Fractional Lion 's Lemma

Lemma 1. Let $\left\{u_{\epsilon}\right\}_{\epsilon>0} \subset H_{0}^{s}(\Omega)$ be a sequence of functions such that $\| ( - \Delta ) ^ { s } u \| _ { L^2 ( \Omega ) } \leq 1$ "and $u_{\epsilon} \rightharpoonup u_{0}$ weakly in $H_{0}^{s}(\Omega)$ . Then for any $p < 1 / ( 1 - \| ( - \Delta ) ^ { s /2} u _ { \epsilon} \| _ { 2 } ^ { 2 } )$ .
$\underset { \epsilon \rightarrow 0 } { \operatorname { lim sup } } \int _ { \Omega } e ^ { {\alpha _ { n , 2 }} p u _ { \epsilon } ^ { 2 } } d x < + \infty$

Proof.
If $u_{0}=0$ , then nothing need to be proved because of
$\left. \begin{array} { c } { \operatorname { Sup } _ { u \in { H } ^ { { s } , p } ( \Omega ) , \| ( - \Delta ) ^ { \frac { s } { 2 } } u \| _ { L ^ { p } ( \Omega ) }\leq 1 } \int _ { \Omega } e ^ { \alpha _ { n , p } | u | ^ { \frac { p } { p - 1 } } } d x = c _ { n , p } | \Omega | \leq + \infty} \end{array} \right.$
It's easy to check when $p=2$ , also Testimonies theorem.The Lemma 1 is obvious.

If $u_{0}\neq0$ , we can see that
${ \| (-\Delta)^{\frac { s } { 2 }} ( u _ { \epsilon } - u _ { 0 } ) \| _ { 2 } ^ { 2 } } \rightarrow 1 - { \| (-\Delta)^{\frac { s } { 2 }} u _ { 0 } \| _ { 2 } ^ { 2 } }< 1$
Then when $p < 1 / ( 1 - \| ( - \Delta ) ^ { s /2} u _ { \epsilon} \| _ { 2 } ^ { 2 } )$ ,we can judge the range of inenqulity in Lemma 1.
At first, we have to check $\delta u _ { \epsilon } ^ { 2 } + ( 2 + \delta + \frac { 1 } { \delta } ) u _ { 0 } ^ { 2 } - ( 1 + \delta ) \cdot 2 \cdot u _ { 0 } u _ { \varepsilon } > 0$ always hold , which is simply easy.Then apply Holder’s inequality :

$\begin{aligned}\int_{\Omega} e^{ \alpha _ { n , 2 } p u_{\epsilon}^{2}} d x &\leqslant \int_{\Omega} e^{ \alpha _ { n , 2 }p(u_{\epsilon}^{2}+\delta u _ { \epsilon } ^ { 2 } + ( 2 + \delta + \frac { 1 } { \delta } ) u _ { 0 } ^ { 2 } - ( 1 + \delta ) \cdot 2 \cdot u _ { 0 } u _ { \varepsilon } )} d x \\ &= \int_{\Omega} e^{ \alpha _ { n , 2 }p(1+\delta)\left(u_{\epsilon}-u_{0}\right)^{2}+ \alpha _ { n , 2 } p(1+1 / \delta) u_{0}^{2}} d x\\ &\leqslant ( \int _ { \Omega } (e ^ { \alpha _ { n , 2} ( u _ { \epsilon } - u _ { 0 } ) ^ { 2 } (1+\delta) p})^{r} d x ) ^ { 1 / r } ( \int _ { \Omega } e ^ { \alpha _ { n , 2} p ^ { \prime } u _ { 0 } ^ { 2 } } d x ) ^ { 1 / s }\\ &\leqslant ( \int _ { \Omega } e ^ { \alpha _ { n , 2} \frac { ( u _ { \epsilon } - u _ { 0 } ) ^ { 2 } } { \| (-\Delta)^{\frac { s } { 2 }} ( u _ { \epsilon } - u _ { 0 } ) \| _ { 2 } ^ { 2 } } } d x ) ^ { 1 / r } ( \int _ { \Omega } e ^ { \alpha _ { n , 2} p ^ { \prime } u _ { 0 } ^ { 2 } } d x ) ^ { 1 / s }\\ &= G1 \times S2 \end{aligned}$
for some $\delta>0$ ,and here $p ^ { \prime }=(p+1/\delta) \frac { 1 } {1-p( 1+\delta )}$ ,where $1/r+1/s=1$ , $r= \frac { 1 } {p( 1+\delta )}$ and $s=\frac { 1 } {1-p( 1+\delta )}$
Tt's easy to check $G1$ and $S2$ both are finite.So prove the Lemma.