On stable entire solutions of sub-elliptic system involving advection terms with negative exponents and weights

where Gu = xu + |x|2α yu, z = (x, y) ∈Rn :=Rn1 ×Rn2 is the Grushin operator, α ≥ 0, p ≥ q > 1, ‖z‖2(α+1) = |x|2(α+1) + |y|2, γ ≥ 0 and a is a smooth divergence-free vector that we will specify later. Inspired by recent progress in the study of the Lane–Emden system, we establish some Liouville-type results for bounded stable positive solutions of the system. In particular, we prove the comparison principle to establish our result. As consequences, we obtain a Liouville-type theorem for the weighted Grushin equation involving advection terms

We start by noting that, in the case a ≡ 0 and γ = 0, the system (1.1) and Eq. (1.2) reduce to ⎧ ⎨ and G u = u -p in R n . (1.5) In the case α = 0, Eq. (1.5) arises in many branches of applied sciences and has been studied in a number of recent works; see [6,20] and the references therein. The nonexistence of positive stable classical solutions of (1.5) was examined in [20]. This result was then generalized in [6] to positive stable weak solutions of a weighted equation. More precisely, the authors of [6] figured out the critical exponent and established an optimal Liouvilletype theorem for this class of solutions. When α > 0, the Liouville-type theorem for a special class of solutions of (1.4) and (1.5). "the so-called stable solutions" has been studied by Duong, Lan, Le and Nguyen [10]. We summarize here some results in [10]. We remark that for Eq. (1.5) (with α = 0), the critical exponent on the right-hand side of (1.6) was first found in [20]. This exponent has been shown to be optimal in the class of positive stable weak solutions; see [6].
Similar to the celebrated Lane-Emden system in the case of positive exponents, the system (1.4) is also a natural extension of Eq. (1.5). It is worth to remark that there are many papers developing various useful tools to study the nonexistence of positive stable solutions (see for example [2,12,13,17,22,26] and the references therein. For other results on Grushin operators, Wei et al. [25] established a Liouville-type theorem for weak stable solutions of weighted p-Laplace-type Grushin equation in the case of negative exponent nonlinearity. Some important and interesting results can be found in [21].
Recently, elliptic problems involving advection terms, i.e. a = 0, have received considerable attention [3,4,8,11]. In particular, Duong and Nguyen [11] studied the equation (1.7) Taking advantage of the variational structure, and using the approach of Farina [13], he established some Liouville-type theorems for the class of stable sign-changing weak solutions. Now, we state this result as follows.
Theorem 1.2 ([11]) Suppose that there is a nonnegative constant θ such that
In the general case where a = 0, elliptic problems with advections have no variational structure and this requires another approach to obtaining a classification of stable solutions. Recall that, in this case, see e.g. [3], a positive classical solution u of u + a · ∇u = u p in R n (1.8) is called stable if there is a smooth positive function F such that Recently, relying on Farina's approach [13] and the generalized Hardy inequality, Cowan [3] established a Liouville-type theorem for stable positive solution of (1.8) under the smallness condition imposed on the divergence-free a.
On the other direction, the Liouville-type theorem for the class of stable solutions for system (1.9) was examined by Duong [8]. He established a Liouville-type result for stable positive solutions of the system in the case p ≥ q ≥ 1 and pq > 1. In particular, when p = q, his result is a natural extension of Cowan [3] to the equation with advection. Furthermore, we would also like to mention that when pq ≤ 1, the system (1.9) has no positive supersolutions (see Theorem 1.3 [9]).
For the general equation or system with γ = 0, the Liouville property is less understood and is more delicate to deal with than γ = 0. There exist many excellent papers using Farina's approach to the Hardy-Hénon equation and the weighted nonlinear elliptic equations. We refer to [7,22,24] and the references therein. Inspired by the ideas in [2,16], Hu [18] adopt the new approach of a combination of second order stability, Souplet's inequality [23] and a bootstrap iteration to establish Liouville-type theorems for the semi-stable and of the scalar equation In particular, Hu [18] has obtained the following result.
Then there is no classical positive semi-stable solution of (1.10). In particular, there is no classical positive semi-stable solution of (1.10) for any 2 ≤ p ≤ q if n ≤ 10 + 4γ . 2. Let p > 4 3 , γ > 0 and Then there does not exist a classical positive semi-stable solution of (1.11).
In this paper, we propose to study the system (1.1) which can be regarded as a natural generalization of the scalar equation (1.2). Motivated by [8,10,18], we give the classification of bounded stable positive solutions of (1.1) under the assumption (1.3). Before stating our main results, let us recall the definition of such solutions motivated by [8,10].
The main result in this paper is the following.
The key in our proof is the comparison principle and nonlinear integral estimates. However, the techniques used to prove the comparison principle in [14,18] for the Laplace operator do not seem applicable to the system (1.1) because the operator G no longer has symmetry and it degenerates on the manifold {0} × R n 2 . Then, in this paper, we establish the comparison principle for Grushin operators by developing the idea in [1,10,12,15]. In addition, the L 1 -estimate to the bootstrap iteration in [2] does not work in the case of Grushin operator, we instead switch to the L 2 -estimate in the bootstrap argument. We also employ the idea in [1,10,12,15] to prove the "inverse" comparison principle which is crucial to proving our result.
Recall that a classical solution of (1.2) is called stable if We remark also that the method used in the present paper can be applied to study the weighted systems, and to more general class of degenerate operator, such as the λ oper-ator (see [19,22]) of the form Here λ i : R n → R, i = 1, . . . , n are nonnegative continuous functions satisfying some properties such that λ is homogeneous of degree two with respect to a group dilation in R n .
The organization of this paper is as follows. In Sect. 2, we establish the stability inequality and the comparison principle for the system (1.1) and then prove an a priori estimate of the solutions. In Sect. 3, we give the proof of the main result.

Stability inequality and comparison principle 2.1 Stability inequality
Proof We follow the idea in [2,8].
Using integration by parts and Young's inequality: 2zzz 2 ≤ z 2 , we obtain Consequently, combining (2.2) and (2.3), it follows that By the same argument, we also have We now add the inequalities (2.4) and (2.5) to obtain Putting this back into (2.6) gives the desired result.

Comparison principle
In this subsection, we shall prove the comparison principle for the system (1.1) without stability assumption.

Lemma 2.2
Suppose that (u, v) is a bounded positive solution of (1.1). Assume that 1 < q ≤ p and (1.3) hold. Then = u d-1 v pl p u dp u dp v p . This implies that ∇ G w z * = 0 and G w z * ≤ 0.
However, the right-hand side of (2.9) at z * is positive. This is a contradiction. Case 2. If the supremum of w is attained at infinity. Take a cut-off function χ ∈ C ∞ c (R n , [0, 1]) verifying χ = 1 on B 1 × B 1 and χ = 0 outside Here m > 0 will be chosen later. A simple calculation yields This implies that In what follows, all the estimates are taken at the point z R . First, using ∇ G w R (z R ) = 0, we have Hence, (2.14) Thus, Recall that the constant C is independent of R. Consequently, Choosing m = 2 p-1 (or p = m+2 m ), we get It follows from (2.19), the boundedness of (u, v) and d ≤ 1 that Finally, letting R → +∞, we get M = 0, which contradicts (2.10). The proof is complete.
Combining the proof of Lemma 2.2 with the idea in [1,10,12,15], we have the inverse comparison principle as follows.

Lemma 2.3
Suppose that (u, v) is a bounded positive solution of (1.1). Assume that 1 < q ≤ p and (1.3) hold. Then we have In order to obtain the proof, it suffices to use the arguments as in Lemma 2.2 by noting that (2.9) is replaced by (2.21). The details are then omitted.
In what follows, the constant C does not depend on a positive parameter R and may change from line to line.

Lemma 2.4
Suppose that (u, v) be a bounded stable positive solution of (1.1). Assume that 1 < q ≤ p and (1.3) hold. Then for R > 0 there exists C > 0 independent of R such that Take a cut-off function φ ∈ C ∞ c (R n , [0, 1]) verifying φ = 1 on B 1 × B 1 and φ = 0 outside where m ≥ 2 which is fixed. Then there exists C > 0 independent of R such that (2.25) By virtue of (2.1) and (2.24), we derive Recall that pq-1 pq-q > 1. Then, by combining the Hölder inequality, (2.25) and (2.26), we get Hence, the desired integral estimate (2.22) follows. Finally, (2.23) follows from using the same argument as above where we use (2.20) instead of (2.7).

Proof of the main result 3.1 Beginning of the proof
In this subsection, we give a preparation for the bootstrap iteration. Using Lemmas 2.2 and 2.1, we get the following.
Then we have where in the last equality, we have used div G a = 0.

End of the proof
The bootstrap argument in this subsection is quite similar to that in [10,12]. For completeness, we present the details.
From a scaling argument it follows that Suppose that (u, v) is a positive stable solution of (1.1). Set A simple calculation gives where φ is given in (3.4). Then Multiplying the second equation in (1.1) by v -2t-1 φ 2 R and using integration by parts, we obtain Inserting this into (3.6), using Lemma 3.1, we obtain w 2 dx dy.
To simplify notations below, we use R n = 2 n R. By using (3.7) and an induction argument, we obtain Substituting (3.10) into the last inequality of (3.9), one has (3.11) Since r ∈ [1, κ] is chosen such that t m is close to τ + 0 , the exponent in the right-hand side of (3.11) is negative. Letting R → +∞, we obtain a contradiction.