数学物理学报(英文版) ›› 2020, Vol. 40 ›› Issue (6): 1808-1830.doi: 10.1007/s10473-020-0613-8

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THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN RN WITH A HARDY TERM

李工宝, 杨涛   

  1. Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • 收稿日期:2019-06-17 修回日期:2020-07-24 出版日期:2020-12-25 发布日期:2020-12-30
  • 通讯作者: Tao YANG,E-mail:yangt@mails.ccnu.edu.cn E-mail:yangt@mails.ccnu.edu.cn
  • 作者简介:Gongbao LI,E-mail:ligb@mail.ccnu.edu.cn
  • 基金资助:
    This work was supported by Natural Science Foundation of China (11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University # IRT17R46.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN RN WITH A HARDY TERM

Gongbao LI, Tao YANG   

  1. Hubei Key Laboratory of Mathematical Sciences and School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China
  • Received:2019-06-17 Revised:2020-07-24 Online:2020-12-25 Published:2020-12-30
  • Contact: Tao YANG,E-mail:yangt@mails.ccnu.edu.cn E-mail:yangt@mails.ccnu.edu.cn
  • Supported by:
    This work was supported by Natural Science Foundation of China (11771166), Hubei Key Laboratory of Mathematical Sciences and Program for Changjiang Scholars and Innovative Research Team in University # IRT17R46.

摘要: In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:

(1)(Δ)suγu|x|2s=|u|2s(β)2u|x|β+[IμFα(,u)](x)fα(x,u),  uH˙s(Rn),(0.1)
where s(0,1), 0α,β<2s<n, μ(0,n), γ<γH, Iμ(x)=|x|μ, Fα(x,u)=|u(x)|2μ#(α)|x|δμ(α), fα(x,u)=|u(x)|2μ#(α)2u(x)|x|δμ(α), 2μ#(α)=(1μ2n)2s(α), δμ(α)=(1μ2n)α, 2s(α)=2(nα)n2s and γH=4sΓ2(n+2s4)Γ2(n2s4). We show that problem (0.1) 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
(2)H˙s(Rn)L2s(α)(Rn,|y|α)Lp,n2s2p+pr(Rn,|y|pr),(0.2)
where s(0,1), 0<α<2s<n, p[1,2s(α)) and r=α2s(α). We also establish an improved Sobolev inequality,
(3)(Rn|u(y)|2s(α)|y|αdy)12s(α)C||u||H˙s(Rn)θ||u||Lp,n2s2p+pr(Rn,|y|pr)1θ,    uH˙s(Rn),(0.3)
where s(0,1), 0<α<2s<n, p[1,2s(α)), r=α2s(α), 0<max{22s(α),2s12s(α)}<θ<1, 2s=2nn2s and C=C(n,s,α)>0 is a constant. Inequality (0.3) is a more general form of Theorem 1 in Palatucci, Pisante [1].
By using the mountain pass lemma along with (0.2) and (0.3), we obtain a nontrivial weak solution to problem (0.1) in a direct way. It is worth pointing out that (0.2) and 0.3) could be applied to simplify the proof of the existence results in [2] and [3].

关键词: existence of a weak solution, fractional Laplacian, double critical exponents, Hardy term, weighted Morrey space, improved Sobolev inequality

Abstract: In this paper, we consider the existence of nontrivial weak solutions to a double critical problem involving a 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(\mathbb{R}^n),        (0.1)\end{equation}
where s(0,1), 0α,β<2s<n, μ(0,n), γ<γH, Iμ(x)=|x|μ, Fα(x,u)=|u(x)|2μ#(α)|x|δμ(α), fα(x,u)=|u(x)|2μ#(α)2u(x)|x|δμ(α), 2μ#(α)=(1μ2n)2s(α), δμ(α)=(1μ2n)α, 2s(α)=2(nα)n2s and γH=4sΓ2(n+2s4)Γ2(n2s4). We show that problem (0.1) 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(\mathbb{R}^n) \hookrightarrow  {L}^{2^*_{s}(\alpha)}(\mathbb{R}^n,|y|^{-\alpha}) \hookrightarrow L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}), (0.2)\end{equation}
where s(0,1), 0<α<2s<n, p[1,2s(α)) and r=α2s(α). We also establish an improved Sobolev inequality,
\begin{equation} \label{eq0.3}  \Big( \int_{ \mathbb{R}^n }  \frac{ |u(y)|^{ 2^*_{s}(\alpha)} }  {  |y|^{\alpha} }{\rm d}y  \Big)^{ \frac{1}{  2^*_{s} (\alpha)  }}  \leq C ||u||_{{\dot{H}}^s(\mathbb{R}^n)}^{\theta} ||u||^{1-\theta}_{  L^{p,\frac{n-2s}{2}p+pr}(\mathbb{R}^n,|y|^{-pr}) },~~~~\forall u \in {\dot{H}}^s(\mathbb{R}^n), (0.3)\end{equation}
where s(0,1), 0<α<2s<n, p[1,2s(α)), r=α2s(α), 0<max{22s(α),2s12s(α)}<θ<1, 2s=2nn2s and C=C(n,s,α)>0 is a constant. Inequality (0.3) is a more general form of Theorem 1 in Palatucci, Pisante [1].
By using the mountain pass lemma along with (0.2) and (0.3), we obtain a nontrivial weak solution to problem (0.1) in a direct way. It is worth pointing out that (0.2) and 0.3) could be applied to simplify the proof of the existence results in [2] and [3].

Key words: existence of a weak solution, fractional Laplacian, double critical exponents, Hardy term, weighted Morrey space, improved Sobolev inequality

中图分类号: 

  • 35A01