This paper is concerned with the Liouville type theorem for stable weak solutions to the following weighted Kirchhoff equations:

where \(M(t)=a+bt^{k}\), \(t\geq0\), with \(a,b,k\geq0\), \(a+b>0\), \(k=0\) if and only if \(b=0\). Let \(N=N_{1}+N_{2}\geq2\), \(p>1+2k\) and \(\xi(z),\eta(z)\in L^{1}_{\mathrm{loc}}(\mathbb{R}^{N})\setminus\{ 0\}\) be nonnegative functions such that \(\xi(z)\leq C\|z\|_{G}^{\theta}\) and \(\eta(z)\geq C'\|z\|_{G}^{d}\) for large \(\|z\|_{G}\) with \(d>\theta-2\). Here \(\alpha\geq0\) and \(\|z\|_{G}=(|x|^{2(1+\alpha)}+|y|^{2})^{\frac{1}{2(1+\alpha)}}\). \(\operatorname{div}_{G}\) (resp., \(\nabla_{G}\)) is Grushin divergence (resp., Grushin gradient). Under some appropriate assumptions on k, θ, d, and \(N_{\alpha}=N_{1}+(1+\alpha)N_{2}\), 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.

本文关注有关下列加权Kirchhoff方程的稳定弱解的Liouville型定理：

其中\（M（t）= a + bt ^ {k} \），\（t \ geq0 \），带有\（a，b，k \ geq0 \），\（a + b> 0 \），\ （k = 0 \）当且仅当\（b = 0 \）。令\（N = N_ {1} + N_ {2} \ geq2 \），\（p> 1 + 2k \）和\（\ xi（z），\ eta（z）\在L ^ {1} _ {\ mathrm {loc}}（\ mathbb {R} ^ {N}）\ setminus \ {0 \} \）是非负函数，例如\（\ xi（z）\ leq C \ | z \ | _ {G } ^ {\ theta} \）和\（\ eta（z）\ geq C'\ | z \ | __ {G} ^ {d} \）对于大\（\ | z \ | _ {G} \）与\（d> \ theta-2 \）。这里\（\ alpha \ geq0 \）和\（\ | z \ | __ {G} =（| x | ^ {2（1+ \ alpha）} + | y | ^ {2}）^ {\ frac { 1} {2（1+ \ alpha）}} \）。\（\ operatorname {div} _ {G} \）（分别是\（\ nabla_ {G} \））是Grushin发散度（分别是Grushin梯度）。在关于k，θ，d和\（N _ {\ alpha} = N_ {1} +（1+ \ alpha）N_ {2} \）的一些适当假设下，获得了对该问题的稳定弱解的不存在。本文的一个显着特征是基尔霍夫函数M可以为零，这意味着上述问题是退化的。