Liouville‐type theorem for nonlinear elliptic equations involving p‐Laplace–type Grushin operators

Abstract

z = (x, ) ∈ R N = R N 1 × R N 2 and ||z|| G = (|x| 2(1+ ) + | | 2 ) 1 2(1+ ) . The results hold true for N < 0 (p, b, ) in (1) and q > q c (p, N , b, ) in (2). Here, 0 and q c are new exponents, which are always larger than the classical critical ones and depend on the parameters p, b and . N = N 1 + (1 + )N 2 is the homogeneous dimension of R N .

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 $$$$ -{\Delta}_Gu&#x0003D;{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-1}u $$$$ was also established in Duong and Nguyen 17 . Concerning the Liouville‐type theorem for the class of stable or finite Morse index solutions of elliptic equations involving the Grushin opeartor or $$$ {\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&#x0003D;{\left&#x0007C;x\right&#x0007C;}_{\lambda}&#x0005E;a{\left&#x0007C;u\right&#x0007C;}&#x0005E;{p-2}u $$$ has been proved in Rahal, 23 where the critical exponents are respectively given by $$$$ {p}_c\left(Q,a\right)&#x0003D;\left\{\begin{array}{ll}&#x0002B;\infty &amp; \kern0.1em \mathrm{if}\kern0.3em Q\le 10&#x0002B;4a\\ {}\frac{{\left(Q-2\right...$$…”

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

“…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].…”

Nonlinear Liouville-type theorem for a differential inequality involving Dunkl-Baouendi-Grushin operator

“…Notice that in [9] the choice = 1 is not admissible. Other references related to nonlinear Liouville-type theorems for problems involving the Baouendi-Grushin operator can be found in [16,30,32,34,47,51,53,54] (see also the references therein). Dunkl operators were introduced by Dunkl [18].…”