IMPROVED SOBOLEV INEQUALITIES AND CRITICAL PROBLEMS
IMPROVED SOBOLEV INEQUALITIES AND CRITICAL PROBLEMS
复制标题
改进的索博列夫不等式和关键问题
DOI：
10.3934/cpaa.2020162
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发表时间：
2020
影响因子：
1
通讯作者：
Yang Jianfu
中科院分区：
文献类型：
--
作者：
Chen Xiaoli;Yang Jianfu
In this paper, we establish two refinement of Sobolev-Hardy inequalities in terms of Morrey spaces. Then, with help of these inequalities, we show the existence of nontrivial solutions for doubly critical problems in \begin{document}$ \mathbb{R}^N $\end{document} involving \begin{document}$ p $\end{document} -Laplacian \begin{document}$ \begin{equation*} \label{eq:1} -\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1 and \begin{document}$ p^\ast_{\alpha} = \frac{p(N-\alpha)}{N-p} $\end{document} is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian \begin{document}$ \begin{equation*} (-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}}, \end{equation*} $\end{document} where \begin{document}$ 0 and \begin{document}$ 2^\ast_{s,\alpha} = \frac{2(N-\alpha)}{N-2s} $\end{document} .
