# 20211020 杨健夫 Improved Sobolev inequalities and critical problems

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 R^N involving p-Laplacian

-\Delta_pu = \frac{|u|^{p^\ast_{\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{p^\ast_{\beta}-2}u}{|y|^{\beta}},

where $x=(y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K}, 1\leq K\leq N, 1<p<N,0<\alpha,\beta<\frac{NKp}{N^2-(N-K)p}$ and $p^\ast_{\alpha}=\frac{p(N-\alpha)}{N-p}$ is the critical Hardy-Sobolev exponent, and critical problems involving fractional Laplacian

(-\Delta)^{s}u = \frac{|u|^{2^\ast_{s,\alpha}-2}u}{|y|^{\alpha}}+\frac{|u|^{2^\ast_{s,\beta}-2}u}{|y|^{\beta}},

where $0<s<\frac{N}{2},0<\alpha,\beta<\frac{2NKs}{N^2-2s(N-K)}$ and $2^\ast_{s,\alpha}=\frac{2(N-\alpha)}{N-2s}$.

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