The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\mathbb{R}}^n$ with a Hardy term

Abstract

In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving fractional Laplacian with a Hardy term: \begin{equation} \label{eq0.1} (-Δ)^{s}u-γ {\frac{u}{|x|^{2s}}}= {\frac{{|u|}^{ {2^{*}_{s}}(β)-2}u}{|x|^β}}+ \big [ I_μ* F_α(\cdot,u) \big](x)f_α(x,u), \ \ u \in {\dot{H}}^s(\R^{n}) \end{equation} where $s\in (0,1)$, $0\le \alpha ,\beta <2s<n$, $\mu \in (0,n)$, $\gamma <{\gamma }_H$, $I_{\mu }(x)=|x{|}^{-\mu }$, $F_{\alpha }(x,u)=\frac{{|u(x)|}^{2_{\mu }^{\#}(\alpha )}}{{|x|}^{{\delta }_{\mu }(\alpha )}}$, $f_{\alpha }(x,u)=\frac{{|u(x)|}^{2_{\mu }^{\#}(\alpha )-2}u(x)}{{|x|}^{{\delta }_{\mu }(\alpha )}}$, $2_{\mu }^{\#}(\alpha )=(1-\frac{\mu }{2n})\cdot 2_s^{\ast }(\alpha )$, ${\delta }_{\mu }(\alpha )=(1-\frac{\mu }{2n})\alpha$, $2_s^{\ast }(\alpha )=\frac{2(n-\alpha )}{n-2s}$ and ${\gamma }_H=4^s\frac{{\Gamma }^2(\frac{n+2s}{4})}{{\Gamma }^2(\frac{n-2s}{4})}$. We show that problem (\ref{eq0.1}) admits at least a weak solution under some conditions. To prove the main result, we develop some useful tools based on a weighted Morrey space. To be precise, we discover the embeddings \begin{equation} \label{eq0.2} {\dot{H}}^s(\R^{n}) \hookrightarrow {L}^{2^*_{s}(α)}(\R^{n},|y|^{-α}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\R^{n},|y|^{-pr}) \end{equation} where $s\in (0,1)$, $0<\alpha <2s<n$, $p\in [1,2_s^{\ast }(\alpha ))$, $r=\frac{\alpha }{2_s^{\ast }(\alpha )}$; We also establish an improved Sobolev inequality. By using mountain pass lemma along with an improved Sobolev inequality, we obtain a nontrivial weak solution to problem (\ref{eq0.1}) in a direct way. It is worth while to point out that the improved Sobolev inequality could be applied to simplify the proof of the main results in \cite{NGSS} and \cite{RFPP}.