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THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN $\mathbb{R}^N$ WITH A HARDY TERM
Gongbao LI, Tao YANG
Acta mathematica scientia,Series B. 2020, 40 (6):
1808-1830.
DOI: 10.1007/s10473-020-0613-8
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 $s \in(0,1)$, $0\leq \alpha,\beta < 2s < n$, $\mu \in (0,n)$, $\gamma < \gamma_{H}$, $I_{\mu}(x)=|x|^{-\mu}$, $F_{\alpha}(x,u)=\frac{ {|u(x)|}^{ {2^{\#}_{\mu} }(\alpha)} }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $f_{\alpha}(x,u)=\frac{ {|u(x)|}^{{ 2^{\#}_{\mu} }(\alpha)-2}u(x) }{ {|x|}^{ {\delta_{\mu} (\alpha)} } }$, $2^{\#}_{\mu} (\alpha)=(1-\frac{\mu}{2n})\cdot 2^{*}_{s} (\alpha)$, $\delta_{\mu} (\alpha)=(1-\frac{\mu}{2n})\alpha$, ${2^{*}_{s}}(\alpha)=\frac{2(n-\alpha)}{n-2s}$ and $\gamma_{H}=4^s\frac{\Gamma^2(\frac{n+2s}{4})} {\Gamma^2(\frac{n-2s}{4})}$. 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 $s \in (0,1)$, $0 < \alpha < 2s < n$, $p\in[1,2^*_{s}(\alpha))$ and $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$. 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 $s \in (0,1)$, $0 < \alpha < 2s < n$, $p\in[1,2^*_{s}(\alpha))$, $r=\frac{\alpha}{ 2^*_{s}(\alpha) }$, $0 < \max \{ \frac{2}{2^*_{s}(\alpha)}, \frac{2^*_{s}-1}{2^*_{s}(\alpha)} \} < \theta < 1$, ${2^{*}_{s}}=\frac{2n}{n-2s}$ and $C=C(n,s,\alpha) > 0$ 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].
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