On the classification of solutions to a weighted elliptic system\n involving the Grushin operator

2020/07/06 by Foued Mtiri, Mtiri, Foued · 1 citation
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2007.03009

openalex publication_date 2020/07/06 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We investigate here the following weighted degenerate elliptic system\n\-
Delta_s u =\n
Big(1+
|
mathbfx
|^2(s+1)
Big)^
frac
alpha2(s+1) v^p,
quad\n-
Delta_s v =\n
Big(1+
|
mathbfx
|^2(s+1)
Big)^
frac
alpha2(s+1)u^
theta,
quadν,vgt;0
quad
mboxin
;
mathbbR^N:=
mathbbR^N₁
times
mathbbR^N₂.\n where \Δ_s=\Δ_x+|x|^2s\Δ_y, is the Grushin\noperator, s \≥ 0, \α \≥ 0 and 1<p\≤\θ. Here\n
|
mathbfx
|=
Big(|x|^2(s+1)+|y|²
Big)^
frac12(s+1),\n
;
mboxand
;
;
mathbfx:=(x, y)
in
mathbbR^N:=
mathbbR^N₁
times\n
mathbbR^N₂. In particular, we establish some new Liouville-type\ntheorems for stable solutions of the system, which recover and considerably\nimprove upon the known results citecow, Hfh, HU, Fa, DP. As a consequence,\nwe obtain a nonexistence result for the weighted Grushin equation\n\-
Delta_s u\n=
Big(1+
|
mathbfx
|^2(s+1)
Big)^
frac
alpha2(s+1) u^p,
;
;
quad ugt;0\n
quad
mboxin
;
;
mathbbR^N. n

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• Liouville type theorems for stable solutions of the weighted system involving the Grushin operator with negative exponents

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