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Stable weak solutions to weighted Kirchhoff equations of Lane–Emden type
Advances in Difference Equations ( IF 3.1 ) Pub Date : 2021-01-07 , DOI: 10.1186/s13662-020-03189-5
Yunfeng Wei , Hongwei Yang , Hongwang Yu

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

$$\begin{aligned}& -M \biggl( \int_{\mathbb{R}^{N}}\xi(z) \vert \nabla_{G}u \vert ^{2}\,dz \biggr){ \operatorname{div}}_{G} \bigl(\xi(z) \nabla_{G}u \bigr) \\& \quad=\eta(z) \vert u \vert ^{p-1}u,\quad z=(x,y) \in \mathbb{R}^{N}=\mathbb{R}^{N_{1}}\times\mathbb{R}^{N_{2}}, \end{aligned}$$

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.



中文翻译:

Lane-Emden型加权Kirchhoff方程的稳定弱解

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

$$ \ begin {aligned}&-M \ biggl(\ int _ {\ mathbb {R} ^ {N}} \ xi(z)\ vert \ nabla_ {G} u \ vert ^ {2} \ ,, dz \ biggr ){\ operatorname {div}} _ {G} \ bigl(\ xi(z)\ nabla_ {G} u \ bigr)\\&\ quad = \ eta(z)\ vert u \ vert ^ {p-1 } u,\ quad z =(x,y)\ in \ mathbb {R} ^ {N} = \ mathbb {R} ^ {N_ {1}} \ times \ mathbb {R} ^ {N_ {2}} ,\ end {aligned} $$

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\(N = N_ {1} + N_ {2} \ geq2 \)\(p> 1 + 2k \)\(\ xi(z),\ eta(z)\在L ^ {1} _ {\ mathrm {loc}}(\ mathbb {R} ^ {N})\ setminus \ {0 \} \)是非负函数,例如\(\ xi(z)\ leq C \ | z \ | _ {G } ^ {\ theta} \)\(\ eta(z)\ geq C'\ | z \ | __ {G} ^ {d} \)对于大\(\ | z \ | _ {G} \)\(d> \ theta-2 \)。这里\(\ alpha \ geq0 \)\(\ | z \ | __ {G} =(| x | ^ {2(1+ \ alpha)} + | y | ^ {2})^ {\ frac { 1} {2(1+ \ alpha)}} \)\(\ operatorname {div} _ {G} \)(分别是\(\ nabla_ {G} \))是Grushin发散度(分别是Grushin梯度)。在关于kθd\(N _ {\ alpha} = N_ {1} +(1+ \ alpha)N_ {2} \)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。

更新日期:2021-01-07
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