Optimal decay of extremals for the fractional Sobolev inequality

Abstract

We obtain the sharp asymptotic behavior at infinity of extremal functions for the fractional critical Sobolev embedding.

Appendix A: Power functions

We have the following result on power functions.

Lemma A.1

Let $0<(N-\textit{sp})/p<\beta <N/(p-1)$. For every $R>0$, the function $x\mapsto |x|^{-\beta }$ belongs to $\widetilde{D}^{s,p}(\overline{B_R}^{\,c})$.

Proof

A direct computation shows that $x\mapsto |x|^{-\beta }$ belongs to $L^{p-1}_{\mathrm {loc}}(\mathbb {R}^N)\cap L^{p^*}(B_R^c)$, when $\beta$ is as in the statement. We take $r<R$, then $E=\overline{B_r}^{\,c}\supset \overline{B_R}^{\,c}$ and we need to show

$$\begin{aligned} \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_r^c)}<+\infty ,\quad \text {for }\,\, \frac{N-\textit{sp}}{p}<\beta . \end{aligned}$$

(A.1)

We compute in polar coordinates

$$\begin{aligned}&\int _{B_r^c\times B_r^c}\frac{||x|^{-\beta }-|y|^{-\beta }|^p}{|x-y|^{N+\textit{sp}}}\, dx\, dy\\&\quad = \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\int _{r}^{+\infty }\int _r^{+\infty }\frac{|\varrho ^{-\beta }-t^{-\beta }|^p\,\varrho ^{N-1}\,t^{N-1}}{|\varrho \,\omega _1-t\,\omega _2|^{N+\textit{sp}}}\, d\varrho \, dt\, d\omega _1\, d\omega _2\\&\quad =2\int _{r}^{+\infty }\frac{\varrho ^{-\beta \, p}\,\varrho ^{2\,N-2}}{\varrho ^{N+\textit{sp}}}\int _r^{\varrho }\left| 1-\left( \frac{t}{\varrho }\right) ^{-\beta }\right| ^p\\&\qquad \times \, \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-(t\,\omega _2)/\varrho |^{N+\textit{sp}}}\left( \frac{t}{\varrho }\right) ^{N-1}\, dt\, d\varrho \\&\quad =2\int _r^{+\infty }\frac{\varrho ^{-\beta \, p}\,\varrho ^{2\,N-1}}{\varrho ^{N+\textit{sp}}}\int _{r/\varrho }^1|1-\xi ^{-\beta }|^p\,\xi ^{N-1}\int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}\, d\xi \, d\varrho . \end{aligned}$$

Let us now prove that for $0<\xi <1$ it holds

$$\begin{aligned} \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}\le \frac{C}{(1-\xi )^{1+\textit{sp}}}. \end{aligned}$$

Without loss of generality, we may assume that $\xi \ge 1/2$, since for $0<\xi <1/2$ the integral is uniformly bounded. By rotational invariance, we have

$$\begin{aligned} \int _{\mathbf {S}^{N-1}\times \mathbf {S}^{N-1}}\frac{d\omega _1\, d\omega _2}{|\omega _1-\xi \,\omega _2|^{N+\textit{sp}}}= |\mathbf {S}^{N-1}|\int _{\mathbf {S}^{N-1}}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}, \end{aligned}$$

where $\mathbf{e}_1=(1,0,\dots ,0)$. By changing variable $\omega _2=(t,z)$ with

$$\begin{aligned} t=\pm \sqrt{1-|z|^2},\quad z\in B_1'\subset \mathbb {R}^{N-1}, \end{aligned}$$

we therefore get (the constant C may vary from a line to another)

$$\begin{aligned} \int _{\mathbf {S}^{N-1}}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}&= \int _{\mathbf {S}^{N-1}{\setminus }B_1(\mathbf{e_1})}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}+\int _{\mathbf {S}^{N-1}\cap B_1(\mathbf{e_1})}\frac{d\omega _2}{|\mathbf{e}_1-\xi \,\omega _2|^{N+\textit{sp}}}\\&\le C\left( 1+ \int _{B_1'}\frac{dz}{((1-\xi \, t)^2+\xi ^2\,|z|^2)^{\frac{N+\textit{sp}}{2}}}\right) \\&\le C\left( 1+\int _{B_1'}\frac{dz}{((1-\xi )^2+\xi ^2\,|z|^2)^{\frac{N+\textit{sp}}{2}}} \right) \\&\le C\left( 1+\frac{1}{(1-\xi )^{1+\textit{sp}}}\int _{B_{\frac{\xi }{1-\xi }}'}\frac{1}{ (1+|y|^2)^{\frac{N+\textit{sp}}{2}}}dy\right) \\&\le C\left( 1+\frac{1}{(1-\xi )^{1+\textit{sp}}}\int _{\mathbb {R}^{N-1}}\frac{1}{ (1+|y|^2)^{\frac{N+\textit{sp}}{2}}}dy\right) \end{aligned}$$

which proves the claim. Taking into account that for $0< \xi <1$ it also holds

$$\begin{aligned} \frac{|1-\xi ^{-\beta }|^p}{|1-\xi |^{1+\textit{sp}}}\le C\,(\xi ^{-\beta \, p}+|1-\xi |^{p\,(1-s)-1}) \end{aligned}$$

we therefore get

$$\begin{aligned} \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_r^c)}^p\le C\int _r^{+\infty }\varrho ^{N-1-p\,(s+\beta )}\,d\varrho \int _{r/\varrho }^1 \xi ^{N-1}\,\big (\xi ^{-\beta \, p}+|1-\xi |^{p\,(1-s)-1}\big )\, d\xi . \end{aligned}$$

All the integrals are now explicitly computable and one can readily get (A.1).$\square$

Lemma A.2

Let $0<(N-\textit{sp})/p<\beta <N/(p-1)$. For every $R>0$, it holds

$$\begin{aligned} (-\Delta _p)^s |x|^{-\beta }=C(\beta )\, |x|^{-\beta \,(p-1)-\textit{sp}}\quad \hbox { weakly in } \overline{B_R}^{\,c}, \end{aligned}$$

where the constant $C(\beta )$ is given by

$$\begin{aligned} C(\beta )=2\,\int _0^1 \varrho ^{\textit{sp}-1}\,\left[ 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right] \,\left| 1-\varrho ^{\beta }\right| ^{p-1}\, \Phi (\varrho )\,d\varrho , \end{aligned}$$

(A.2)

and

$$\begin{aligned} \Phi (\varrho )=\mathcal {H}^{N-2}(\mathbf {S}^{N-2})\,\int _{-1}^1 \frac{(1-t^2)^\frac{N-3}{2}}{\Big (1-2\,t\,\varrho +\varrho ^2\Big )^\frac{N+\textit{sp}}{2}}\, dt. \end{aligned}$$

(A.3)

Proof

Observe that

$$\begin{aligned} |x|^{-\beta \,(p-1)-\textit{sp}} \in L^{(p*)'}(B_R^c), \,\,\quad \text {for any }\beta >(N-\textit{sp})/p. \end{aligned}$$

Then, by Theorem 2.1 and Proposition 2.5 it suffices to show that

$$\begin{aligned} \int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dx\,dy=C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx, \end{aligned}$$

for an arbitrary $\varphi \in C^{\infty }_c(\overline{B_R}^{\,c})$. For every such a $\varphi$ we consider the double integral

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dx\,dy. \end{aligned} \end{aligned}$$

We observe that this is absolutely convergent, indeed

$$\begin{aligned} \begin{aligned} \int _{\mathbb {R}^{2N}}&\frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (x)-\varphi (y)|\,dx\,dy\\&=\int _{B_R^c\times B_R^c}\frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (x)-\varphi (y)|\,dx\,dy\\&\quad +2\,\int _{B_R} \int _{\mathrm {supp}(\varphi )} \frac{|J_p(|x|^{-\beta }-|y|^{-\beta })|}{|x-y|^{N+\textit{sp}}}\,|\varphi (y)|\,dx\,dy\\&\le \Big [|x|^{-\beta }\Big ]_{W^{s,p}(B_R^c)}\,[\varphi ]_{W^{s,p}(B_R^c)}+C\,\Vert \varphi \Vert _{L^\infty }\,|\mathrm {supp}(\varphi )|\,\int _{B_R} |x|^{-\beta \,(p-1)}\,dx, \end{aligned} \end{aligned}$$

and both terms are finite, thanks to Lemma A.1. For $\delta >0$ we consider the conical set

$$\begin{aligned} \mathcal {O}_\delta =\{(x,y)\in \mathbb {R}^{2\,N}\, :\, (1-\delta )\,|x|\le |y|\le (1+\delta )\,|x|\}, \end{aligned}$$

then by the Dominated Convergence Theorem

$$\begin{aligned}&\lim _{\delta \searrow 0}\int _{\mathcal {O}_\delta ^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dy\,dx\nonumber \\&\quad =\int _{\mathbb {R}^{2N}}\frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x) -\varphi (y))\,dx\,dy. \end{aligned}$$

We now observe that

$$\begin{aligned} \begin{aligned}&\int _{\mathcal {O}_\delta ^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,(\varphi (x)-\varphi (y))\,dy\,dx\\&\quad =2\,\int _{\mathbb {R}^N}\left( \int _{\mathcal {K}_\delta (x)^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,dy\right) \,\varphi (x)\,dx, \end{aligned} \end{aligned}$$

where for every $x\in \mathbb {R}^N$

$$\begin{aligned} \mathcal {K}_\delta (x)=\{y\in \mathbb {R}^N\, :\, (1-\delta )\,|x|\le |y|\le (1+\delta )\,|x|\}, \end{aligned}$$

and of course $\mathcal {K}_\delta (x)=\mathcal {K}_\delta (x')$ whenever $|x|=|x'|$. We set

$$\begin{aligned} f_\delta (x)=2\,\int _{\mathcal {K}_\delta (x)^c} \frac{J_p(|x|^{-\beta }-|y|^{-\beta })}{|x-y|^{N+\textit{sp}}}\,dy,\quad x\in \mathbb {R}^N{\setminus }\{0\}, \end{aligned}$$

it is easily seen that $f_\delta$ is a radial function, homogeneous of degree $-\beta \,(p-1)-\textit{sp}$ (see [4, Lemma 6.2]). Thus for $x\not =0$ we have

$$\begin{aligned} f_\delta (x)=|x|^{-\beta \,(p-1)-\textit{sp}}\,f_{\delta }(\omega ),\quad \hbox { for }\quad \omega =\frac{x}{|x|}\in \mathbf {S}^{N-1}. \end{aligned}$$

(A.4)

We set

$$\begin{aligned} C(\beta ;\delta ):=f_{\delta }(\omega )=2\,\int _{\mathcal {K}_\delta (\omega )^c} \frac{J_p(1-|y|^{-\beta })}{|\omega -y|^{N+\textit{sp}}}\,dy,\quad \omega \in \mathbf {S}^{N-1}, \end{aligned}$$

which is independent of the direction $\omega$, by radiality of $f_\delta$. By taking the average over $\mathbf {S}^{N-1}$ and proceeding as in [4, Lemma B.2], we get

$$\begin{aligned} C(\beta ;\delta )=2\,\int _{|\varrho -1|\ge \delta } \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-2}\,(1-\varrho ^{-\beta })\,\Phi (\varrho )\,d\varrho , \end{aligned}$$

where $\Phi$ is defined in (A.3). We now decompose the integral defining $C(\beta ;\delta )$ and perform a change of variables, i.e.

$$\begin{aligned} \begin{aligned} C(\beta ;\delta )&=-2\,\int _0^{1-\delta } \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-1}\,\Phi (\varrho )\,d\varrho +2\,\int _{1+\delta }^\infty \varrho ^{N-1}\,|1-\varrho ^{-\beta }|^{p-1}\,\Phi (\varrho )\,d\varrho \\&=-2\,\int _0^{1-\delta } \varrho ^{N-1-\beta \,(p-1)}\,|\varrho ^{\beta }-1|^{p-1}\,\Phi (\varrho )\,d\varrho \\&\quad +2\,\int _0^{1/(1+\delta )} \varrho ^{-N-1}\,|1-\varrho ^{\beta }|^{p-1}\,\Phi (1/\varrho )\,d\varrho . \end{aligned} \end{aligned}$$

Finally, observe that

$$\begin{aligned} \Phi (1/\varrho )=\varrho ^{N+\textit{sp}}\,\Phi (\varrho ), \end{aligned}$$

thus the quantity $C(\beta ;\delta )$ can be written as

$$\begin{aligned} \begin{aligned} C(\beta ;\delta )&=2\,\int _0^{1-\delta } \left( 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right) \,\varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho \\&\quad +2\,\int _{1-\delta }^{1/(1+\delta )} \varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho . \end{aligned} \end{aligned}$$

(A.5)

Recall that $\varphi$ is compactly supported in $\overline{B_R}^{\,c}$, thus by using (A.4) we can estimate

$$\begin{aligned} \begin{aligned}&\left| \int _{\Omega } f_\delta \,\varphi \,dx-C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx\right| \\&\quad \le \Vert \varphi \Vert _\infty \,R^{-\beta \,(p-1)-\textit{sp}}\,|\mathrm {supp}(\varphi )|\,\Big |C(\beta ;\delta )-C(\beta )\Big |. \end{aligned} \end{aligned}$$

In order to prove that $C(\beta ;\delta )$ converges to $C(\beta )$ as $\delta$ goes to 0, we decompose the function $\Phi$ defined in (A.3) as follows

$$\begin{aligned} \begin{aligned} \Phi (\varrho )&=\int _{-1}^{1/2} \frac{(1-t^2)^\frac{N-3}{2}}{(1-2\,t\,\varrho +\varrho ^2)^\frac{N+\textit{sp}}{2}}\, dt+\int _{1/2}^{1} \frac{(1-t^2)^\frac{N-3}{2}}{(1-2\,t\,\varrho +\varrho ^2)^\frac{N+\textit{sp}}{2}}\, dt=:\Phi _1(\varrho )+\Phi _2(\varrho ), \end{aligned} \end{aligned}$$

where we omitted the dimensional constant $\mathcal {H}^{N-2}(\mathbf {S}^{N-2})$ for simplicity. Let us start estimating $\Phi _1$. If we use that

$$\begin{aligned} 1-2\,t\,\varrho +\varrho ^2=(\varrho -t)^2+(1-t^2)\ge \frac{3}{4},\quad \hbox { if }-1\le t\le \frac{1}{2}, \end{aligned}$$

we get

$$\begin{aligned} 0\le \Phi _1(\varrho )\le C,\quad 0<\varrho <1. \end{aligned}$$

(A.6)

We now consider $\Phi _2(\varrho )$, discussing separately the cases $0<\varrho <1/2$ and $1/2\le \varrho <1$. We observe that for $0<\varrho <1/2$ we have

$$\begin{aligned} 1-2\,t\,\varrho +\varrho ^2=(1-\varrho )^2+2\,\varrho \,(1-t)\ge \frac{1}{4},\quad \hbox { if } \frac{1}{2}\le t\le 1. \end{aligned}$$

Then we get again

$$\begin{aligned} 0\le \Phi _2(\varrho )\le C,\quad \hbox { if } 0<\varrho <\frac{1}{2}. \end{aligned}$$

(A.7)

We are left with the term $\Phi _2(\varrho )$ for $1/2\le \varrho <1$. With simple manipulations⁴ we can write it as

$$\begin{aligned} \Phi _2(\varrho )=\frac{(2\,\varrho )^{-\frac{N-1}{2}}}{(1-\varrho )^{1+\textit{sp}}}\int _{0}^{\frac{\varrho }{(1-\varrho )^2}} \frac{\left( 2-\frac{(1-\varrho )^2}{2\,\varrho }\,\tau \right) ^\frac{N-3}{2}\,\tau ^\frac{N-3}{2}}{(1+\tau )^\frac{N+\textit{sp}}{2}}\, d\tau . \end{aligned}$$

In particular, we get

$$\begin{aligned} 0\le \Phi _2(\varrho )\le C\,(1-\varrho )^{-1-\textit{sp}},\quad \hbox { if }\frac{1}{2}\le \varrho <1. \end{aligned}$$

(A.8)

By using (A.6), (A.7) and (A.8), we thus obtain for the first integral in (A.5)

$$\begin{aligned} \lim _{\delta \searrow 0}2\,\int _0^{1-\delta } \left( 1-\varrho ^{N-\textit{sp}-\beta \,(p-1)}\right) \,\varrho ^{\textit{sp}-1}\,(1-\varrho ^{\beta })^{p-1}\,\Phi (\varrho )\,d\varrho =C(\beta ), \end{aligned}$$

and observe that the latter is finite, thanks to (A.8). It is only left to show that the other integral in (A.5) converges to 0. Still by (A.6) and (A.8), we obtain

where we assumed for simplicity that $p-1-\textit{sp}\not =0$. If $p-1-\textit{sp}>0$, the last term converges to 0. If $p-1-\textit{sp}<0$, we have

$$\begin{aligned} \left( \frac{\delta }{1+\delta }\right) ^{p-1-\textit{sp}}-\delta ^{p-1-\textit{sp}}= & {} \delta ^{p-1-\textit{sp}}\,\Big [(1+\delta )^{\textit{sp}+1-p}-1\Big ]\\\simeq & {} (\textit{sp}+1-p)\,\delta ^{p-\textit{sp}},\quad \hbox { as } \delta \searrow 0, \end{aligned}$$

and thus the integral converges to 0 again. Finally, the borderline case $p-1-\textit{sp}=0$ is treated similarly, we leave the details to the reader.

In conclusion, we get

$$\begin{aligned} \lim _{\delta \searrow 0}\int _{\Omega } f_\delta \,\varphi \,dx=C(\beta )\,\int _\Omega |x|^{-\beta \,(p-1)-\textit{sp}}\,\varphi \,dx, \end{aligned}$$

as desired.$\square$

Remark A.3

The previous result was proved in [15, Lemma 3.1] for the limit case $\beta =(N-\textit{sp})/p$. Our argument is different, since we rely on elementary estimates for the function $\Phi$, rather than on special properties of hypergeometric and beta functions like in [15].

Observe that the choice $\beta =(N-\textit{sp})/(p-1)$ is feasible in the previous results, since

$$\begin{aligned} \frac{N-\textit{sp}}{p}<\frac{N-\textit{sp}}{p-1}<\frac{N}{p-1}. \end{aligned}$$

Moreover, with such a choice we have $C(\beta )=0$ in (A.2). Then from Lemmas A.1 and A.2, we get the following.

Theorem A.4

For any $R>0$, $\Gamma (x)=|x|^{-\frac{N-\textit{sp}}{p-1}}$ belongs to $\widetilde{D}^{s,p}(\overline{B_R}^{\,c})$ and weakly solves $(-\Delta _p)^su=0$ in $\overline{B_R}^{\,c}$.