Liouville-type theorem for Kirchhoff equations involving Grushin operators

Abstract

The aim of this paper is to prove the Liouville-type theorem of the following weighted Kirchhoff equations:-M R N ω(z)|∇ G u| 2 dz div G (ω(z)∇ G u) = f (z)e u , z = (x, y) ∈ R N = R N 1 × R N 2 (0.1) and M R N ω(z)|∇ G u| 2 dz div G (ω(z)∇ G u) = f (z)u-q , z = (x, y) ∈ R N = R N 1 × R N 2 , (0.2) where M(t) = a + bt k , t ≥ 0, with a > 0, b, k ≥ 0, k = 0 if and only if b = 0. q > 0 and ω(z), f (z) ∈ L 1 loc (R N) are nonnegative functions satisfying ω(z) ≤ C 1 z θ G and f (z) ≥ C 2 z d G as z G ≥ R 0 with d > θ-2, R 0 , C i (i = 1, 2) are some positive constants, here α ≥ 0 and z G = (|x| 2(1+α) + |y| 2) 1 2(1+α) is the norm corresponding to the Grushin distance. N α = N 1 + (1 + α)N 2 is the homogeneous dimension of R N. div G (resp., ∇ G) is Grushin divergence (resp., Grushin gradient). Under suitable assumptions on k, θ , d, and N α , the nonexistence of stable weak solutions to equations (0.1) and (0.2) is investigated.

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 $$$ has been proved in Rahal, 23 where the critical exponents are respectively given by $$$$ {p}_c\left(Q,a\right)=\left\{\begin{array}{ll}+\infty & \kern0.1em \mathrm{if}\kern0.3em Q\le 10+4a\\ {}\frac{{\left(Q-2\right...$$…”