Liouville theorems for stable solutions of the weighted Lane-Emden system

We examine the general weighted Lane-Emden system \begin{align*} -\Delta u = \rho(x)v^p,\quad -\Delta v= \rho(x)u^\theta, \quad u,v>0\quad \mbox{in }\;\mathbb{R}^N \end{align*} where $1<p\leq\theta$ and $\rho: \mathbb{R}^N\rightarrow \mathbb{R}$ is a radial continuous function satisfying $\rho(x)\geq A(1+|x|^2)^{\frac{\alpha}{2}}$ in $\mathbb{R}^N$ for some $\alpha\geq 0$ and $A>0$. We prove some Liouville type results for stable solution and improve the previous works \cite{co, Fa, HU}. In particular, we establish a new comparison property (see Proposition 1.1 below) which is crucial to handle the case $1<p \leq \frac{4}{3}$. Our results can be applied also to the weighted Lane-Emden equation $-\Delta u = \rho(x)u^p$ in $\mathbb{R}^N$.

As a consequence of Theorem 1.1, we obtain the following classification result for stable solution of the Lane-Emden equation Recalling that for the autonomous case, i.e. when ρ ≡ 1, the stable solutions of the corresponding Lane-Emden equation and system, or the biharmonic equation (corresponding to p = 1) have been widely studied by many authors. See for instance [8,18,2,11,1,6] and the references there in.
For the second order Lane-Emden equation (p > 1) Farina classified completely in [8] all finite Morse index classical solutions for 1 < p < p JL , where p JL stands for the Joseph-Lundgren exponent [13] (see also [10] Dávila-Dupaigne-Wang-Wei [6] recently gave a complete classification of finite Morse index solutions. They derived a monotonicity formula for the solutions of (1.7) and reduced the problem to the nonexistence of stable homogeneous solutions.
It is worthy to mention that Chen-Dupaigne-Chergu [1] proved an optimal Liouville type result for the radial stable solutions of (1.1) for θ ≥ p > 1 and ρ ≡ 1.
For the weighted equation or system with positive weights, the Liouville type results are less understood.
• Using also Farina's approach, Cowan-Fazly [4] established a Liouville type result for classical stable sub-solutions of (1.4) for N satisfying (1.5), p > 1 and • Adopting the new approach in [5], Hu proved the following Liouville theorem for classical stable solutions of (1.1) for ρ = ρ 0 and θ ≥ p ≥ 2 or θ = p > 4 3 , obtaining a direct extension of Theorem 1 in [2] for ρ ≡ 1. More precisely, let t + 0 and t − 0 be the quantities used in [2] : Hu proved in [12] : then there is no classical stable solution of (1.1). In particular there is no classical stable solution of (1.1) for any 2 ≤ p ≤ θ and N ≤ 10 + 4α.
ii) If p > 4 3 and N satisfies (1.5), then there is no classical stable solution of (1.4).
Using the above remark, we see that Theorem A (hence Theorem 1 in [2]) can be extended immediately for 4 3 < p ≤ θ.
• We can show that 2t + 0 θ+1 pθ−1 < x 0 for any 1 < p ≤ θ (see Lemma 2.4 below), where x 0 is the largest root of the polynomial H given by (1.3). So Theorem 1.1 improves the bound given in Theorem A.
• Our approach permits to prove a Liouville type result for θ ≥ p > 1. To the best of our knowledge, no general Liouville type result was known for stable solution of (1.1) with positive weight for 1 < p ≤ 4 3 .
To prove Theorem 1.1, we will use the following Souplet type estimate [17]. Its proof is the same as for Lemma 2.3 in [12] where we replace just ρ 0 by ρ, so we omit the details. Lemma 1.1. Let θ ≥ p > 1 and ρ satisfy (⋆). Then any classical solution of (1.1) verifies However, to handle the case 1 < p ≤ 4 3 , we need the following new comparison property between u and v. It is somehow an inverse version of Souplet's estimate (1.9), and has its own interest.
Our paper is organized as follows. In section 2, we prove some preliminaries results, in particular we give the proof of Proposition 1.1. The proofs of Theorem 1.1 and Corollary 1.1 are given in section 3.

Preliminaries
In order to prove our results, we need some technical lemmas. In the following, C denotes always a generic positive constant independent on (u, v), which could be changed from one line to another. The ball of center 0 and radius r > 0 will be denoted by B r .

Comparison property
In this subsection, we give the proofs of Proposition 1.1. First, we can adapt the proof of Lemma 2.1 in [9] (which was inspired by the previous works [16,14]), to obtain the following integral estimates for all classical solutions of (1.1).
Lemma 2.1. Let p ≥ 1, θ > 1 and suppose that ρ satisfies (⋆). For any classical solution (u, v) of (1.1) there exists C > 0 such that for any R ≥ 1, there holds Multiplying the equation −∆u = ρ(x)v p by ψ m and integrating by parts, there holds then By Hölder's inequality, we get Take now k and m large verifying m ≤ (k − 2)p and k ≤ (m − 2)θ. Combining the two above inequalities, we get Similarly, we obtain the estimate for u.
Now we are in position to prove the inverse comparison property.
It follows that ∆w ≥ 0 in the set {w ≥ 0}. Consider w + := max(w, 0). Next, we split the proof into two cases.
Hereafter, S N −1 denotes by the unit sphere in R N . By Lemma 2.1, we derive that This implies that lim inf r→∞ g(r) = 0, hence lim inf r→∞ f (r) = 0. Consequently, there exist Case 2 : 1 < p < 2. For any R > 0 and ǫ > 0, we have Letting ǫ → 0 (passing to limit in the l.h.s. via monotone convergence and use the dominated convergence on the r.h.s.), we get always the estimate (2.1), which will lead to the same conclusion : w + ≡ 0 in R N .

Consequence of stability
With the ideas in [5,7], we can proceed similarly as the proof of Lemma 2.1 in [12] and claim Lemma 2.2. If (u, v) is a nonnegative classical stable solution of (1.1), then 2) The following Lemma is a consequence of the stability inequality (2.2) and Proposition 1.1. It plays a crucial role to handle the case 1 < p ≤ 4 3 . Here we use also some ideas coming from [18,11]. Let (u, v) be a stable solution to (1.1) with 1 < p ≤ min( 4 3 , θ). Assume that u is bounded and ρ satisfies (⋆), there holds Proof. Let (u, v) be a stable solution of (1.1), where u is bounded. Take η ∈ C ∞ c (R N ). Multiplying −∆v = ρ(x)u θ by vη 2 and integrating by parts, there holds Using Lemma 1.1, we get Set φ = vη in (2.2) and integrating by parts, we deduce that Combining the two last inequalities, we obtain Using Proposition 1.1, there exists a positive constant C such that Remark that p < 2 < θ+p+2 2 for 1 < p ≤ 4 3 and θ ≥ p. A direct calculation yields Take m large such that mλ > 1. By Hölder's inequality, Lemma 2.1 and (2.4), we get so we are done.
The following lemma plays an important role in dealing with Theorems 1.1 and Corollary 1.1, where we use some ideas from [11]. Here and in the following, we define R k = 2 k R for all R > 0 and integers k ≥ 1.
We need also the following L 1 elliptic regularity result, see Lemma 5 in [2].
Lemma 3.2. For any 1 ≤ β < N N −2 , there exists C > 0 such that for any smooth non-negative function w, we have wdx.
Applying the above two lemmas, we establish the following result which plays an essential role in iteration process.
Proof of Theorem 1.1 completed. Let (u, v) be a classical stable solution of (1.1) with ρ satisfying (⋆). We split the proof into two cases.
Case 2 : 1 < p ≤ 4 3 and u is bounded. Let now 2 > q > 0, using (3.10), with p is replaced by 2 and applying Lemma 2.3, it follows that for any R > 1, Proceeding as above, we can apply Corollary 3.1, with 2t − 0 < q < 2 and q < β < N s 0 N −2 to complete the proof of Theorem 1.1.
Then, 2t + 0 is the largest root of L as t + 0 > p > 1. Therefore is the largest root of H, and we can check easily that x 0 > 4 for all p > 1. The result follows immediately by applying Theorem 1.1.