A Liouville theorem for subcritical Lane-Emden system

In this paper, we present a necessary and sufficient condition to the Lane-Emden conjecture. This condition is an energy type of integral estimate on solutions to subcritical Lane-Emden system. To approach the long standing and interesting conjecture, we believe that one plausible path is to refocus on establishing this energy type estimate.


Introduction
This paper is devoted to the nonexistence of solution to Lane-Emden system, where u, v ≥ 0, 0 < p, q < +∞. The hyperbola [8,9] 1 p + 1 + 1 q + 1 = n − 2 n is called critical curve because it is known that on or above it, i.e.
For n = 3, the conjecture is solved by two papers. First, Serrin and Zou [15] proved that there is no positive solution with polynomial growth. Then Poláčik, Quittner and Souplet [12] proved that no bounded positive solution implies no positive solution. Among others, 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 is that the Lane-Emden conjecture is equivalent to non-existence of no bounded positive solution. We always assume that (u, v) are bounded in this paper .
In Serrin and Zou [15], the authors used the integral estimates to derive the non-existence results. Later, in an important paper [17], Souplet further developed the approach of integral estimates and solved the conjecture for n = 4 along the case n = 3. As for 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. Souplet has proved that if  [17] is also effective on non-existence of solution to Hardy-Hénon type equations and systems (cf. Phan and Souplet [11], Phan [10]): This approach can also be applied to more general elliptic systems, for further details, we refer to [18] and [13]. Moreover, a natural extension and application of this tool is the high order Lane-Emden system which was done by Arthur, Yan and Zhao [1].
It is also worthy to point out that in Lane-Emden system the coefficient "1" of the source terms is not essential in the proof of Souplet [17]. In fact, if we consider the following Lane-Emden type system x · ∇c 2 (x) ≥ 0 for some positive constants a, b > 0. We can also have the following Rellich-Pohožaev type identity for some constants (1.9) By the constrains on c 1 (x), c 2 (x), we can have the left terms (LT) in (1.9) as The arguments in [17] is also valid in this case, and we still can prove non-existence for n ≤ 4 and for max(α, β) > n − 3, n ≥ 5. There is also an example that for some c 1 (r), c 2 (r) with c ′ 1 (r), c ′ 2 (r) < 0, we can construct solutions for (1.8) in some subcritical cases. For details, see Lei and Li [5]. It indicates that the monotonicity of c 1 (x), c 2 (x) in the radial direction plays an important role in the nonexistence.
Let us point out another observation that the key to the Lane-Emden conjecture is energy estimates in a certain form. In fact, we shall see that the decay rate of the energy estimates depending proportionally on the power of the integrand is crucial. Here is our result, Theorem 1.1. Let n ≥ 3 and (u, v) be a non-negative bounded solution to (1.1). Assume there exists an s > 0 satisfying n − sβ < 1 such that Remark 1.2. If p ≥ q, the assumption on v is weaker than the corresponding assumption on u due to a comparison principle (i.e. Lemma 2.5) between u and v.
In other words, if p ≥ q, and we assume for some r > 0, such that n − rα < 1, ∂r , ∂c 2 (x) ∂r ≥ 0. And the proof is more or less the same. So in this paper, we only prove the case c 1 = c 2 = 1.
Notice that Lane-Emden conjecture in the case n = 3 or 4 follows from Theorem 1.1 directly by taking s = p, as it includes Souplet's result for higher dimensions. Conjecture 1.4. Let n ≥ 3 and (u, v) be a non-negative bounded solution to (1.1). Then there exists an s > 0 satisfying n − sβ < 1 such that The paper is organized as follows. In section 2, we provide a few preliminary results. Some simplified proofs are given for the completeness and readers' convenience for readers. In section 3, we give the proof of Theorem 1.1.

Preliminaries
Throughout this paper, we often use the standard Sobolev embedding on S n−1 . We also make some conventions about the notations here. Let D denote the gradient with respect to standard metric on manifold. Let n ≥ 2, j ≥ 1 be integers and and C = C(j, z 1 , n) > 0. Although C may be different from line to line, we always denote the universal constant by C. For simplicity, in what follows, for a function f (r, θ), we define if no risk of confusion arises. Also let s, p, q be defined as in Theorem 1.1 and By Remark 1.2 and Lemma 2.5, throughout the paper, we always assume p ≥ q. Last we set

Basic Inequalities
Let us start with a basic yet important fact. Considering L r -norm on B 2R , we can write then by a standard measurement argument (cf. [15], [17]) one can prove that, The following lemma is a varied W 2,p -estimate which seems not to appear in any literature, so we shall give a simple proof.
, 2] such that on BR, u can be written as u = w 1 +w 2 , where respectively w 1 and w 2 are solutions to and additionally, By standard W 2,p -estimate with homogeneous boundary condition, we have Since w 2 can be solved explicitly by Poisson formula on BR, we see that by (2.4) for any x ∈ B 3 2 BR, w 2 (x) can be bounded pointwisely by So, Hence, Therefore, (2.3) becomes Then the lemma follows from a dilation and approximation argument.

Pohožaev Identity, Comparison Principle and Energy Estimates
For system (1.1) we have a Rellich-Pohožaev type of identity, which is the starting point of the proof of Theorem 1.1, We will show that ω ≤ 0.
Remark 2.6. For general Lane-Emden type system (1.8), we can choose By the same arguments, we can also get the desired comparison principle.
Proof. Without loss of generality, we can assume that p ≥ q. Let φ ∈ C ∞ (B R (0)) be the first eigen-function of −∆ in B R and λ be the eigenvalue. By definition and rescaling, it is easy to see that φ | ∂B R = 0 and λ ∼ 1 R 2 . By normalizing, one gets φ ≥ c 0 on B R/2 for some constant c 0 independent of R, φ(0) = φ ∞ = 1. So, multiplying (1.1) with φ then integrating by parts on B R we have, By Hopf's Lemma we know that ∂φ ∂n < 0 on ∂B R , so Similarly, we have Applying Lemma 2.5 to the inequality above, we have Notice that q(p+1) q+1 > 1 as pq > 1, so by Hölder inequality Therefore, by (2.7) and by Hölder inequality Last, by (2.8)

Key Estimates on S n−1
Now that we have energy inequalities (2.6), in our assumption (1.11) we can always assume s ≥ p.
Since l = s p , we have l ≥ 1. The following estimates for quantities on sphere S n−1 are necessary to the proof. Proposition 2.8. For R ≥ 1, there existsR ∈ [R, 2R] such that for l = s p ≥ 1, k = p+1 p and m = q+1 q , we have In the view of Lemma 2.1, to prove Proposition 2.8, we shall give the corresponding estimates on B 2R first. We use the varied W 2,p -estimate (i.e. Lemma 2.2) to achieve this.
For estimates (2.11), by Lemma 2.3, The second estimate in (2.11) can be obtained by a similar process. Last, the fact that n−(p+1)β < 0 gives Proof. Suppose not, then for any M > 0 and any {R j } → +∞ we have Take M > 5 n and R j+1 = 4R j with R 0 > 1. Therefore,

Proof of Liouville Theorem
. By the Rellich-Pohožaev type identity in Lemma 2.4, we can denote Heuristically, we are aiming for estimate as Then by Lemma 2.10 there exists a sequence {R j } → +∞ such that Suppose there are infinitely many R j 's such that G 1 (R j ) ≥ G 2 (R j ), then take that subsequence of {R j } and still denote as {R j }. We do the same if there are infinitely many R j 's such that G 1 (R j ) ≤ G 2 (R j ). So, there are only two cases we shall deal with: there exists a sequence {R j } → +∞ such that Then we conclude that F (R) ≡ 0. Surprisingly, for both cases a i ≈ (α + β + 2 − n)δ i . Indeed, we have Hence, Theorem 1.1 is a direct consequence of Theorem 3.1, and we only need to prove Theorem 3.1 for case 1 and 2.
In addition to our assumption that n − sβ < 1, since we have energy inequalities (2.6), we can assume s ≥ p. Also, if n − sβ < 0, (1.11) implies v ≡ 0 and hence u ≡ 0. So, we assume n − sβ ≥ 0. Let l = s p , then l ≥ 1, and n − 1 pβ < l ≤ n pβ . (3.6) It is worthy to point out that, what the proof of Lane-Emden conjecture really needs is a "breakthrough" on the energy estimate (2.6). s in (1.11) needs not be very large but enough to satisfy n − sβ < 1. In other words, s can be very close to n−1 β , and it is sufficient to prove Theorem 1.1.
The strategies of attacking G 1 and G 2 are the same. Basically, first by Hölder inequality we split the quantities on sphere S n−1 into two parts. One has a lower (than original) index after embedding, and the other has a higher one. Then we estimate the latter part by F (R), and thus we get a feedback estimate as (3.4). Let k = p + 1 p .
To get desired estimate, we have requirements in form of inequalities involving parameters, such as α, β, ǫ and etc. To verify those requirements very often we just verify strict inequalities with ǫ = 0 because such inequalities continuously depend on ǫ.
Thus, we have proved Case 1.