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On the Hardy-Littlewood-Sobolev type systems

In this paper, we study some qualitative properties of Hardy-Littlewood-Sobolev type systems. The HLS type systems are categorized into three cases: critical, supercritical and subcritical. The critical case, the well known original HLS system, corresponds to the Euler-Lagrange equations of the fundamental HLS inequality. In each case, we give a brief survey on some important results and useful methods. Some simplifications and extensions based on somewhat more direct and intuitive ideas are presented. Also, a few new qualitative properties are obtained and several open problems are raised for future research.

The critical case of the HLS type system, also known as the HLS system, was studied in an elegant paper of E. Lieb [13]. The existence of solution is proved (see [13] and details in Section 3), so the remaining question is the classification/uniqueness of positive solutions. We conjecture that all positive solutions of (1.4) or (1.5) are given by a single family of solutions via translation and scaling. In other words, up to some translations and scalings, the positive solutions of (1.4) or (1.5) are unique. This kind of uniqueness is called essential uniqueness. This problem is solved in some special cases but is still open in general.
In the supercritical case the existence of solution is also established (see Section 3). The existence proof is based on a relatively new method which combines shooting method with degree theory. We give an outline of the method to prove the existence for an even more general system in Section 4.1, which contains supercrtical and critical HLS type systems as special cases.
We are also interested in asymptotic analysis of the solutions of supercritical HLS type systems. For example, what are the asymptotic expansions of the radial solutions? Are all radial solutions scaling related? These questions are addressed in Section 4.2.
Last, we consider the subcritical HLS type systems, in particular, the Lane-Emden system. The so-called Lane-Emden conjecture states that, for 0 < p, q < ∞, 1 has u = 0 and v = 0 as the unique locally bounded solution.
The conjecture naturally generalize to the systems (1.4) or (1.5) in the subcritical cases with an additional condition, pq > 1. Notice that pq > 1 is a necessary condition for this conjecture to hold in high order HLS type systems. For example if p = q = 1 and γ = 4 we have solution u = v = e w·x to (1.5) for w ∈ R n with |w| = 1.
This paper is organized as following. In Section 2 we provide some useful estimates. In Section 3, we discuss about the classification of solution to critical HLS type system. In Section 4, existence and asymptotic analysis of solutions to supercritical cases are investigated. In Section 5, we focus on subcritical cases, in particular, the Lane-Emden system. Throughout this paper, we assume that0 < p, q < ∞ and pq > 1, and all positive solution are assumed locally bounded, unless specified otherwise.

Some basic estimates
For pq > 1, denote the scaling component of system (1.9) by Remark 2.1. For pq > 1, the critical hyperbola has a new form in terms of α and β, Hence, supercritical condition 1 p+1 + 1 q+1 < n−2 n ⇔ α + β < n − 2, and subcritical condition Here we present some useful estimates including comparison principle and energy estimates for (1.9). These estimates are valid for all three cases, i.e., critical, supercritical and subcritical. They are useful in many aspects, such as in asymptotic analysis for solutions to critical and supercritical cases, and to prove Liouville type theorem in subcritical case (Actually, to prove Lane-Emden conjecture a proper energy estimate is the key, which is detailed in Section 5). These estimates are: Lemma 2.2. Let p, q > 0 with pq > 1. For any positive (locally bounded) solution (u, v) of (1.9) and by Maximum-principle we get, Lemma 2.3 (Comparison principle). Let p ≥ q > 0 with pq > 1. Let (u, v) be a positive bounded solution of (1.9). Then we have the following comparison principle, Do similar estimates exist for general HLS type systems (1.4) and (1.5)? This is also an interesting question, yet as far as we know there has not been any answer to it. Proof of comparison principle Lemma 2.3.
Lemma 2.2 is first obtained by Serrin and Zou [23] (1996). In [7] a simpler proof is given, to which we refer readers for detail. Here we only sketch the proof. First, we multiply (1.9) with φ, the first eigenfunction of −∆ operator on B R with eigenvalue λ ∼ O( 1 R 2 ). Then integrate the equations by parts and use the fact that φ is positive in B R (which leads to ∂φ ∂n < 0 on ∂B R by Hopf's Lemma), and we obtain the following inequalities, Apply comparison principle lemma 2.3 to the first inequality above to get Notice that q(p+1) q+1 > 1 as pq > 1, so by Hölder inequality Therefore, by (2.4) and Case 1: q ≥ 1, by Hölder's inequality: and by the second inequality of (2.4), Case 2: q < 1, this case is more complex than the first one. By the first equation of (1.9) we have −∆u ≤ 0, then multiply it with η 2 u γ where η ∈ C ∞ 0 (R n ) and η ∈ (0, 1) and integrate over whole space, we get Now we use Poincaré inequality to induce an embedding inequality on the support of η. Then we use Hölder inequality and estimate for B R u q in previous proof to obtain, Then the proof is finished by taking γ ≤ q then use Hölder inequality to get for any θ ∈ (0, n n−2 ).

Existence and classification of solutions for critical HLS type systems
The existence of solution to critical HLS type systems, for both integral equations (1.4) and PDE (1.5), is completely revolved. In [13] E. Lieb (1983) established the existence of ground state solution (i.e. the optimizer of variational problem (1.3)). Later, people find that shooting method is a powerful tool to prove existence of solution to both critical and supercritical cases of (1.5) with integer power of Laplacian. We sketch the proof of the shooting method in Section 4.1.
In the following system, Liu, Guo and Zhang [14] (2006) have obtained the existence for both critical and supercritical cases, admits a positive solution in the critical and super-critical cases 1 p+1 + 1 q+1 ≤ n−2k n for any integer k with 2k < n. In fact, given pq > 1, the above system admits a positive radial solution if and only if 1 p+1 + 1 q+1 ≤ n−2k n . In Section 4.1, we outline the proof of a more general theorem which contains the result above. In the scalar case: where p > 1, 2k < n. Lei and Li (2013) [10] showed that Moreover, Lei and Li (2013) [10] showed that Theorem 3.4. Assume pq > 1, then the HLS type system (1.4) has a pair of positive solutions if and only if it is critical: Now we shall focus on the classification of the solutions. As mentioned in the introduction, for critical HLS type systems, we conjecture that all positive solutions of (1.4) or (1.5) are given by a single family of solutions via translation and scaling. As a special case of this conjecture, E. Lieb raised an open problem in [13] (1983) that asks for p = q if (1.8) has unique solution up to scaling and translating. Lieb's problem was completely resolved with the introduction of the integral form of method of moving planes by Chen, Li and Ou [6] (2006). The conjecture is still open in general.
The method of moving planes was introduced by A.D. Alexandrov in 1950s, and then developed by J. Serrin [22] (1971) and Gidas, Ni, Nirenberg [9] (1981). Caffarelli, Gidas and Spruck classified all the solutions to (1.7) in [3] (1989) by the method of moving planes. Then Chen and Li simplified their proof [4] (1991). Wei and Xu generalized this result to higher order conformally invariant equations [26]  3. all radial solutions are finite energy solution (the converse is known to be true, see [6]); 4. all solutions decaying to zero at infinity are radial; 5. all bounded solutions are radial.

The super-critical HLS type systems
In this section, we study the existence and asymptotic analysis of solution to supercritical HLS type systems.

Existence of solution to critical and supercritical HLS type systems
For the Lane-Emden system (1.9), Serrin and Zou (1998) [24] used shooting method to obtain the existence of solution. Liu, Guo and Zhang (2006) [14] introduced a degree approach to shooting method (see also Li (2011) [12]) to obtain radial positive (locally bounded) solution for both critical and supercritical cases of HLS type system (1.5) in a uniform way.
In short, the degree approach of shooting method combines three ingredients together: degree theory, target map (i.e. shooting) and non-existence on balls with Dirichlet boundary condition (Pohozaév identities). Approach of this kind, relating the existence of solutions in R n to the nonexistence to a corresponding Dirichlet problem on balls, is implemented by many mathematicians, for instance earlier by Berestycki, Lions and Peletier [2] (1981).

4)
where C is a non-negative constant that depends only on α.
Notice that the nonlinear terms of (3.1) (i.e. the HLS type system (1.5) with γ = 2k) satisfy the assumptions (4.3)-(4.4). So Theorem 4.1 recovers Theorem 3.1 given the non-existence of solution to the corresponding Dirichlet problems. For nonexistence part, people usually apply Pohozaév inequalities [18] (see also [20]). This is now more or less standard, a large amount of literature can be found in related topics, for example [16] and [21].
In what follows we outline the proof of Theorem 4.1, which should exhibit the standard procedure of the degree approach of shooting method.
In the view of seeking radial solutions of (4.1) we solve the following initial value problem with any initial value α = (α 1 , · · · , α L ) with α i > 0, i = 1, 2, ....L. Denote the solution as u(r, α): (4.5) We want to find the suitable initial value α so that u i (r, α) > 0 for all r > 0. When L = 1 the question is simple, the assumption that (4.2) admits no solution is equivalent to u 1 (r, α) > 0 for all r > 0. Then there exists a global solution for any initial value, and we are done.
When L ≥ 2, instead of one dimensional initial value which scales to each other, we are encountered with multi-dimensional initial value. Among α i 's, in many critical cases as well as in many supercritical cases, there is at most one scaling class (one-dimensional) of initial values from which we can shoot to a global solution. To show the existence of positive solutions of (4.5), up to a simple scaling, we have to find the special 1-D initial values. This is the main reason why there are so many results in the scalar case but very little about (4.1) for a long time period.
The degree theory approach for the shooting method gives a simple solution to this difficult problem. It can be used to solve a much larger class of problems. The argument starts with defining the target map ψ. By α > 0 we mean α is an interior point of R L + , and let r 0 be the smallest value of r for which u i (r, α) = 0 for some i. Define a map Then we need to show that ψ is continuous from R L + to ∂R L + . Assumptions (4.3)-(4.4) guarantee this.
In the next step, applying the degree theory, we show that ψ is onto from A a to B a where: for any a > 0. In particular, there exists at least one α a ∈ A a for sufficiently small a > 0 such that ψ(α a ) = 0. Shooting from the initial value α a , by the assumption that the system (4.2) admits no radially symmetric solution, we obtain a solution of (4.1) which decays to 0 at infinity ( lim r→∞ u(r, α a ) = ψ(α a ) = 0). Notice that we did not rule out the radial solution that does not decay at infinity for system with sign-changing nonlinearity, however for positive nonlinear source term such as HLS type system (1.5), the radial solution must monotone decrease, hence its radial solution must decay to zero at infinity.

Asymptotic analysis
In the asymptotic analysis we try to answer two questions: What are the asymptotic expansions of the radial solutions? Are all radial solutions scaling related?

Definition 4.2.
A function u is asymptotic to B |x| t near x = ∞, that is for a positive number t and B = 0 (∞).
The existence of the positive radial symmetric solutions are already known by [14] via shooting method. We consider the asymptotic property of these radial solutions in the supercritical cases.
Theorem 4.4. Suppose (u, v) be locally bounded non-negative radial solutions of (1.9) with p, q satisfying (4.12). Then either for some positive constants C 1 , C 2 > 0 and r large or u = v ≡ 0.
A more interesting open problem is if u(r)r 2(p+1) pq−1 converge as r → ∞. Proof. Recall that we define α = 2(p+1) pq−1 , β = 2(q+1) pq−1 (see Section 2). Considering the radial solutions of (1.9), we have the following ODE system, (4.14) Since we consider the radial solutions, by Lemma 2.2, the upper bound of (4.13) follows easily. It remains to prove that u, v are positive solutions, then lim inf Suppose not, we may assume u(r) = w(r)r −α and lim inf where σ = p+1 p+1 q+1 . As discussed in [15], we have The same conclusion also holds for −rv ′ (r) such that −rv ′ (r) ≤ (N − 2)σw q+1 p+1 (r)r −β . Recall the following Pohozaev's identity for radial solutions, (4.16) We distinguish two cases: 1. There exists r 0 > 0 such that for r > r 0 , w ′ (r) ≤ 0. Now consider Pohozaev's identity at r l and a 1 = n p+1 . By the supercritical property of p, q, we have where we have used q+1 p+1 < q. By comparison principle, we also have Now we return to the first equation of (4.14), Combine (4.17) and (4.19) we get From w(r l ) = 0, we obtain This contradicts with w(r l ) → 0. Hence u(r) ∼ r −α near ∞. Then integrate the second equation of (4.14) from 0 to r, Dividing the above inequality by r n−1 and integrating from r to ∞, we derive v(r) ≥ cr −β . This ends the proof of the asymptotics of u, v at ∞.

Subcritical HLS type systems: Liouville type theorem and the Lane-Emden conjecture
For subcritical HLS type systems, various Liouville type theorems are obtained. This kind of results are often based on the study of the Lane-Emden system (1.9). The Lane-Emden conjecture has been lasting unsolved for decades. Many mathematicians have contributed in this question, for example, the pioneer job done by Mitidieri (1992) [15] (see also [17]) which solves the Lane-Emden conjecture in radial case. For expository reference about the Lane-Emden conjecture, readers can check [7,25] and reference therein. Among these mathematical works we mention a couple of results below.
For n = 3, the conjecture is solved by two papers. First, Serrin and Zou (1996) [23] proved that there is no positive solution with polynomial growth at infinity. Theorem 5.1 (Serrin-Zou-1996). Let n = 3. Lane-Emden system (1.9) admits no solution given the solution has at most polynomial growth at infinity.
Then Poláčik,   [19] removed the growth condition. In fact, they proved that no bounded positive solution implies no positive solution.
Theorem 5.2 (Poláčik-Quittner- Souplet-2007 ). Let pq > 1. Assume that (1.9) does not admit any bounded nontrivial (nonnegative) solution in R n . Let Ω = R n be a domain. There exists C = C(n, p, q) > 0 such that any solution (u, v) of (1.9) in Ω satisfies Remark 5.3. In [19], p, q were assumed to be both > 1, however, their proof is valid for pq > 1 and can be verified directly.
This result has two important consequences. One is that combining with Serrin and Zou's result, one can prove the conjecture for n = 3. The other consequence is that one can reduce the Lane-Emden conjecture to the nonexistence of bounded positive solution. Thus, hereinafter we always assume that (u, v) are bounded.
For n = 4, the conjecture is recently solved by Souplet (2009) [25]. In [23], Serrin and Zou used the integral estimates to derive the nonexistence results. Souplet further developed the approach of integral estimates and solved the conjecture for n = 4 along the case n = 3. In higher dimensions, this approach provides a new subregion where the conjecture holds, but the problem of full range in high dimensional space still seems stubborn. For higher order HLS type systems (1.5), Arthur, Yan and Zhao (2014) [1] have proved a Liouville theorem with an adapted idea of measure and feedback argument developed by Souplet (2009) [25].
Existence of solutions to subcritical Lane-Emden system with double bounded coefficients can be extended from low dimension to system in higher dimension which fails the integral estimates (2.2). This may imply that, integral estimates are essential to problem of nonexistence of solutions. In [7], Cheng, Huang and Li raised the conjecture below, which is proven to be equivalent to the Lane-Emden conjecture.
Conjecture 5.6. For solution (u, v) to the Lane-Emden system with p ≥ q, there exist an s > 0 such that n − sβ < 1 and B R v s ≤ CR n−sβ .
Remark 5.8. Energy estimate (5.2) is a necessary condition to the Lane-Emden conjecture. One just needs to notice that when u, v ≡ 0, (5.2) is obviously satisfied. The key to the proof of Theorem 5.7 is to show (5.2) is sufficient.
Remark 5.9. The assumption on v is weaker than the corresponding assumption on u due to a comparison principle between u and v.
Indeed, we can replace (5.2) by: for some r > 0, n − rα < 1, 3) The proof of Theorem 5.7 is also based on the feedback argument developed by Souplet, though some basic estimates are adapted as needed. Here a sketch of the proof of Theorem 5.7 is presented. 2. Heuristically, we prove that there exists a sequence {R j } → ∞ such that the following estimate holds, , with a > 0 and b < 1. Then F (R) ≡ 0.
We start with estimate on G 1 (R). By Hölder's inequality and Sobolev embedding on S n−1 , So, Then by W 2,p -estimate, energy estimates in Lemma 2.2 together with the assumed integral estimate we have, ∃R ∈ (R, 2R) such that which is true under our assumption.