On a doubly critical system involving fractional Laplacian with partial weight

Abstract

In this paper, we establish a new improved Sobolev inequality based on a weighted Morrey space. To be precise, there exists $C=C(n,m,s,\alpha )>0$ such that for any $u,v\in {\dot{H}}^s({\mathbb{R}}^n)$ and for any $\theta \in (\bar{\theta },1)$, it holds that \begin{equation} \label{eq0.3} \Big( \int_{ \mathbb{R}^{n} } \frac{ |(uv)(y)|^{\frac{2^*_{s}(α)}{2} } } { |y'|^α } dy \Big)^{ \frac{1}{ 2^*_{s} (α) }} \leq C ||u||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||v||_{\dot{H}^s(\mathbb{R}^{n})}^{\fracθ{2}} ||(uv)||^{\frac{1-θ}{2}}_{ L^{1,n-2s+r}(\mathbb{R}^{n},|y'|^{-r}) }, \end{equation} where $s\in (0,1)$, $0<\alpha <2s<n$, $2s<m<n$, $\bar{\theta }=\max \{\frac{2}{2_s^{\ast }(\alpha )},1-\frac{\alpha }{s}\cdot \frac{1}{2_s^{\ast }(\alpha )},\frac{2_s^{\ast }(\alpha )-\frac{\alpha }{s}}{2_s^{\ast }(\alpha )-\frac{2\alpha }{m}}\}$, $r=\frac{2\alpha }{2_s^{\ast }(\alpha )}$ and $y=(y^{\prime },y^{\prime \prime })\in {\mathbb{R}}^m\times {\mathbb{R}}^{n-m}$. By using mountain pass lemma and (\ref{eq0.3}), we obtain a nontrivial weak solution to a doubly critical system involving fractional Laplacian in ${\mathbb{R}}^n$ with partial weight in a direct way. Furthermore, we extend inequality (\ref{eq0.3}) to more general forms on purpose of studying some general systems with partial weight, involving p-Laplacian especially.

arXiv admin note: text overlap with arXiv:1908.02536

References in corpus (2)

• The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\mathbb{R}}^n$ with a Hardy term
• Approximation by regular functions in Sobolev spaces arising from doubly elliptic problems

Cited by in corpus (1)

• Normalized solutions to fractional mass supercritical NLS systems with Sobolev critical nonlinearities