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

Advances in Difference Equations

article

Wei, Yunfeng¹  Yang, Hongwei²  Yu, Hongwang¹ 

[1] School of Statistics and Mathematics, Nanjing Audit University;College of Mathematics and Systems Science, Shandong University of Science and Technology

关键词: Liouville type theorem;    Stable weak solutions;    Weighted Kirchhoff equations;    Grushin operator;    Lane–Emden nonlinearity;   

DOI  :  10.1186/s13662-020-03189-5

学科分类：航空航天科学

来源: SpringerOpen

【 摘 要 】

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.

【 授权许可】

CC BY