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

Abstract

We examine the weighted Grushin system involving advection terms given by G u-a • ∇ G u = (1 + z 2(α+1)) γ 2(α+1) v-p in R n , G v-a • ∇ G v = (1 + z 2(α+1)) γ 2(α+1) u-q in R n , where G u = x u + |x| 2α y u, z = (x, y) ∈ R n := R n 1 × R n 2 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 G u-a • ∇ G u = (1 + z 2(α+1)) γ 2(α+1) up in R n. The main tools in the proof of the main result are the comparison principle, nonlinear integral estimates via the stability assumption and the bootstrap argument. Our results generalize and improve the previous work in (Duong et al. in Complex Var.

Liouville‐type theorem for finite Morse index solutions to the Choquard equation involving Δλ‐Laplacian

“…17 Concerning the Liouville-type theorem for the class of stable or finite Morse index solutions of elliptic equations involving the Grushin opeartor or Δ 𝜆 -Laplacian, we refer the readers to previous studies. [18][19][20][21][22][23][24][25][26] In particular, the nonexistence of stable or finite Morse index solutions to the equation −Δ 𝜆 u = |x| a 𝜆 |u| p−2 u has been proved in Rahal, 23 where the critical exponents are respectively given by…”

“…The nonexistence of stable solutions to the equation $$$$ -{\Delta}_Gu={\left|u\right|}^{p-1}u $$$$ was also established in Duong and Nguyen 17 . Concerning the Liouville‐type theorem for the class of stable or finite Morse index solutions of elliptic equations involving the Grushin opeartor or $$$ {\Delta}_{\lambda } $$$‐Laplacian, we refer the readers to previous studies 18–26 . In particular, the nonexistence of stable or finite Morse index solutions to the equation $$$ -{\Delta}_{\lambda }u={\left|x\right|}_{\lambda}^a{\left|u\right|}^{p-2}u $$$ has been proved in Rahal, 23 where the critical exponents are respectively given by $$$$ {p}_c\left(Q,a\right)=\left\{\begin{array}{ll}+\infty & \kern0.1em \mathrm{if}\kern0.3em Q\le 10+4a\\ {}\frac{{\left(Q-2\right...$$…”