ABSTRACT

The existence of at least one positive radial solution of the generalized Kohn–Laplacian problem (1) $\begin{array}{rl}-{\mathrm{\Delta }}_{{\mathbb{H}}^n}u+R\left(\xi \right)u& =\sum _{i=1}^k{a}_i\left(|\xi {|}_{{\mathbb{H}}^n}\right)|u{|}^{p_i-2}u\\ & -\sum _{j=1}^m{b}_j\left(|\xi {|}_{{\mathbb{H}}^n}\right)|u{|}^{q_j-2}u\xi \in \mathrm{\Omega },\\ u>0& \xi \in \mathrm{\Omega },\\ \frac{\mathrm{\partial }u}{\mathrm{\partial }n}=0& \xi \in \mathrm{\partial }\mathrm{\Omega },\end{array}$(1) is proved, where ${\mathrm{\Delta }}_{{\mathbb{H}}^n}$ is the Kohn–Laplacian (Heisenberg–Laplacian) operator, Ω is a Korányi ball, $a_i,1\le i\le k$ and $b_j,1\le j\le m$ are nonnegative radial functions and $R\left(\xi \right)$ satisfies some suitable conditions.

中文翻译：

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Kohn-Laplacian 问题的径向解的存在性

摘要

广义 Kohn-Laplacian 问题的至少一个正径向解的存在(1)$\begin{array}{rl}-{\mathrm{\Delta }}_{{\mathbb{H}}^n}你+R\left(\xi \right)你& =\sum _{\mathrm{一世}=1}^{ķ}{\mathrm{一种}}_{\mathrm{一世}}\left(|\xi {|}_{{\mathbb{H}}^n}\right)|你{|}^{p_{\mathrm{一世}}-2}你\\ & -\sum _{j=1}^{米}{b}_j\left(|\xi {|}_{{\mathbb{H}}^n}\right)|你{|}^{q_j-2}你\xi \in \mathrm{\Omega },\\ 你>0& \xi \in \mathrm{\Omega },\\ \frac{\mathrm{\partial }你}{\mathrm{\partial }n}=0& \xi \in \mathrm{\partial }\mathrm{\Omega },\end{array}$(1)被证明，其中${\mathrm{\Delta }}_{{\mathbb{H}}^n}$是 Kohn-Laplacian (Heisenberg-Laplacian) 算子，Ω 是 Korányi 球，${\mathrm{一种}}_{\mathrm{一世}},1\le \mathrm{一世}\le ķ$和$b_j,1\le j\le 米$是非负径向函数和$R\left(\xi \right)$满足一些合适的条件。