r = s. Substituting r = s into (4.2) and (4.8) yields Clearly, c 0 > 0. Using (1.1), for any x ∈ R n we have This implies c 0 − 1 = , which leads to c 0 = 1. Thus, from (4.13) it follows u ≡ v in R n . Now, (1.1) is reduced to (1.2) with (4.1). According to (R1), all the solutions are classified as the form (1.3). Theorem 1.2 is proved. 5. System with double bounded coefficients. In this section, we prove Theorem 1.3.
In order to ensure I 3 < ∞, we require and hence both the integral orders in I 2 and I 3 are negative: Thus, combining the estimates of I 1 , I 2 , I 3 , we see that Let θ 1 = θ 2 = λ. Then (5.2) ensures that (5.4) holds. By (5.1) and (5.5), we obtain 1
If β ≤ 0, we see (6.7) immediately. If β > 0, then By the estimates of v 1 , v 2 , v 3 , we obtain three increasing rates for v. We compare those rates in three cases.
Here ε > 0 is sufficiently small. By an analogous calculation of (6.9), we know u also blows up. It is impossible and Case (ii) does not happen.
When y ∈ R n \ B 2|x| (0), |x − y| ≥ |y| − |x| ≥ |y|/2. By Step 3, we have Since u is an entire solution, we also see from the result above Therefore, we can estimate the upper bound of u by the same process for v, and still obtain u(x) |x| λ−α (6.12) when |x| → ∞. In addition, the conclusions of Steps 2 and 3 and (6.11) show that (Rt6) is true.
Next, We consider the asymptotic rates of u, v when |x| << 1. From (6.10) we can find R > 0 sufficiently large such that v(y) ≤ C|y| λ−β , f or |y| > R.
Remark 6.1. From the kelvin type transformation (3.2), we also see the rates of u and v are −α and −β respectively. In fact, when |x| → 0, we have |y| → ∞ if setting y = x |x| 2 . Thus, from (3.2) and (6.12), we deduce Similarly, v has an analogous conclusion.
In addition, by (6.15), we have The proof of (Rt8) is complete. This is (6.18). By the same calculation, we also deduce (6.17) from (6.18). Similarly, u has also the analogous results.