On the H\'enon-Lane-Emden conjecture

We consider Liouville-type theorems for the following H\'{e}non-Lane-Emden system \hfill -\Delta u&=&|x|^{a}v^p \text{in} \mathbb{R}^N, \hfill -\Delta v&=&|x|^{b}u^q \text{in} \mathbb{R}^N, when $pq>1$, $p,q,a,b\ge0$. The main conjecture states that there is no non-trivial non-negative solution whenever $(p,q)$ is under the critical Sobolev hyperbola, i.e. $ \frac{N+a}{p+1}+\frac{N+b}{q+1}>{N-2}$. We show that this is indeed the case in dimension N=3 provided the solution is also assumed to be bounded, extending a result established recently by Phan-Souplet in the scalar case. Assuming stability of the solutions, we could then prove Liouville-type theorems in higher dimensions. For the scalar cases, albeit of second order ($a=b$ and $p=q$) or of fourth order ($a\ge 0=b$ and $p>1=q$), we show that for all dimensions $N\ge 3$ in the first case (resp., $N\ge 5$ in the second case), there is no positive solution with a finite Morse index, whenever $p$ is below the corresponding critical exponent, i.e $ 1<p<\frac{N+2+2a}{N-2}$ (resp., $ 1<p<\frac{N+4+2a}{N-4}$). Finally, we show that non-negative stable solutions of the full H\'{e}non-Lane-Emden system are trivial provided \label{sysdim00} N<2+2(\frac{p(b+2)+a+2}{pq-1}) (\sqrt{\frac{pq(q+1)}{p+1}}+ \sqrt{\frac{pq(q+1)}{p+1}-\sqrt\frac{pq(q+1)}{p+1}}).


Introduction and main results
We consider the following weighted system where pq > 1 and p, q, a, b ≥ 0 and Ω is a subset of R N , N ≥ 1.
We start by noting that in the case of the Lane-Emden equation (i.e., when p = q and a = b = 0), the Pohozaev inequality shows that there is no positive solution satisfying the Dirichlet boundary condition, whenever Ω is a bounded star-shaped domain and p ≥ N +2 N −2 , the critical Sobolev exponent. On the other hand, a celebrated theorem by Gidas-Spruck [11] states that there is no solution whenever Ω = R N and p < N +2 N −2 for N ≥ 3. This non-existence result is also optimal as shown by Gidas, Ni and Nirenberg in [10] under the assumption that u = O(|x| 2−N ), and by Caffarelli, Gidas and Spruck in [3] without the growth assumption. See also Chen and Li [4] for an easier proof based on the moving planes method. Also, Lin [13] using moving plane methods proved similar optimal non-existence results for p < N +4 N −4 , N > 4 in the case of the fourth order Lane-Emden equation (i.e., when p > 1 = q and a = b = 0).
Note that while there is no positive solution (for the Sobolev critical and super-critical exponents) in a bounded star-shaped domain, the non-existence result (for Sobolev sub-critical exponents) on the whole space is sharper as it does not assume the positivity of the solution.
In the case of the system (1), one can again use the Pohozaev identity whenever Ω is a bounded starshaped domain in R N , to establish the following non-existence result.
Theorem A. [8,19] Let N ≥ 3 and let Ω ⊂ R N be a star-shaped bounded domain. If N + a p + 1 then there is no positive solution for (1) that satisfy the Dirichlet boundary conditions.
By noting that the curve N +a p+1 + N +b q+1 = N − 2 is the critical hyperbola or Sobolev hyperbola, the above theorem states that the Liouville-type result for positive solutions on bounded star-shaped domain holds when (p, q) is above the critical hyperbola. It is therefore expected that -just like the case of the scalar Lane-Emden equation (p = q and a = b = 0) -the non-existence of solutions on the whole space R N should occur exactly when (p, q) is in the complementary domain, that is when it is under the critical hyperbola. This is the statement of the following Hénon-Lane-Emden conjecture.
Conjecture 1. Suppose (p, q) is under the critical hyperbola, i.e., N + a p + 1 If Ω = R N , then there is no positive solution for system (1).
Proving such a non-existence result seems to be challenging even for the Lane-Emden conjecture (i.e., when a = b = 0) for systems. The case of radial solutions was solved by Mitidieri [14] in any dimension, and both Mitidieri [14] and Serrin-Zou [23] constructed positive radial solutions on and above the critical hyperbola, i.e. 1 p+1 + 1 q+1 ≤ N −2 N , which means that the non-existence theorem is optimal for radial solutions. For non-radial solutions of the Lane-Emden system, there are the results of Souto [24], Mitidieri [14] and Serrin-Zou [22] who proved the non-existence of solutions in dimensions N = 1, 2, while in dimension N = 3, Serrin-Zou [22] gave a proof for the non-existence of polynomially bounded solutions, an assumption that was removed later by Poláčik, Quittner and Souplet [18]. More recently, Souplet [21] settled completely the conjecture in dimension N = 4, while providing in dimensions N ≥ 5, a more restrictive new region for the exponents (p, q) that insures non-existence.
(i) Let N = 4 and p, q > 0. If (p, q) satisfies then system (1) with Ω = R N has no positive classical solutions.
The Lane-Emden conjecture in dimensions N ≥ 5 is still open. The Hénon-Lane-Emden conjecture is even less understood. Even for the scalar case a = b and p = q ( i.e., the Hénon equation), Gidas and Spruck in [11] solved the conjecture only for radial solutions, also showing that in this case, the non-existence result is optimal. For non-radial solutions, they proved some partial results such as the non-existence of positive solutions for a ≥ 2 and p ≤ N +2 N −2 (the Sobolev critical exponent for a = 0). Recently, Phan and Souplet [17] showed among other things that the Hénon-Lane-Emden conjecture for the scalar case holds for bounded positive solutions in dimension N = 3.
Theorem C. ) Let N = 3, a = b > 0 and p = q > 1. Assume (p, q) satisfies (3), then there is no positive bounded solution for the Hénon equation, i.e., For systems, Mitidieri [14] gave a partial solution to the conjecture for radial solutions by showing the following.
Recently, Bidaut-Veron-Giacomini [2] used a Pohozaev type argument and a suitable change of variables to improve the above result by proving the following result.
if and only if (p, q) is above or on the critical hyperbola, i.e., when (2) holds.
In this note, we shall prove that the Hénon-Lane-Emden conjecture holds in dimension N = 3 for bounded positive solutions, extending the result of Phan-Souplet [17] mentioned in Theorem C above 1 . We also give a few partial results for the Hénon equation whether of second order or fourth order in all dimensions N ≥ 3 or N ≥ 5. Here are our main results: We note that Miditieri and Pohozaev [16] have shown that the above result holds in higher dimension provided the following stronger condition holds: . For that they used a rescaled test-function method (as in Lemma 1 below) to prove the result for p, q ≥ 1. More recently, Armstrong and Sirakov [1] proved -among other things-similar results for p, q > 0, by developing new maximum principle type arguments. We are thankful to P. Souplet for informing us of these latest developments by Armstrong and Sirakov.
We shall also consider in the scalar case the question of existence of solutions with finite Morse index solutions (as opposed to bounded solutions). We get the following counterpart to the Phan-Souplet result in higher dimensions (N ≥ 3).
We also have the following result for the fourth order equation, Theorem 3. Let a ≥ 0, p > 1 and N ≥ 5. Then, for any Sobolev sub-critical exponent, i.e., (7) has no positive solution with finite Morse index.
There are also various results for the cases where −2 < a, b < 0 and pq ≤ 1. For that we refer to [2,8,16,9,11,12,17]. In [8], similar Liouville-type theorems in the notion of stability have been proved for positive solutions of system (1) on R N provided p > q = 1 and a = b in dimensions Note that this range of dimensions falls under the corresponding critical hyperbola, i.e. N < 4 + a + 8+4a p−1 .

Proofs
The main tools used in our proof are Pohozaev-type identities for both systems and equations as well as various integral estimates.

Proof of Theorem 1
The proof of Theorem 1 is heavily inspired by ideas of Souplet [21] and Serrin-Zou [22]. We use Pohozaevtype identities, various integral estimates, as well as some elliptic estimates on the sphere. Throughout this subsection, all norms refer to functions defined on the unit sphere, i.e. ||u|| m := ||u|| L m (S N −1 ) . We start with the following estimate on the non-linear terms. Note that for a = b = 0, this was proved by Serrin and Zou [22] via ODE techniques, and by Miditieri and Pohozaev [16] who used the following rescaled test functions approach for a, b > −2. For the sake of convenience of readers, we recall the proof. Interested readers can find more details for both scalar and system cases in [20].
where the positive constant C does not depend on R.
Proof: Fix the following function For m ≥ 2, test the first equation of (1) by ζ m R and integrate to get Applying Hölder's inequality we get By a similar calculation for k ≥ 2, we obtain By collecting the above inequalities we get for pq > 1, and ✷ By using Hölder's inequality, we can now get the following L 1 -estimates. Corollary 1. With the same assumptions as Lemma 1, we have where the positive constant C does not depend on R.

Lemma 2. (Sobolev inequalities on the sphere S
By applying Lemma 1, Corollary 1 and Lemma 4, we obtain the following estimates on the derivatives of u and v.
where the positive constant C does not depend on R.
For a = b = 0, the following Pohozaev identity has been obtained by Mitidieri [15], Serrin and Zou [22]. It has also been used by Souplet in [21].
is a positive solution of (1), then it necessarily satisfy Now, we are in the position to prove Theorem 1. Proof of Theorem 1: Since (p, q) satisfy (3), then we can choose λ and γ such that N +a p+1 > λ and N +b q+1 > γ. Now, for all R > 0 define where Step 1. Upper bounds for G 1 and G 2 . Set m = ∞ in Lemma 2 to get for either t = p + 1 or t = q + 1 , where ǫ > 0 is small enough and will be chosen later. So, We now look for the same type bounds for G 2 . Apply Schwarz's inequality to get Then, using Lemma 2 we obtain the following upper bounds.
Step 2. The following L t -estimates hold in the annulus domain To prove (15)-(18), we just apply Corollary 1 and Lemma 5. Here is for example the proof for (20). Apply Lemma 3, Corollary 1 and Lemma 1 to get The proof of (19) is similar.

Proof of Theorem 2
We recall that a critical point u ∈ C 2 (Ω) of the energy functional is said to be • a stable solution of (6) if for any φ ∈ C 1 c (Ω), we have • a solution with Morse index m if there exist φ 1 , ..., φ m such that X m = Span{φ 1 , ..., φ m } ⊂ C 1 c (Ω) and I uu (φ) < 0 for all φ ∈ X m \ {0}.
Note that if u is of Morse index m, then for all φ ∈ C 1 c (Ω\Σ) we have I uu (φ) ≥ 0, where Σ = ∪ m i=1 supp(φ i ), and therefore u is stable outside the compact set Σ ⊂ Ω.
We shall need the following lemma.
Proof: The following proof also holds true for weak solutions. The ideas are adapted from [5,6,7]. Note first that for any stable solution of (6) and η ∈ C 1 c (Ω), we have the following: Test (26) on η = u t φ 2 for φ ∈ C 1 c (Ω) for an appropriate t ∈ R that will be chosen later, to get Apply Young's inequality 2 to |∇u|u Now, test (25) on u t+1 2 φ to get where again we have used Young's inequality in the last estimate. Combine now this inequality with (27) to see For an appropriate choice of t, given in the assumption, we see that the coefficient in L.H.S. is positive for ǫ small enough. Therefore, replacing φ with φ m for large enough m and applying Hölder's inequality with exponents t+p t+1 and t+p p−1 we obtain Note that both exponents are greater than 1 for t given in (i) and (ii).
On the other hand, combining (27) and (28) gives us Similarly, replace φ by φ m and apply Hölder's inequality with exponents t+p t+1 and t+p p−1 to get This inequality and (29) finish the proof of (24). ✷ Now, we are in the position to prove the theorem.
Proof of Theorem 2: We proceed in the following steps.
Step 1: We have the following standard Pohozaev type identity on any Ω ⊂ R N .
To get (30), just multiply both sides of (6) by x · ∇u, do integration by parts and collect terms.
Step 3: The following equality holds Then, integrate over B 2R to get By Hölder's inequality, we have the following upper bound for R.H.S. of (32), . Therefore, from Step 2, there exists a positive constant C independent of R such that .
Step 4: we have Apply Lemma 7 for t = 1 with the following test function Now, define the following sets for large enough M ; From (33), we have Similarly, one can show |θ 1 (R)| ≤ R/M . By choosing M large enough we conclude |θ i (R)| ≤ R/3 for i = 1, 2. Therefore, for each R ≥ 1, we can findR Now, apply Pohozaev identity, (30), with Ω = BR to see that R.H.S. converges to zero if R → ∞ for subcritical p, i.e. N < 2(p+a+1) p−1 . Hence, From this and (31), we finish the proof of Step 4.
✷ Remark: For the Sobolev critical case p = N +2+2a N −2 , using the change of variable w := u(r 1+ a 2 ) and applying well-known classifying-type results mentioned in the introduction for the Lane-Emden equation, one can see all radial solutions of (6) are of the following form where k(ǫ) = (ǫ(N + a)(N − 2)) N −2 2(2+a) . Then, from the classical Hardy's inequality it is straightforward to see u ǫ is stable outside a compact set B R0 , for an appropriate R 0 . Note that for −2 < a ≤ 0, by Schwarz symmetrization (or rearrangement), it is shown in [9] that all radial solutions of (6) with p = N +2+2a N −2 and N > 2 are of the form (34).

Proof of Theorem 3:
We recall that a critical point u ∈ C 4 (Ω) of the energy functional is said to be a stable solution of (7), if for any φ ∈ C 4 c (Ω), we have Similarly to the second order case, one can define the notion of stability outside a compact set, which contains the notion of solutions with finite Morse index. We first prove the following estimate.
Step 1: We have the following standard Pohozaev type identity on any Ω ⊂ R N .
To get (43), just multiply both sides of (7) by x · ∇u, do integration by parts and collect terms.
By Hölder's inequality, we have the following upper bound for I 1 (R), . Therefore, from Step 2, there exists a positive constant C independent of R such that .
Since N < 2(p+a+1) p−1 , we have lim R→∞ |I 1 (R)| = 0. Now, we consider the second term in R.H.S. of (45). Apply Young's inequality for a given ǫ > 0 (we choose it later) to get Using Green's theorem we get By the same discussion as given for I 1 (R) one can see lim R→∞ |I 3 (R)| = 0. For the term I 4 (R), we apply Hölder's inequality again
Step 4: The following equality holds Apply Lemma 8 with the following test function Since R N |x| a u p+1 < ∞ and N < 2(2p+2+a) Similarly for the second term, J 2 (R), using Lemma 3, i.e., In the following, we shall find a bound for the measure of the above sets. From (46), we have Similarly, from (48) and (46) we get |Λ 1 (R)|, |Λ 4 (R)| ≤ R/M . By choosing M large enough we conclude |Λ i (R)| ≤ R/5 for i = 1, · · · , 4. Therefore, for each R ≥ 1, we can find Then, from the definition ofR and Λ i for i = 1, · · · , 4, we have