Liouville type theorems for stable solutions of elliptic system involving the Grushin operator

2020/12/20 by Mtiri, Foued
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2012.11023

Abstract

We examine the degenerate elliptic system -Δ_s u = v^p, -Δ_s v= u^θ, u,vgt;0 \quadin ℝ^N=ℝ^N₁× ℝ^N₂, \quadwhere s ≥ 0 and p,θgt;0. We prove that the system has no smooth stable solution provided p,θ>0 and N_s< 2 + α+ β, where α= (2(p+1))/(pθ- 1) \quadand β= (2(θ+1))/(pθ- 1). This result is an extension of some result in \cite MY. In particular, we establish a new the integral estimate for u and v (see Proposition 1.1), which is crucial to deal with the case 0 < p < 1.

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