The existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${\R}^n$ 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 $s \in(0,1)$, $0\leq α,β<2s<n$, $μ\in (0,n)$, $γ<γ_{H}$, $I_μ(x)=|x|^{-μ}$, $F_α(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_μ }(α)} }{ {|x|}^{ {δ_μ (α)} } }$, $f_α(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_μ }(α)-2}u(x) }{ {|x|}^{ {δ_μ (α)} } }$, $2^{\#}_μ (α)=(1-\fracμ{2n})\cdot 2^{*}_{s} (α)$, $δ_μ (α)=(1-\fracμ{2n})α$, ${2^{*}_{s}}(α)=\frac{2(n-α)}{n-2s}$ and $γ_{H}=4^s\frac{Γ^2(\frac{n+2s}{4})} {Γ^2(\frac{n-2s}{4})}$. 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 $s \in (0,1)$, $0<α<2s<n$, $p\in[1,2^*_{s}(α))$, $r=\fracα{ 2^*_{s}(α) }$; 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}.