Stable weak solutions to weighted Kirchhoff equations of Lane–Emden type

This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations: -M ( ∫_ℝ^Nξ(z) |∇_Gu | ^2 dz )div_G(ξ(z) ∇_Gu ) =η(z) | u | ^p-1u, z=(x,y) ∈ℝ^N=ℝ^N_1×ℝ^N_2, where M(t)=a+bt^k , t≥0 , with a,b,k≥0 , a+b>0 , k=0 if and only if b=0 . Let N=N_1+N_2≥2 , p>1+2k and ξ(z),η(z)∈ L^1_loc(ℝ^N)∖{ 0} be nonnegative functions such that ξ(z)≤ Cz_G^θ and η(z)≥ C'z_G^d for large z_G with d>θ-2 . Here α≥0 and z_G=(|x|^2(1+α)+|y|^2)^1/2(1+α) . div_G (resp., ∇_G ) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k , θ , d , and N_α=N_1+(1+α)N_2 , the nonexistence of stable weak solutions to the problem is obtained. A distinguished feature of this paper is that the Kirchhoff function M could be zero, which implies that the above problem is degenerate.