اشترك بالحزمة الذهبية واحصل على وصول غير محدود شمرا أكاديميا
تسجيل مستخدم جديدThe existence of a nontrivial weak solution to a double critical problem involving fractional Laplacian in ${R}^n$ with a Hardy term
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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} (-Delta)^{s}u-{gamma} {frac{u}{|x|^{2s}}}= {frac{{|u|}^{ {2^{*}_{s}}(beta)-2}u}{|x|^{beta}}}+ big [ I_{mu}* F_{alpha}(cdot,u) big](x)f_{alpha}(x,u), u in {dot{H}}^s(R^{n}) end{equation} where $s in(0,1)$, $0leq alpha,beta<2s<n$, $mu in (0,n)$, $gamma<gamma_{H}$, $I_{mu}(x)=|x|^{-mu}$, $F_{alpha}(x,u)=frac{ {|u(x)|}^{ {2^{#}_{mu} }(alpha)} }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $f_{alpha}(x,u)=frac{ {|u(x)|}^{{ 2^{#}_{mu} }(alpha)-2}u(x) }{ {|x|}^{ {delta_{mu} (alpha)} } }$, $2^{#}_{mu} (alpha)=(1-frac{mu}{2n})cdot 2^{*}_{s} (alpha)$, $delta_{mu} (alpha)=(1-frac{mu}{2n})alpha$, ${2^{*}_{s}}(alpha)=frac{2(n-alpha)}{n-2s}$ and $gamma_{H}=4^sfrac{Gamma^2(frac{n+2s}{4})} {Gamma^2(frac{n-2s}{4})}$. 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}(alpha)}(R^{n},|y|^{-alpha}) hookrightarrow L^{p,frac{n-2s}{2}p+pr}(R^{n},|y|^{-pr}) end{equation} where $s in (0,1)$, $0<alpha<2s<n$, $pin[1,2^*_{s}(alpha))$, $r=frac{alpha}{ 2^*_{s}(alpha) }$; 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}.
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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
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