Existence and asymptotics for solutions of a non-local Q-curvature equation in dimension three

Abstract

We study conformal metrics on ${\mathbb {R}}^{3}$, i.e., metrics of the form $g_u=e^{2u}|dx|^2$, which have constant $Q$-curvature and finite volume. This is equivalent to studying the non-local equation

$$\begin{aligned} (-\Delta )^\frac{3}{2} u = 2 e^{3u}\text { in }{\mathbb {R}}^{3},\quad V:=\int _{{\mathbb {R}}^{3}}e^{3u}\,dx<\infty , \end{aligned}$$

where $V$ is the volume of $g_u$. Adapting a technique of A. Chang and W-X. Chen to the non-local framework, we show the existence of a large class of such metrics, particularly for $V\le 2\pi ^2=|{\mathbb {S}}^3|$. Inspired by previous works of C-S. Lin and L. Martinazzi, who treated the analogue cases in even dimensions, we classify such metrics based on their behavior at infinity.

Appendix A: The fractional Laplacian in \({\mathbb {R}}^{n}\)

If $\sigma \in (0,1)$ and $u$ belongs to the Schwarz space $\mathcal S$ of rapidly decreasing smooth functions in ${\mathbb {R}}^{n}$, then $(-\Delta )^\sigma u$ is defined by

$$\begin{aligned} \widehat{(-\Delta )^\sigma u}(\xi )=|\xi |^{2\sigma }{\hat{u}}(\xi ), \end{aligned}$$

where

$$\begin{aligned} {\hat{f}}(\xi )=\mathcal {F}(f)(\xi ):=\frac{1}{(2\pi )^{n/2}}\int _{{\mathbb {R}}^{n}}f(x)e^{- ix\cdot \xi }\,dx \end{aligned}$$

denotes the Fourier transform. An equivalent definition is the following:

$$\begin{aligned} (-\Delta )^\sigma u(x):=C_{n,\sigma } P.V.\int _{{\mathbb {R}}^{n}}\frac{u(x)-u(y)}{|x-y|^{n+2\sigma }}\,dy, \end{aligned}$$

(33)

where the right-hand side is defined in the sense of the principal value. One can see that (33) makes sense for classes of functions larger than the Schwarz space, for instance for functions in $C^{2\sigma +\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n})\cap L_{\sigma }({\mathbb {R}}^{n})$ for some $\alpha >0$, where

$$\begin{aligned} L_{\sigma }({\mathbb {R}}^{n}):=\left\{ u\in L^1_{{{\mathrm{loc}}}}({\mathbb {R}}^{n}):\int _{{\mathbb {R}}^{n}}\frac{|u(x)|}{1+|x|^{n+2\sigma }}\,dx< \infty \right\} , \end{aligned}$$

and $C^{2\sigma +\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n}):=C^{0,2\sigma +\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n})$ for $2\sigma +\alpha \le 1$ and $C^{2\sigma +\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n}):=C^{1,2\sigma -1+\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n})$ for $2\sigma +\alpha >1$. We denote $\Vert u\Vert _{L_{\sigma }({\mathbb {R}}^{n})}=\int _{{\mathbb {R}}^{n}}\frac{|u(x)|}{1+|x|^{n+2\sigma }}dx$. Observing that

$$\begin{aligned} \sup _{{\mathbb {R}}^{n}}(1+|x|^{n+2\sigma })|(-\Delta )^\sigma \varphi (x)|< +\infty \quad \text {for }\varphi \in {\mathcal {S}}, \end{aligned}$$

(34)

and that $(-\Delta )^{\sigma }:{\mathcal {S}}\rightarrow {\mathcal {S}}$ is symmetric, as shown in [32], one can define $(-\Delta )^{\sigma } u$ by duality for functions $u\in L_{\sigma }({\mathbb {R}}^{n})$ as a tempered distribution via the relation

$$\begin{aligned} \langle (-\Delta )^{\sigma } u,\varphi \rangle =\int _{{\mathbb {R}}^{n}} u(x) (-\Delta )^{\sigma } \varphi (x)\,dx\quad \text {for every }\varphi \in {\mathcal {S}}. \end{aligned}$$

(35)

That for $u\in C^{2\sigma +\alpha }_{{{\mathrm{loc}}}}({\mathbb {R}}^{n})\cap L_\sigma ({\mathbb {R}}^{n})$ the definitions (33) and (35) coincide is shown in [32, Proposition 2.4].

The following lemma is well-known, but we include a proof here for convenience and completeness.

Lemma 21

The function $K(x):=\frac{1}{2\pi ^2|x|^2}$ is a fundamental solution of $(-\Delta )^\frac{1}{2}$ in ${\mathbb {R}}^{3}$ in the sense that for every $f\in L^1({\mathbb {R}}^{3})$ we have $K*f\in L_{1/2}({\mathbb {R}}^{3})$ and

$$\begin{aligned} (-\Delta )^\frac{1}{2} (K*f)=f, \end{aligned}$$

(36)

in the sense of (35).

Proof

First of all, it follows easily from Theorem 5.9 in [16] that (36) holds if we assume $f\in {\mathcal C}^\infty _c({\mathbb {R}}^{3})$.

Secondly, we notice that, if $f\in L^{1}$ then $K*f\in L_{1/2}({\mathbb {R}}^{3})$. Indeed,

$$\begin{aligned} K(x)=\frac{1}{2\pi ^2|x|^{2}}\chi _{B_{1}} + \frac{1}{2\pi ^2|x|^{2}}\chi _{{\mathbb {R}}^{3}\setminus B_{1}}=: K_1(x)+K_2(x), \end{aligned}$$

and $K_{1}\in L^{\frac{3}{2}-\varepsilon }({\mathbb {R}}^{3})$, $K_{2}\in L^{\frac{3}{2}+\varepsilon }({\mathbb {R}}^{3})$ for any $\varepsilon >0$. Hence, by Young’s inequality

$$\begin{aligned} K*f\in L^{\frac{3}{2}-\varepsilon } ({\mathbb {R}}^{3})+L^{\frac{3}{2}+ \varepsilon }({\mathbb {R}}^{3})\subset L_{1/2}({\mathbb {R}}^{3}), \end{aligned}$$

where the last inclusion follows from Hölder’s inequality.

Lastly, if $f\in L^1({\mathbb {R}}^{3})$ we take a sequence $(f_k)\subset C^\infty _c({\mathbb {R}}^{3})$ with $f_k\rightarrow f$ in $L^1({\mathbb {R}}^{3})$. Then for every $\varphi \in {\mathcal S}$ we have

$$\begin{aligned} (I)_k:=\langle (-\Delta )^\frac{1}{2}(K*f_k),\varphi \rangle =\langle f_k,\varphi \rangle \rightarrow \langle f,\varphi \rangle \quad \text {as }k\rightarrow \infty . \end{aligned}$$

Since $K*f=K_{1}*f+K_{2}*f\in L^{\frac{3}{2}-\varepsilon }+L^{\frac{3}{2}+\varepsilon }$, we have $K_{1}*f_{k}\rightarrow K_{1}*f$ in $L^{\frac{3}{2}-\varepsilon }$, and thus, in $L_{1/2}({\mathbb {R}}^{3})$ by Hölder’s inequality. Similarly, $K_{2}*f_{k}\rightarrow K_{2}*f$ in $L_{1/2}({\mathbb {R}}^{3})$, and $K*f_k\rightarrow K*f$ in $L_{1/2}({\mathbb {R}}^{3})$. By (34), we find

$$\begin{aligned} (I)_k=\langle K*f_k,(-\Delta )^\frac{1}{2}\varphi \rangle \rightarrow \langle K*f,(-\Delta )^\frac{1}{2}\varphi \rangle . \end{aligned}$$

Hence, we conclude that $(-\Delta )^\frac{1}{2} (K*f)= f$ in the sense of (35). $\square$

1.1 A.1 Schauder estimates

The following proposition should be well-known, but we include here an elementary proof of the estimate (37) which was used in Sect. 4.

Let $\Omega$ be a domain in ${\mathbb {R}}^{n}$ and $f\in L^1(\Omega )$. We say that $u\in L_{\sigma }({\mathbb {R}}^{n})$ is a solution of $(-\Delta )^\sigma u=f$ in $\Omega$ if

$$\begin{aligned} \int _{{\mathbb {R}}^{n}} u\, (-\Delta )^\sigma \varphi \,dx=\int _{{\mathbb {R}}^{n}} f\, \varphi \,dx\quad \text {for every }\varphi \in C_c^{\infty }(\Omega ). \end{aligned}$$

Proposition 22

If $u\in L_{\sigma }({\mathbb {R}}^{n})$ for some $\sigma \in (0,1)$ and $(-\Delta )^\sigma u=0$ in $B_{2r}$ for some $r>0$, then $u\in C^\infty (B_{2r})$. Moreover, for every $k\in {\mathbb {N}}$ the following estimate holds:

$$\begin{aligned} \Vert \nabla ^k u\Vert _{L^\infty (B_r)}\le \frac{C_{n,\sigma ,k}}{r^{k}} \left( r^{2\sigma }\int _{{\mathbb {R}}^{n}\setminus B_{2r}}\frac{|u(x)|}{|x|^{n+2\sigma }}\,dx + \frac{\Vert u\Vert _{L^1(B_{2r})}}{r^n}\right) , \end{aligned}$$

(37)

where $C_{n,\sigma ,k}$ is a positive constant depending only on $n,\sigma$ and $k$.

Notice that the right-hand sides of (37) are equivalent to $C_{n,\sigma ,k,\alpha ,r}\Vert u\Vert _{L_\sigma }$ for every fixed $r$ and, although this term is more compact, it is not scale invariant with respect to $r$.

For the proof of this proposition we will use a couple of results from [32]. Following the notations of Silvestre [32] we set $\Phi (x)= \frac{C_{n,\sigma }}{|x|^{n-2\sigma }}$ the fundamental solution of $(-\Delta )^{\sigma }$ and we construct $\Gamma$ from $\Phi$ by modifying $\Phi$ only in $B_1$ so that $\Gamma \in C^{\infty }({\mathbb {R}}^{n})$. Via a rescaling, we consider for $\lambda >0$ the function

$$\begin{aligned} \Gamma _{\lambda }(x)=\frac{1}{\lambda ^{n-2\sigma }}\Gamma \left( \frac{x}{\lambda }\right) , \end{aligned}$$

and also define $\gamma _{\lambda }(x):=(-\Delta )^{\sigma } \Gamma _{\lambda }(x)$. Notice that

$$\begin{aligned} \gamma _\lambda (x)=\frac{1}{\lambda ^n}\gamma _1\left( \frac{x}{\lambda }\right) . \end{aligned}$$

(38)

By [32, Prop. 2.7] $\gamma _\lambda \in C^\infty ({\mathbb {R}}^{n})$. We will need the following two results:

Proposition 23

([32], Prop. 2.12) For $|x|>\lambda$, we have

$$\begin{aligned} \gamma _{\lambda }(x)=\int _{B_{\lambda }(0)}\frac{\Phi (y)-\Gamma _{\lambda }(y)}{|x-y|^{n+2\sigma }}\,dy. \end{aligned}$$

(39)

Proposition 24

([32], Prop. 2.22) Assume that $u\in L_{\sigma }({\mathbb {R}}^{n})$ such that $(-\Delta )^{\sigma }u=0$ in $\Omega \subset {\mathbb {R}}^{n}$. Then $u\in C^0(\Omega )$ and $u(x)=u*\gamma _{\lambda }(x)$ for every $x\in \Omega$ and $\lambda \in (0,\mathrm{{dist}}(x,\partial \Omega ))$.

We remark that, although our definition of $\Gamma$ (hence of $\Gamma _\lambda$ and $\gamma _\lambda$) is slightly different from the one in [32], the proofs of the above propositions go through with almost no change.

Proof of Proposition 22

The proof uses Proposition 24 and a standard convolution argument. For every $k\in {\mathbb {N}}\cup \{0\}$, we have from Proposition 24 that $\nabla ^k u=u*\nabla ^k \gamma _\lambda$ (we use the notation that $\nabla ^0$ is the identity operator) in $B_r$ for $\lambda = r/2$. Hence, for $x\in B_r$,

$$\begin{aligned} |\nabla ^k u(x)|\le \int _{{\mathbb {R}}^{n}\setminus B_{2r}}|u(y)| |\nabla ^k\gamma _{\lambda }(x-y)|\,dy+\int _{B_{2r}}|u(y)| |\nabla ^k\gamma _{\lambda }(x-y)|\,dy=:I+II. \end{aligned}$$

Notice that

$$\begin{aligned} \frac{1}{|x-y-z|^{n+2\sigma +k}}\le \frac{1}{(|y|-r-\lambda )^{n+2\sigma +k}}\le \frac{C_{n,\sigma ,k}}{|y|^{n+2\sigma +k}}, \quad |y|>2r, |x|<r,|z|<\lambda = \frac{r}{2}. \end{aligned}$$

Then we have, by differentiating (39),

$$\begin{aligned} |\nabla ^k\gamma _\lambda (x-y)|\le C_{n,\sigma ,k}\int _{B_\lambda }\frac{|\Phi (z)-\Gamma _\lambda (z)|}{|x-y-z|^{n+2\sigma +k}}dz\le \frac{C_{n,\sigma ,k} \lambda ^{2\sigma }}{|y|^{n+2\sigma +k}},\quad |y|>2r, |x|<r,\lambda = \frac{r}{2}. \end{aligned}$$

It follows that

$$\begin{aligned} I\le C_{n,\sigma } r^{2\sigma -k} \int _{{\mathbb {R}}^{n}\setminus B_{2r}}\frac{|u(y)|}{|y|^{n+2\sigma }}\,dy. \end{aligned}$$

As for II, notice that (38) implies $\nabla ^k \gamma _\lambda =\lambda ^{-n-k}\nabla ^k\gamma _1\big (\frac{x}{\lambda }\big )$, from which one bounds

$$\begin{aligned} II\le C_{n,\sigma ,k}\Vert \nabla ^k\gamma _{r/2}\Vert _{L^\infty }\int _{B_{2r}} |u(y)|\,dy \le \frac{C_{n,\sigma ,k}}{r^{n+k}} \Vert u\Vert _{L^1(B_{2r})}. \end{aligned}$$

The proof of (37) is completed. $\square$

Corollary 25

Suppose $u\in L_\sigma ({\mathbb {R}}^{n})$ for some $\sigma \in (0,1)$ and $(-\Delta )^\sigma u=f$ in $B_2$ for some $f\in C^{k,\alpha }(B_2)$, where $\alpha \in (0,1), k\in {\mathbb {N}}\cup \{0\}$ and $\alpha +2\sigma$ is not an integer. Then $u\in C^{k,\alpha +2\sigma }(B_1)$ $(C^{k,\beta }(B_1)= C^{k+1, \beta -1}(B_1)$ if $\beta >1).$ Moreover,

$$\begin{aligned} \Vert u\Vert _{C^{k,\alpha +2\sigma }(B_1)}\le C_{n,\sigma ,k}\left( \int _{{\mathbb {R}}^{n}}\frac{|u(x)|}{1+|x|^{n+2\sigma }}\,dx+\Vert f\Vert _{C^{k,\alpha }(B_2)}\right) , \end{aligned}$$

where $C_{n,\sigma ,k}$ is a positive constant depending only on $n,\sigma$ and $k$.

Proof

This can be proven similarly as in Proposition 2.8 of [32], by using the estimates in Proposition 22. $\square$