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 $ \mathbb{R}^N $ involving $ p $-Laplacian

$ \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*} $

where $ x = (y,z)\in\mathbb{R}^K\times\mathbb{R}^{N-K},1\leq K\leq N,1<p<N,0<\alpha,\beta<\min\{K,\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

$ \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*} $

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