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

In this article, we prove the Liouville‐type theorem for stable solutions of weighted p‐Laplace–type Grushin equations 1−divG(a(z)|∇Gu|p−2∇Gu)=h(z)eu,z=(x,y)∈RN=RN1×RN2 and 2divG(a(z)|∇Gu|p−2∇Gu)=h(z)u−q,z=(x,y)∈RN=RN1×RN2,where p ≥ 2, q>0 and a(z),h(z)∈Lloc1(RN) are nonnegative functions satisfying a(z)≤C1‖z‖Gb and h(z)≥C2‖z‖Gθ as ‖z‖G ≥ R0 with p−Nγ<b<θ+p, R0,Ci(i=1,2) are some positive constants. divG(f,g)=∑i=1N1∂fi∂xi+(1+γ)|x|γ∑j=1N2∂gj∂yj,(f,g)∈C1(RN,RN1×RN2),∇G=(∇x,(1+γ)|x|γ∇y),γ ≥ 0, z=(x,y)∈RN=RN1×RN2 and ‖z‖G=(|x|2(1+γ)+|y|2)12(1+γ). The results hold true for Nγ<μ0(p,b,θ) in and q>qc(p,Nγ,b,θ) in. Here, μ0 and qc are new exponents, which are always larger than the classical critical ones and depend on the parameters p,b and θ. Nγ=N1+(1+γ)N2 is the homogeneous dimension of RN. [ABSTRACT FROM AUTHOR]