Nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations

Xuemei Li Xuemei Li
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China. [email protected] , Chenxi Liu Chenxi Liu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China. [email protected] , Xingdong Tang Xingdong Tang
School of Mathematics and Statistics,
Nanjing Univeristy of Information Science and Technology,
Nanjing, 210044, People’s Republic of China. [email protected] and Guixiang Xu Guixiang Xu
Laboratory of Mathematics and Complex Systems,
Ministry of Education,
School of Mathematical Sciences,
Beijing Normal University,
Beijing, 100875, People’s Republic of China. [email protected]

Abstract.

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH)

-{\Delta u}\left(x\right)-{\bm{\alpha}}\left(N,\lambda\right)\int_{\mathbb{R}^{N}}{\frac{u^{p}\left(y\right)}{\left|\,x-y\,\right|^{\lambda}}}\mathrm{d}y\,u^{p-1}\left(x\right)=0,\quad x\in\mathbb{R}^{N}

where $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ , $\displaystyle p=\frac{2N-\lambda}{N-2}$ and $\displaystyle{\bm{\alpha}}\left(N,\lambda\right)$ is a normalized constant such that $\displaystyle u(x)=\left(1+|x|^{2}\right)^{-\frac{N-2}{2}}$ is a bubble solution of the equation (NLH). It solves an open nondegeneracy problem in [41, 27] and generalizes the partial nondegeneracy results in [15, 29, 34] to the full range $\displaystyle 0<\lambda<N$ . The key observation is that by use of the stereographic projection $\displaystyle\mathcal{S}$ , the weighted pushforward map $\displaystyle\mathcal{S}_{*}$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\displaystyle\mathcal{H}_{1}^{N+1}$ of degree one.

Key words and phrases:

Bubble solution; Funk-Hecke formula; Hartree equation; Nondegeneracy; Spherical harmonic functions; Stereographic projection

2010 Mathematics Subject Classification:

Primary 35B09, 47A74. Secondly 35P10, 42B37

1. Introduction

In this paper we consider the generalized energy-critical Hartree equations in $\displaystyle\mathbb{R}^{N}$

-{\Delta u}\left(x\right)-{\bm{\alpha}}\left(N,\lambda\right)\int_{\mathbb{R}^{N}}{\frac{u^{p}\left(y\right)}{\left|\,x-y\,\right|^{\lambda}}}\mathrm{d}y\,u^{p-1}\left(x\right)=0,\quad x\in\mathbb{R}^{N}

(NLH)

where $\displaystyle u$ is a real-valued function, $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ , $\displaystyle p=\frac{2N-\lambda}{N-2}$ and the normalized constant

{\bm{\alpha}}\left(N,\lambda\right)=\frac{N\left(N-2\right)}{\pi^{\frac{N}{2}}}\cdot\frac{\Gamma\left(N-\frac{\lambda}{2}\right)}{\Gamma\left(\frac{N-\lambda}{2}\right)}.

(1.1)

The equation (NLH) is left invariant under the scaling transform

u(x)\longmapsto u_{\delta}(x)=\delta^{\frac{N-2}{2}}u(\delta x),

which preserves the $\displaystyle\dot{H}^{1}(\mathbb{R}^{N})$ norm. That is the reason why the equation (NLH) is called the energy-critical equation.

The equation (NLH), which is also called nonlinear Choquard or Choquard-Pekar equation, has several physical motivations. In the subcritical case $\displaystyle N=3,$ $\displaystyle p=2$ and $\displaystyle\lambda=1$ , the equation (NLH) firstly appeared in the context of Fröhlich and Pekar’s polaron model, which describes the interaction between one single electron and the dielectric polarisable continuum, see [22, 23, 1, 46]. Later, Choquard proposed the equation (NLH) to describe an approximation to the Hartree-Fock theory of a plasma, and then attracted the substantial attention in the field of nonlinear elliptic equations, see [37, 40, 43]. The equation (NLH) also arises as a model problem in the study of stationary solutions to nonlinear Schrödinger equation with nonlocal nonlinearity:

iu_{t}-{\Delta u}-{\bm{\alpha}}\left(N,\lambda\right)\int_{\mathbb{R}^{N}}{\frac{\left|u\left(y\right)\right|^{p}}{\left|\,x-y\,\right|^{\lambda}}}\mathrm{d}y\,\left|u\right|^{p-2}u=0,\quad p\geqslant 2.

(1.2)

Physically, the equation (1.2) effectively describes the mean field limit of quantum many-body systems, see e.g., [26, 33, 25], and references therein.

The existence and uniqueness of positive bubble solutions of the equation (NLH) has been known for some time, see e.g., [15, 28, 31, 41, 42, 43] and references therein.

The existence of bubble solutions of the equation (NLH) is closely related to the sharp \HLSinequality in $\displaystyle\mathbb{R}^{N}$

\displaystyle\displaystyle\left(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\left|f\left(x\right)\right|^{p}\left|f\left(y\right)\right|^{p}}{\left|x-y\right|^{\lambda}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{p}}\leqslant\;\mathrm{C}\left(N,\lambda\right)\left\|\nabla f\right\|_{L^{2}(\mathbb{R}^{N})}^{2},

(1.3)

where $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ and $\displaystyle p=\frac{2N-\lambda}{N-2}$ . In fact, by use of the sharp \HLSinequality and the sharp \Sbinequality in $\displaystyle\mathbb{R}^{N}$ in [39], Miao, Wu and the fourth author firstly showed that the sharp constant $\displaystyle\mathrm{C}\left(N,\lambda\right)$ in (1.3) is obtained in the classical energy-critical case $\displaystyle p=2$ , $\displaystyle\lambda=4<N$ if and only if

f\left(x\right)=c\cdot\left(\delta^{2}+\left|x-x_{0}\right|^{2}\right)^{-\frac{N-2}{2}}

(1.4)

for some $\displaystyle c\in\mathbb{R}\setminus\{0\},\leavevmode\nobreak\ \delta>0$ and $\displaystyle x_{0}\in\mathbb{R}^{N}$ in [41]. Later, Du, Yang and Gao generalized the result in the general energy-critical case in [15, 28].

The existence of the extremizer for the sharp Hardy-Littlewood-Sobolev inequality (1.3) is more subtle than the fact that the inequality (1.3) holds. The rearrangement inequalities, the conformal transform and the stereographic projection are useful arguements to show the existence of the extremizer of (1.3), see [38, 39]. In fact, we have

Theorem 1.1.

[20, 28, 41] Let $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ , and $\displaystyle p=\frac{2N-\lambda}{N-2}$ . Then for any $\displaystyle f,g\in\dot{H}^{1}\left(\mathbb{R}^{N}\right)\setminus\left\{0\right\}$ , the inequality

\left(\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{\left|f\left(x\right)\right|^{p}\left|g\left(y\right)\right|^{p}}{\left|x-y\right|^{\lambda}}\mathrm{d}x\mathrm{d}y\right)^{\frac{1}{p}}\leqslant\mathrm{C}\left(N,\lambda\right)\left\|\nabla f\right\|_{2}\left\|\nabla g\right\|_{2}

(1.5)

holds with sharp constant

\mathrm{C}\left(N,\lambda\right)=\left(\frac{\Gamma\left(N\right)}{\Gamma\left(\frac{N}{2}\right)\left(4\pi\right)^{\frac{N}{2}}}\right)\left(\frac{\Gamma\left(\frac{N}{2}\right)\Gamma\left(\frac{N-\lambda}{2}\right)}{\Gamma\left({N}\right)\Gamma\left(N-\frac{\lambda}{2}\right)}\left(4\pi\right)^{N}\right)^{\frac{N-2}{2N-\lambda}}.

(1.6)

Moreover, the equality in (1.5) holds if and only if

f(x)=c\cdot\left(\delta^{2}+\left|x-x_{0}\right|^{2}\right)^{-\frac{N-2}{2}},\quad\text{and}\quad g\left(x\right)=c^{\prime}\cdot f(x),

where $\displaystyle c,c^{\prime}\in\mathbb{R}\setminus\left\{0\right\}$ , $\displaystyle\delta>0$ , and $\displaystyle x_{0}\in\mathbb{R}^{N}$ .

As for the rigidity classification of the positive solution to the equation (NLH) in $\displaystyle L^{\frac{2N}{N-2}}(\mathbb{R}^{N})$ , Miao, Wu and the fourth author firstly showed that any nontrivial solutions to the equation (NLH) with constant sign in the case $\displaystyle p=2$ , $\displaystyle\lambda=4<N$ must be the form of (1.4) by use of the Kelvin transform and the moving plane method in [41]. Later, Du, Yang [15] and Guo, Hu, Peng, Shuai [31] independently generalized the result in the general case. More precisely, we have

Proposition \theproposition ([15, 31, 41]).

Suppose that $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ and $\displaystyle p=\frac{2N-\lambda}{N-2}$ . Let $\displaystyle u$ be a nontrivial solution of the equation (NLH) with constant sign, then there exist $\displaystyle c\in\mathbb{R}\setminus\{0\},\leavevmode\nobreak\ \delta>0$ and $\displaystyle x_{0}\in\mathbb{R}^{N}$ such that $\displaystyle u\left(x\right)=U_{c,\delta,x_{0}}\left(x\right)$ , where

U_{c,\delta,x_{0}}\left(x\right)=c\cdot\left(\delta^{2}+\left|x-x_{0}\right|^{2}\right)^{-\frac{N-2}{2}}.

(1.7)

Motivated by the nondegeneracy results of eigenfunctions and ground state in [18, 24, 52, 53], a natural question arises in the study of bubble solutions to the equation (NLH) is that

Are the positive solutions of (NLH) non-degenerate?

Since the equation (NLH) is invariant under the scaling and spatial translations, i. e., any solution $\displaystyle{v}$ solves the equation (NLH) if and only if $\displaystyle v_{\delta,x_{0}}\left(x\right)=\delta^{\frac{N-2}{2}}v\left(\delta x+x_{0}\right)$ satisfies

-{\mathop{}\!\mathbin{\bigtriangleup}v_{\delta,x_{0}}}\left(x\right)-{\bm{\alpha}}\left(N,\lambda\right)\int_{\mathbb{R}^{N}}{\frac{v_{\delta,x_{0}}^{p}\left(y\right)}{\left|\,x-y\,\right|^{\lambda}}}\mathrm{d}y\,v_{\delta,x_{0}}^{p-1}\left(x\right)=0

(NLH*)

for any $\displaystyle\delta>0$ and $\displaystyle x_{0}\in\mathbb{R}^{N}$ . Hence, for simplicity, it suffices to consider the solution (1.7) with the normalized parameters $\displaystyle c=1$ , $\displaystyle\delta=1$ and $\displaystyle x_{0}=0,$ i.e.

u\left(x\right):=U_{1,1,0}\left(x\right)=\left(1+\left|x\right|^{2}\right)^{-\frac{N-2}{2}}.

(1.8)

That is the reason why we choose the normalized constant $\displaystyle{\bm{\alpha}}\left(N,\lambda\right)$ in (NLH) and (NLH*). By differentiating (NLH*) with respect to $\displaystyle\delta$ and $\displaystyle x_{0}$ at $\displaystyle\left(\delta,x_{0}\right)=\left(1,0\right)$ , we know that the generator

\displaystyle\displaystyle\varphi_{j}\left(x\right):=\frac{\partial u}{\partial x_{j}}\left(x\right)=\left(2-N\right)u\left(x\right)\frac{x_{j}}{1+\left|x\right|^{2}},\;\;\leavevmode\nobreak\ 1\leqslant j\leqslant N,

(1.9)

and

\displaystyle\displaystyle\varphi_{N+1}\left(x\right):=\frac{N-2}{2}u\left(x\right)+x\cdot\nabla u\left(x\right)=\frac{N-2}{2}u\left(x\right)\frac{1-\left|x\right|^{2}}{1+\left|x\right|^{2}},

(1.10)

are $\displaystyle N+1$ linear independent bounded solutions with vanishing at infinity to the following linearized equation

\displaystyle\displaystyle-{\mathop{}\!\mathbin{\bigtriangleup}\varphi}={\bm{\alpha}}\left(N,\lambda\right)\left[{p\left(\frac{1}{\left|\,\cdot\,\right|^{\lambda}}\ast u^{p-1}\varphi\right)u^{p-1}+\left(p-1\right)\left(\frac{1}{\left|\,\cdot\,\right|^{\lambda}}\ast u^{p}\right)u^{p-2}\varphi}\right].

(1.11)

Definition 1.2.

The solution $\displaystyle u$ defined by (1.8) to the equation (NLH) is said to be nondegenerate if any bounded solution vanishing at infinity to the linearized equation (1.11) must be the linear combinations of the functions $\displaystyle\varphi_{1},\cdots,\varphi_{N}$ and $\displaystyle\varphi_{N+1}$ defined by (1.9) and (1.10).

We now state the main result in this paper.

Theorem 1.3.

Let $\displaystyle N\geqslant 3$ , $\displaystyle\lambda\in\left(0,N\right)$ and $\displaystyle p=\frac{2N-\lambda}{N-2}$ . Then the nontrivial solution $\displaystyle U_{c,\delta,x_{0}}$ of the equation (NLH) with constant sign is nondegenerate.

The result in Theorem 1.3 solves an open nondegeneracy problem in [27, 41], and generalizes the partial nondegeneracy results in [15, 29, 34, 41] to the full rang $\displaystyle\lambda\in(0,N)$ . The argument in this paper is different with those in [29, 15] and [32, 34, 56]. In fact, we rewrite (1.11) as an integral form in $\displaystyle\mathbb{R}^{N}$ , and its equivalent integral form on the sphere $\displaystyle\mathbb{S}^{N}$ via the stereographic projection $\displaystyle\mathcal{S}:\mathbb{R}^{N}\longrightarrow\mathbb{S}^{N}$ . The key observation is that together with the spherical harmonic decomposition and the Funk-Hecke formula in [4, 8, 51], the weighted pushforward map $\displaystyle\mathcal{S}_{*}$ related to the stereographic projection $\displaystyle\mathcal{S}$ is one-to-one map between the null space of the linearized operator and the spherical harmonic function subspace $\displaystyle\mathcal{H}_{1}^{N+1}$ of degree one. The idea with use of the stereographic projection and the Funk-Hecke formula is inspired by Frank and Lieb in [21, 20],

Remark 1.4 (Nondegeneracy of positive solutions to nonlinear elliptic equations with local nonlinearity).

There were extensive literatures to show the nondegeneracy of positive solutions to nonlinear elliptic equations. Weinstein [55] and Oh [45] made use of the spherical harmonic expansion to obtain the nondegeneracy of positive solutions to nonlinear elliptic equation with subcritical nonlinearity

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)+u\left(x\right)-u^{p}\left(x\right)=0,\quad x\in\mathbb{R}^{N},\quad 1<p<\frac{N+2}{N-2}.

Rey [48], Dolbeault and Jankowiak [14] made use of the stereographic projection to obtain the nondegeneracy of positive bubble solutions to nonlinear elliptic equation with critical nonlinearity

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)-u^{\frac{N+2}{N-2}}\left(x\right)=0,\quad x\in\mathbb{R}^{N}.

(1.12)

For other applications of the spherical harmonic expansion in the nondegeneracy of positive solutions for other nonlinear elliptic equations with local nonlinearity, please refer to [49, 30, 5, 9, 3, 16, 10, 18, 19, 17, 2, 47, 44, 52] and references therein.

Remark 1.5 (Nondegeneracy of positive solutions to nonlinear Hartree equations).

Let $\displaystyle N\geqslant 3$ , $\displaystyle 0<\lambda<N$ and $\displaystyle 1<p<\frac{2N-\lambda}{N-2}$ . Due to nonlocal nonlinearity of nonlinear Hartree equations, the nondegeneracy of positive solutions to the equation (NLH)

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)+u\left(x\right)-\int_{\mathbb{R}^{N}}{\frac{u^{p}\left(y\right)}{\left|\,x-y\,\right|^{\lambda}}}\mathrm{d}y\,u^{p-1}\left(x\right)=0,\leavevmode\nobreak\ x\in\mathbb{R}^{N},

(1.13)

is more subtle than that of positive solutions to nonlinear elliptic equations with local nonlinearity.

Lenzmann firstly made use of the multipole expansion of the Newtonian potential , and obtained the nondegeneracy of the ground state to

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)+u\left(x\right)-\frac{1}{8\pi}\int_{\mathbb{R}^{3}}{\frac{u^{2}\left(y\right)}{\left|\,x-y\,\right|}}\mathrm{d}y\,u\left(x\right)=0,\leavevmode\nobreak\ x\in\mathbb{R}^{3}

(1.14)

in [32], see also [54]. For the application of the multipole expansion of the Newtonian potential in $\displaystyle\mathbb{R}^{N}(3\leqslant N\leqslant 5)$ to the nondegeneracy of positive solution to (1.13) with $\displaystyle\lambda=N-2$ and $\displaystyle p=2$ , we can refer to [6]. Later, Xiang made use of a perturbation argument to obtain the non-degeneracy of positive solution to (1.13) for the case that $\displaystyle\lambda=1$ and $\displaystyle p$ is slightly larger than $\displaystyle 2$ in $\displaystyle\mathbb{R}^{3}$ in [56], and recently, Li extended the perturbation argument to the non-degeneracy of positive solution to (1.13) for the case that $\displaystyle\lambda$ close to $\displaystyle N-2$ and $\displaystyle p$ slightly larger than $\displaystyle 2$ in $\displaystyle\mathbb{R}^{N}$ in [36] . For other applications of the perturbation argument, please refer to [15, 29] for the case that $\displaystyle\lambda$ is closed to $\displaystyle 0$ or $\displaystyle N$ .

Recently, Li, Tang and Xu made use of the spherical harmonic expansion and the multipole expansion of the Newtonian potential to obtain the non-degeneracy of bubble solutions to the energy-critical Hartree equation in $\displaystyle\mathbb{R}^{6}$

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)-\int_{\mathbb{R}^{6}}{\frac{u^{2}\left(y\right)}{\left|\,x-y\,\right|^{4}}}\mathrm{d}y\,u\left(x\right)=0,\leavevmode\nobreak\ x\in\mathbb{R}^{6}

(1.15)

in [34]. We can also refer to [27].

Recently, the authors make use of the Moser iteration method in [13] to obtain the $\displaystyle L^{\infty}(\mathbb{R}^{N})$ regularity of the energy solution to the linearized equation (1.11), and show that the nontrivial solution $\displaystyle U_{c,\delta,x_{0}}$ of the equation (NLH) with constant sign is nondegenerate in the energy space $\displaystyle\dot{H}^{1}(\mathbb{R}^{N})$ in [35]. At the same time, the authors consider long time dynamics of the radial threshold solution of the equation (1.2) and its rigidity classification in [35], which depends on the nondegeneracy of the bubble solutions in the energy space $\displaystyle\dot{H}^{1}(\mathbb{R}^{N})$ and the spectrum of the linearized operator.

Remark 1.6 (Application of nondegeneracy of positive solution in the construction of multi-bubble solutions).

The existence of bubble solutions to the energy-critical nonlinear Hartree equation has been well-studied recently, see [29, 15, 31, 28, 41]. It is interesting to construct the existence of multi-bubble solutions to the equation (NLH). To the best of our knowledge, there are few results concerning multi-bubble solutions to (NLH) except that in [27]. However, for the limiting case $\displaystyle\lambda=0$ in (NLH), i.e.

-{\mathop{}\!\mathbin{\bigtriangleup}u}\left(x\right)-u^{\frac{N+2}{N-2}}\left(x\right)=0,\quad x\in\mathbb{R}^{N},

(1.16)

the multi-bubble solutions has been constructed by using the \LSargument in [12, 11], where the nondegeneracy of positive solutions plays a crucial role.

Lastly, the rest of this paper is organized as follows. In Section 2, we introduce some notation, the preliminary results about the stererographic projection and the Funk-Hecke formula of the spherical harmonic functions. In Section 3, we prove Theorem 1.3.

Acknowledgements.

The authors were supported by National Key Research and Development Program of China (No. 2020YFA0712900) and by NSFC (No. 12371240, No. 12431008).

2. Notation and Preliminary Results

In this section, we introduce some notation. We denote $\displaystyle\langle x\rangle=\left(1+\left|x\right|^{2}\right)^{\frac{1}{2}},$ and use $\displaystyle\mathbb{S}^{N}$ to denote the unit sphere in $\displaystyle\mathbb{R}^{N+1}$ , i.e.

\mathbb{S}^{N}=\left\{\xi=\left(\xi_{1},\xi_{2},\cdots,\xi_{N+1}\right)\in\mathbb{R}^{N+1}\quad\middle|\quad\sum_{j=1}^{N+1}\xi_{j}^{2}=1\right\},

and $\displaystyle g_{ij}\left(1\leqslant i,\,j\leqslant N+1\right)$ stand for the metric on $\displaystyle\mathbb{S}^{N}$ , which is inherited from $\displaystyle\mathbb{R}^{N+1}$ . For any $\displaystyle 1\leqslant p<+\infty$ , let us denote by $\displaystyle L^{p}\left(\mathbb{R}^{N}\right)$ and $\displaystyle L^{p}\left(\mathbb{S}^{N}\right)$ the space of real-valued $\displaystyle p$ -th power integrable functions on $\displaystyle\mathbb{R}^{N}$ and $\displaystyle\mathbb{S}^{n}$ . Moreover, with a little abuse of notation, we equip $\displaystyle L^{p}\left(\mathbb{R}^{N}\right)$ and $\displaystyle L^{p}\left(\mathbb{S}^{N}\right)$ with the norms:

\left\|f\right\|_{p}=\left(\int_{\mathbb{R}^{N}}\left|f\left(x\right)\right|^{p}\mathrm{d}x\right)^{\frac{1}{p}},\quad\text{\leavevmode\nobreak\ for\leavevmode\nobreak\ }\quad f\in L^{p}\left(\mathbb{R}^{N}\right),

and

\left\|F\right\|_{p}=\left(\int_{\mathbb{S}^{N}}\left|F\left(\xi\right)\right|^{p}\mathrm{d}\xi\right)^{\frac{1}{p}},\quad\text{\leavevmode\nobreak\ for\leavevmode\nobreak\ }\quad F\in L^{p}\left(\mathbb{S}^{N}\right),

where $\displaystyle\mathrm{d}\xi$ is the standard volume element on the sphere $\displaystyle\mathbb{S}^{N}$ .

We denote the stereographic projection $\displaystyle\mathcal{S}:\mathbb{R}^{N}\mapsto\mathbb{S}^{N}\setminus\left\{\left(0,0,\cdots,0,-1\right)\right\}$ by

\displaystyle\displaystyle\mathcal{S}x=\left(\frac{2x}{1+\left|x\right|^{2}},\frac{1-\left|x\right|^{2}}{1+\left|x\right|^{2}}\right),

and its inverse map $\displaystyle\mathcal{S}^{-1}:\mathbb{S}^{N}\setminus\left\{\left(0,0,\cdots,0,-1\right)\right\}\mapsto\mathbb{R}^{N}$ by

\displaystyle\displaystyle\mathcal{S}^{-1}\left(\xi_{1},\xi_{2},\cdots,\xi_{N+1}\right)=\left(\frac{\xi_{1}}{1+\xi_{N+1}},\frac{\xi_{2}}{1+\xi_{N+1}},\cdots,\frac{\xi_{N}}{1+\xi_{N+1}}\right).

Let $\displaystyle\rho\left(x\right)=\left(\frac{2}{1+\left|x\right|^{2}}\right)^{\frac{1}{2}}$ . From [39, 20], we have

\displaystyle\displaystyle g_{ij}=\rho^{4}\left(x\right)\delta_{ij},\;\;{\left|\mathcal{S}x-\mathcal{S}y\right|}={\left|x-y\right|}\rho\left(x\right)\rho\left(y\right),

(2.1)

and

\displaystyle\displaystyle\mathrm{d}\xi=\rho^{2N}\left(x\right)\mathrm{d}x.

(2.2)

Therefore, for any $\displaystyle F\in L^{1}\left(\mathbb{S}^{N}\right)$ , we have the following identity

\int_{\mathbb{S}^{N}}F\left(\xi\right)\mathrm{d}\xi=\int_{\mathbb{R}^{N}}F\left(\mathcal{S}x\right){\rho^{2N}\left(x\right)}\mathrm{d}x.

In order to relate the functions between in $\displaystyle\mathbb{R}^{N}$ and on the sphere $\displaystyle\mathbb{S}^{N}$ , we can compose the functions in $\displaystyle\mathbb{R}^{N}$ and $\displaystyle\mathbb{S}^{N}$ with the maps $\displaystyle\mathcal{S}^{\pm}$ . For any $\displaystyle f:\mathbb{R}^{N}\mapsto\mathbb{R}$ , we denote the weighted pushforward map $\displaystyle\mathcal{S}_{\ast}f:\mathbb{S}^{N}\setminus\left\{\left(0,0,\cdots,0,-1\right)\right\}\mapsto\mathbb{R}$ by

\displaystyle\displaystyle\mathcal{S}_{\ast}f(\xi)=\rho^{2-N}\left(\mathcal{S}^{-1}\xi\right)f\left(\mathcal{S}^{-1}\xi\right),

(2.3)

and for any $\displaystyle F:\mathbb{S}^{N}\setminus\left\{\left(0,0,\cdots,0,-1\right)\right\}\mapsto\mathbb{R}$ , we denote the weighted pullback map $\displaystyle\mathcal{S}^{\ast}F:\mathbb{R}^{N}\mapsto\mathbb{R}$ by

\displaystyle\displaystyle\mathcal{S}^{\ast}F(x)=\rho^{N-2}\left(x\right)F\left(\mathcal{S}x\right).

(2.4)

A simple calculation shows that

Proposition \theproposition.

Let $\displaystyle\varphi_{j}$ , $\displaystyle 1\leqslant j\leqslant N+1$ be defined by (1.9) and (1.10) and $\displaystyle\mathcal{S}_{*}$ be defined by (2.3), then for any $\displaystyle\xi\in\mathbb{S}^{N}$ and $\displaystyle 1\leqslant j\leqslant N$ , we have

\displaystyle\displaystyle\mathcal{S}_{*}\varphi_{j}(\xi)=\frac{2-N}{2}\,2^{\frac{2-N}{2}}\xi_{j},\;\;\text{and}\;\;\mathcal{S}_{*}\varphi_{N+1}(\xi)=\frac{N-2}{2}\,2^{\frac{2-N}{2}}\xi_{N+1}.

(2.5)

Proof.

First of all, for $\displaystyle j=1,2,\cdots,N$ , we have by (2.3) that

\displaystyle\displaystyle\mathcal{S}_{*}\varphi_{j}\left(\mathcal{S}x\right)=\rho^{2-N}\left(x\right)\varphi_{j}\left(x\right),

(2.6)

which, together with (1.9) and (1.10), implies that

\displaystyle\displaystyle\mathcal{S}_{*}\varphi_{j}\left(\mathcal{S}x\right)=\frac{2-N}{2}2^{\frac{2-N}{2}}\frac{2x_{j}}{1+\left|x\right|^{2}}=\frac{2-N}{2}2^{\frac{2-N}{2}}\left(\mathcal{S}x\right)_{j}=\frac{2-N}{2}2^{\frac{2-N}{2}}\xi_{j},

and

\displaystyle\displaystyle\mathcal{S}_{*}\varphi_{N+1}\left(\mathcal{S}x\right)=\frac{N-2}{2}2^{\frac{2-N}{2}}\frac{1-\left|x\right|^{2}}{1+\left|x\right|^{2}}=\frac{N-2}{2}2^{\frac{2-N}{2}}\left(\mathcal{S}x\right)_{N+1}=\frac{N-2}{2}2^{\frac{2-N}{2}}\xi_{N+1},

which implies (2.5) and completes the proof. ∎

We now introduce the spherical harmonic functions, which are related to the spectral properties of the Laplace-Beltrami opertor on the sphere $\displaystyle\mathbb{S}^{N}$ (see [4, 8, 51]). In fact, we have the following orthogonal decomposition:

L^{2}\left(\mathbb{S}^{N}\right)=\bigoplus\limits_{k=0}^{\infty}\mathscr{H}_{k}^{N+1},

(2.7)

where $\displaystyle\mathscr{H}_{k}^{N+1}\left(k\geqslant 0\right)$ denote the mutually orthogonal subspace of the restriction on $\displaystyle\mathbb{S}^{N}$ of real, homogeneous harmonic polynomials of degree $\displaystyle k$ , and

\dim{\mathscr{H}_{k}^{N+1}}:=\begin{dcases}1,&\mbox{if }k=0,\\ N+1,&\mbox{if }k=1,\\ \binom{k+N}{k}-\binom{k-2+N}{k-2},&\mbox{if }k\geqslant 2.\end{dcases}

We will use $\displaystyle\left\{Y_{k,j}\mid 1\leqslant j\leqslant\dim{\mathscr{H}_{k}^{N+1}}\right\}$ to denote an orthonormal basis of $\displaystyle\mathscr{H}_{k}^{N+1}$ . In particular, we have

Y_{1,j}\left(\xi\right)=\sqrt{\frac{\left(N+1\right)\Gamma\left(\frac{N}{2}\right)}{2\pi^{\frac{N}{2}}}}\xi_{j},\quad 1\leqslant j\leqslant N+1,

and

{\mathscr{H}_{1}^{N+1}}=\mathrm{span}\left\{\xi_{j}\,\middle|\,1\leqslant j\leqslant N+1\right\},

(2.8)

which together with Proposition 2 implies that the weighted pushfoward map $\displaystyle\mathcal{S}_{*}$ is one-to-one map from the subspace $\displaystyle\text{Span}\{\varphi_{j},1\leqslant j\leqslant N+1\}\subset L^{\infty}(\mathbb{R}^{N})$ to the subspace $\displaystyle{\mathscr{H}_{1}^{N+1}}$ , and so is the weighted pullback map $\displaystyle\mathcal{S}^{*}:{\mathscr{H}_{1}^{N+1}}\rightarrow\text{Span}\{\varphi_{j},1\leqslant j\leqslant N+1\}$ . This is the key observation in the proof of Theorem 1.3.

We recall the Funk-Hecke formula of the spherical harmonic functions as follows.

Lemma \thelemma ([4, 8]).

Let $\displaystyle\lambda\in\left(0,N\right)$ , the integer $\displaystyle k\geqslant 0$ and $\displaystyle\mu_{k}\left(\lambda\right)$ be defined by (2.10), then for any $\displaystyle Y\in\mathscr{H}_{k}^{N+1}$ , we have

\int_{\mathbb{S}^{N}}\frac{1}{\left|\xi-\eta\right|^{\lambda}}Y\left(\eta\right)\mathrm{d}\eta=\mu_{k}\left(\lambda\right)Y\left(\xi\right).

(2.9)

where

\mu_{k}\left(\lambda\right)=2^{N-\lambda}\pi^{\frac{N}{2}}\frac{\Gamma\left(k+\frac{\lambda}{2}\right)\Gamma\left(\frac{N-\lambda}{2}\right)}{\Gamma\left(\frac{\lambda}{2}\right)\Gamma\left(k+N-\frac{\lambda}{2}\right)}.

(2.10)

In particular, the simple calculation gives that

	$\displaystyle\displaystyle\mu_{0}\left(N-2\right)=\frac{8}{N}\cdot\frac{\pi^{\frac{N}{2}}}{\Gamma\left(\frac{N}{2}\right)},\qquad\mu_{1}\left(N-2\right)=\frac{N-2}{N+2}\cdot\mu_{0}\left(N-2\right),$		(2.11)
	$\displaystyle\displaystyle\mu_{0}\left(\lambda\right)=2^{N-\lambda}\pi^{\frac{N}{2}}\cdot\frac{\Gamma\left(\frac{N-\lambda}{2}\right)}{\Gamma\left(N-\frac{\lambda}{2}\right)},\qquad\mu_{1}\left(\lambda\right)=\frac{\lambda}{2N-\lambda}\cdot\mu_{0}\left(\lambda\right).$		(2.12)

and

\displaystyle\displaystyle\mu_{k}\left(\lambda\right)>\mu_{k+1}\left(\lambda\right),\quad\text{for all}\quad k\geqslant 0,

(2.13)

As a direct consequence of the Funk-Hecke formula, we have

Lemma \thelemma.

	$\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\xi-\eta\right\|^{N-2}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}Y\left(\sigma\right)\mathrm{d}\sigma\mathrm{d}\eta=\mu_{k}\left(N-2\right)\mu_{k}\left(\lambda\right)Y\left(\xi\right),$		(2.14)
	$\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\xi-\eta\right\|^{N-2}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}Y\left(\eta\right)\mathrm{d}\eta\mathrm{d}\sigma=\mu_{k}\left(N-2\right)\mu_{0}\left(\lambda\right)Y\left(\xi\right).$		(2.15)

Proof.

We first show (2.14). Indeed, by Lemma 2, we have, for any $\displaystyle Y\in\mathscr{H}_{k}^{N+1}$ ,

\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\frac{1}{\left|\eta-\sigma\right|^{\lambda}}Y\left(\sigma\right)\mathrm{d}\sigma=\mu_{k}\left(\lambda\right)Y\left(\eta\right).

(2.16)

by Lemma 2 with $\displaystyle\lambda=N-2$ again, we get

\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\frac{1}{\left|\xi-\eta\right|^{N-2}}\left(\int_{\mathbb{S}^{N}}\frac{1}{\left|\eta-\sigma\right|^{\lambda}}Y\left(\sigma\right)\mathrm{d}\sigma\right)\mathrm{d}\eta=\mu_{k}\left(N-2\right)\mu_{k}\left(\lambda\right)Y\left(\xi\right),

which, together with Fubini’s theorem, implies (2.14).

Next, we show (2.15). By Lemma 2 and the fact that $\displaystyle 1\in\mathscr{H}_{0}^{N+1}$ , we have

\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\frac{1}{\left|\eta-\sigma\right|^{\lambda}}\mathrm{d}\sigma=\mu_{0}\left(\lambda\right).

(2.17)

By Lemma 2 with $\displaystyle\lambda=N-2$ again, we obtain, for any $\displaystyle Y\in\mathscr{H}_{k}^{N+1}$ ,

\displaystyle\displaystyle\int_{\mathbb{S}^{N}}\frac{1}{\left|\xi-\eta\right|^{N-2}}Y\left(\eta\right)\left(\int_{\mathbb{S}^{N}}\frac{1}{\left|\eta-\sigma\right|^{\lambda}}\mathrm{d}\sigma\right)\mathrm{d}\eta=\mu_{k}\left(N-2\right)\mu_{0}\left(\lambda\right)Y\left(\xi\right),

which implies (2.15), and completes the proof. ∎

3. Proof of Theorem 1.3

In this section, we will prove Theorem 1.3, which is the main result in this paper. We firstly give an integral estimates, which will be used in next decay estimate.

Lemma \thelemma.

Let $\displaystyle\lambda\in\left(0,N\right)$ and $\displaystyle\theta+\lambda>N$ . Then

\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{\lambda}}\frac{1}{\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\begin{dcases}\langle x\rangle^{N-\lambda-\theta},&\quad\text{if }\quad\theta<N,\\ \langle x\rangle^{-\lambda}\left(1+{\log}{\langle x\rangle}\right),&\quad\text{if }\quad\theta=N,\\ \langle x\rangle^{-\lambda},&\quad\text{if }\quad\theta>N.\end{dcases}

(3.1)

Proof.

The proof will be divided into three cases.

Case 1: $\displaystyle\bm{x=0}$ . By a direct computation, using the assumptions that $\displaystyle N>\lambda$ and $\displaystyle N<\theta+\lambda$ , we have

\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left|y\right|^{\lambda}}\frac{1}{\langle y\rangle^{\theta}}\mathrm{d}y\lesssim 1.

(3.2)

Case 2: $\displaystyle\bm{x\in\mathrm{B}\left(0,1\right)\setminus\left\{0\right\}}$ . On the one hand, for $\displaystyle y\in\mathrm{B}\left(x,2\left|x\right|\right)$ , we have $\displaystyle\langle y\rangle\approx 1$ , which together with $\displaystyle\left|x\right|<1$ and $\displaystyle N>\lambda$ implies that,

\displaystyle\displaystyle\int_{\mathrm{B}\left(x,2\left|x\right|\right)}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\int_{0}^{2}r^{N-\lambda-1}\mathrm{d}r\lesssim 1.

(3.3)

On the other hand, for $\displaystyle y\in\mathbb{R}^{N}\setminus\mathrm{B}\left(x,2\left|x\right|\right)$ , we have $\displaystyle\left|y\right|\approx\left|x-y\right|$ , which, together with the assumption that $\displaystyle N<\lambda+\theta$ , implies that

\displaystyle\displaystyle\int_{\mathbb{R}^{N}\setminus\mathrm{B}\left(x,2\left|x\right|\right)}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\int_{\mathbb{R}^{N}\setminus\mathrm{B}\left(0,2\left|x\right|\right)}\frac{1}{\left|y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\langle x\rangle^{N-\lambda-\theta}.

(3.4)

Combining (3.3) with (3.4), we get

\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim 1,\quad\text{for any}\quad x\in\mathrm{B}\left(0,1\right)\setminus\left\{0\right\}.

(3.5)

Case 3: $\displaystyle\bm{x\in\mathbb{R}^{N}\setminus\mathrm{B}\left(0,1\right)}$ . Firstly, noticing that for any $\displaystyle y\in\mathrm{B}\left(x,\frac{\left|x\right|}{2}\right)$ , we have $\displaystyle\langle y\rangle\approx\langle x\rangle$ . Hence,

\int_{\mathrm{B}\left(x,\frac{\left|x\right|}{2}\right)}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\frac{1}{\langle x\rangle^{\theta}}\int_{0}^{\frac{\left|x\right|}{2}}r^{N-\lambda-1}\mathrm{d}r\lesssim\langle x\rangle^{N-\lambda-\theta},

(3.6)

where we used the assumption that $\displaystyle N>\lambda$ .

Secondly, for any $\displaystyle y\in\mathrm{B}\left(x,2{\left|x\right|}\right)\setminus\mathrm{B}\left(x,\frac{\left|x\right|}{2}\right)$ , we have

\left|y\right|\leqslant 3\left|x\right|,\quad\text{and}\quad\left|x-y\right|\approx\langle x\rangle,

which implies that,

\displaystyle\displaystyle\int_{\mathrm{B}\left(x,2{\left|x\right|}\right)\setminus\mathrm{B}\left(x,\frac{\left|x\right|}{2}\right)}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\frac{1}{\langle x\rangle^{\lambda}}\int_{0}^{3\left|x\right|}\frac{r^{N-1}}{\langle r\rangle^{\theta}}\mathrm{d}r\lesssim\begin{dcases}\langle x\rangle^{N-\lambda-\theta},&\text{if}\;\theta<N,\\ \langle x\rangle^{-\lambda}\left(1+{\log}{\langle x\rangle}\right),&\text{if}\;\theta=N,\\ \langle x\rangle^{-\lambda},&\text{if}\;\theta>N.\end{dcases}

(3.7)

Thirdly, for any $\displaystyle y\in\mathbb{R}^{N}\setminus\mathrm{B}\left(x,2{\left|x\right|}\right)$ , we have $\displaystyle\langle y\rangle\approx\left|y-x\right|$ , which implies that,

\displaystyle\displaystyle\int_{\mathbb{R}^{N}\setminus\mathrm{B}\left(x,2{\left|x\right|}\right)}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\int_{\mathbb{R}^{N}\setminus\mathrm{B}\left(x,2{\left|x\right|}\right)}\frac{1}{\left|x-y\right|^{\lambda+\theta}\mathrm{d}y}\lesssim\left|x\right|^{N-\lambda-\theta}\approx\langle x\rangle^{N-\lambda-\theta},

(3.8)

where we used the assumption that $\displaystyle N<\lambda+\theta$ .

Finally, using (3.6), (3.7) and (3.8), we obtain that, for any $\displaystyle x\in\mathbb{R}^{N}\setminus\mathrm{B}\left(0,1\right)$ ,

\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{\lambda}\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\begin{dcases}\langle x\rangle^{N-\lambda-\theta},&\text{if }\theta<N,\\ \langle x\rangle^{-\lambda}\left(1+{\log}{\langle x\rangle}\right),&\text{if }\theta=N,\\ \langle x\rangle^{-\lambda},&\text{if }\theta>N.\end{dcases}

(3.9)

Combining (3.2), (3.5) with (3.9), we obtain(3.1) and complete the proof. ∎

Next, we denote

	$\displaystyle\displaystyle\mathfrak{N}\left(\varphi\right)\left(x\right)$	$\displaystyle\displaystyle=p\int_{\mathbb{R}^{N}}\frac{u^{p-1}\left(y\right)\varphi\left(y\right)}{\left\|x-y\right\|^{\lambda}}\mathrm{d}y\,u^{p-1}\left(x\right)+\left(p-1\right)\int_{\mathbb{R}^{N}}\frac{u^{p}\left(y\right)}{\left\|x-y\right\|^{\lambda}}\mathrm{d}y\,u^{p-2}\left(x\right)\varphi\left(x\right)$
		$\displaystyle\displaystyle=:\mathfrak{N}_{1}\left(\varphi\right)\left(x\right)+\mathfrak{N}_{2}\left(\varphi\right)\left(x\right),$		(3.10)

where $\displaystyle u$ is defined by (1.8).

Lemma \thelemma.

Let $\displaystyle\lambda\in\left(0,N\right)$ , and $\displaystyle\theta\in[0,N-2]$ . If $\displaystyle\varphi$ satisfies $\displaystyle\left|\varphi\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{\theta}}$ , then we have

\left|\mathfrak{N}\left(\varphi\right)\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{\theta+4}}.

(3.11)

Proof.

First, by Section 3, we have

\displaystyle\displaystyle\left|\mathfrak{N}_{1}\left(\varphi\right)\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{N+2-\lambda}}\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{\lambda}}\frac{1}{\langle y\rangle^{N+2-\lambda}}\frac{1}{\langle y\rangle^{\theta}}\mathrm{d}y\lesssim\begin{dcases}\frac{1}{\langle x\rangle^{N+2}},&\quad\text{if}\quad\theta+2>\lambda,\\ \frac{1+\log\langle x\rangle}{\langle x\rangle^{N+2}},&\quad\text{if}\quad\theta+2=\lambda,\\ \frac{1}{\langle x\rangle^{N+\theta+4-\lambda}},&\quad\text{if}\quad\theta+2<\lambda.\end{dcases}

(3.12)

which, together with the fact that $\displaystyle N+2\geqslant\theta+4$ , $\displaystyle N>\lambda$ , and $\displaystyle\frac{1+\log\langle x\rangle}{\langle x\rangle^{N-\lambda}}\lesssim 1$ , implies that

\left|\mathfrak{N}_{1}\left(\varphi\right)\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{\theta+4}}.

(3.13)

Next, by Section 3 once again, we have

\displaystyle\displaystyle\left|\mathfrak{N}_{2}\left(\varphi\right)\left(x\right)\right|\lesssim\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{\lambda}}\frac{1}{\langle y\rangle^{2N-\lambda}}\mathrm{d}y\frac{1}{\langle x\rangle^{4+\theta-\lambda}}\lesssim\frac{1}{\langle x\rangle^{\theta+4}}.

(3.14)

By combining (3.13) with (3.14), we can obtain the result. ∎

By Section 3 and the classical Riesz potential theory in [7, 50], we can rewrite the linearized equation (1.11) as an integral form.

Lemma \thelemma.

Let $\displaystyle N\geqslant 3$ , $\displaystyle\lambda\in\left(0,N\right)$ . If $\displaystyle\varphi\in L^{\infty}(\mathbb{R}^{N})$ satisfies (1.11), then we have

\varphi\left(x\right)=\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)\int_{\mathbb{R}^{N}}\frac{1}{\left|x-y\right|^{N-2}}\mathfrak{N}\left(\varphi\right)\left(y\right)\mathrm{d}y,

(3.15)

where $\displaystyle\mathfrak{N}\left(\varphi\right)$ is given by (3.10) and the constant $\displaystyle\mathrm{G}\left(N\right)=\frac{\Gamma\left(\frac{N}{2}\right)}{2\left(N-2\right)\pi^{\frac{N}{2}}}.$ Moreover, we have the following improved decay estimate

\left|\varphi\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{N-2}}.

(3.16)

Proof.

(3.15) follows from Section 3 and the classical Riesz potential theory in [7, 50]. Therefore, it suffices to show the decay estimate (3.16). It follows from (3.15) and the bootstrap argument on the decay rate.

Since $\displaystyle\varphi\in L^{\infty}(\mathbb{R}^{N})$ , we have $\displaystyle\left|\varphi\left(x\right)\right|\lesssim 1,$ which together with Section 3 implies that $\displaystyle\left|\mathfrak{N}\left(\varphi\right)\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{4}}.$ Therefore, by Section 3, we have for some $\displaystyle 0<\epsilon\ll 1$ that

\left|\varphi\left(x\right)\right|\lesssim\begin{cases}\frac{1}{\langle x\rangle^{N-2}},&\mbox{if\leavevmode\nobreak\ \leavevmode\nobreak\ }N<4,\\ \frac{1}{\langle x\rangle^{2(1-\epsilon)}},&\mbox{if\leavevmode\nobreak\ \leavevmode\nobreak\ }N\geqslant 4.\end{cases}

which reduces to boost the case $\displaystyle N\geqslant 4$ . Now, we assume by induction for some $\displaystyle 1\leqslant j\leqslant\left[\frac{N-2}{2}\right]$ , which is the integer part of $\displaystyle\frac{N-2}{2}$ that

\displaystyle\left|\varphi\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{2(j-\epsilon)}}.

Repeating a similar argument as above, we can obtain that $\displaystyle\left|\mathfrak{N}\left(\varphi\right)\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{2j+4-\epsilon}}$ and

\left|\varphi\left(x\right)\right|\lesssim\begin{cases}\frac{1}{\langle x\rangle^{N-2}},&\mbox{if\leavevmode\nobreak\ \leavevmode\nobreak\ }N<2j+4-\epsilon,\\ \frac{1}{\langle x\rangle^{2(j+1-\epsilon)}},&\mbox{if\leavevmode\nobreak\ \leavevmode\nobreak\ }N\geqslant 2j+4-\epsilon.\end{cases}

By the induction argument for $\displaystyle j=1,\cdots,\left[\frac{N-2}{2}\right]$ , we can obtain the result. ∎

Remark 3.1.

The decay estimate in the above lemma is optimal since the null element $\displaystyle\varphi_{N+1}$ defined by (1.10) satisfies (3.16).

Now we will make use of the stereographic projection to transform the integral equation (3.15) on $\displaystyle\mathbb{R}^{N}$ to that on the sphere $\displaystyle\mathbb{S}^{N}$ . Let us denote

\mathcal{T}_{\mathbb{S}^{N}}{\Phi}\left(\xi\right)=p\mathcal{T}_{\mathbb{S}^{N},1}\Phi\left(\xi\right)+\left(p-1\right)\mathcal{T}_{\mathbb{S}^{N},2}\Phi\left(\xi\right),

(3.17)

where $\displaystyle\xi\in\mathbb{S}^{N}$ and

	$\displaystyle\displaystyle\mathcal{T}_{\mathbb{S}^{N},1}\Phi\left(\xi\right)$	$\displaystyle\displaystyle=2^{-(p-1)(N-2)}\int_{\mathbb{S}^{N}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\xi-\eta\right\|^{N-2}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}\Phi\left(\sigma\right)\mathrm{d}\sigma\mathrm{d}\eta,$		(3.18)
	$\displaystyle\displaystyle\mathcal{T}_{\mathbb{S}^{N},2}\Phi\left(\xi\right)$	$\displaystyle\displaystyle=2^{-(p-1)(N-2)}\int_{\mathbb{S}^{N}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\xi-\eta\right\|^{N-2}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}\Phi\left(\eta\right)\mathrm{d}\eta\mathrm{d}\sigma.$		(3.19)

Lemma \thelemma.

Let $\displaystyle N\geqslant 3$ , and $\displaystyle\lambda\in\left(0,N\right)$ . If $\displaystyle\varphi$ satisfies (3.15) with $\displaystyle\left|\varphi\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{N-2}}$ , then $\displaystyle{\mathcal{S}_{\ast}\varphi}\in L^{2}\left(\mathbb{S}^{N}\right)$ satisfies

{\mathcal{S}_{\ast}\varphi}\left(\xi\right)=\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)\mathcal{T}_{\mathbb{S}^{N}}{{\mathcal{S}_{\ast}\varphi}}\left(\xi\right).

(3.20)

Proof.

By (2.2), (2.3) and the estimate $\displaystyle\left|\varphi\left(x\right)\right|\lesssim\frac{1}{\langle x\rangle^{N-2}}$ , we have

\int_{\mathbb{S}^{N}}\left|{\mathcal{S}_{\ast}\varphi}\left(\xi\right)\right|^{2}\mathrm{d}\xi=\int_{\mathbb{R}^{N}}\left|\varphi\left(x\right)\right|^{2}\rho^{4}\left(x\right)\mathrm{d}x\lesssim\int_{\mathbb{R}^{N}}\frac{1}{\langle x\rangle^{2N}}\mathrm{d}x<+\infty.

Now we turn to the proof of (3.20). First, by the definition of (3.10) with (1.8), and (2.1), we have

\displaystyle\displaystyle\mathfrak{N}_{1}\left(\varphi\right)\left(x\right)=

\displaystyle\displaystyle p\,2^{-(p-1)(N-2)}\rho^{N+2}\left(x\right)\int_{\mathbb{R}^{N}}\frac{1}{\left|\mathcal{S}x-\mathcal{S}y\right|^{\lambda}}\left[\rho^{2-N}\left(y\right)\varphi\left(y\right)\right]\rho^{2N}\left(y\right)\mathrm{d}y,

which together with (2.3) implies that

\displaystyle\displaystyle\mathfrak{N}_{1}\left(\varphi\right)\left(x\right)=

\displaystyle\displaystyle p\,2^{-(p-1)(N-2)}\rho^{N+2}\left(x\right)\int_{\mathbb{S}^{N}}\frac{1}{\left|\mathcal{S}x-\eta\right|^{\lambda}}{\mathcal{S}_{\ast}\varphi}\left(\eta\right)\mathrm{d}\eta.

(3.21)

By (2.1) and (3.21), we have

		$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left\|x-y\right\|^{N-2}}\mathfrak{N}_{1}\left(\varphi\right)\left(y\right)\mathrm{d}y$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle p\,2^{-(p-1)(N-2)}\rho^{N-2}\left(x\right)\int_{\mathbb{R}^{N}}\frac{1}{\left\|\mathcal{S}x-\mathcal{S}y\right\|^{N-2}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\mathcal{S}y-\sigma\right\|^{\lambda}}{\mathcal{S}_{\ast}\varphi}\left(\sigma\right)\mathrm{d}\sigma\rho^{2N}\left(y\right)\mathrm{d}y,$		(3.22)

which together with the stereographic projection implies that

	$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left\|x-y\right\|^{N-2}}$	$\displaystyle\displaystyle\mathfrak{N}_{1}\left(\varphi\right)\left(y\right)\mathrm{d}y$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle p\,2^{-(p-1)(N-2)}\rho^{N-2}\left(x\right)\int_{\mathbb{S}^{N}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\mathcal{S}x-\eta\right\|^{N-2}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}{\mathcal{S}_{\ast}\varphi}\left(\sigma\right)\mathrm{d}\sigma\mathrm{d}\eta.$		(3.23)

Next, by the definition of (3.10) with (1.8) and (2.1), we have

\displaystyle\displaystyle\mathfrak{N}_{2}\left(\varphi\right)\left(x\right)=

\displaystyle\displaystyle(p-1)\,2^{-(p-1)(N-2)}\rho^{4}\left(x\right)\varphi\left(x\right)\int_{\mathbb{R}^{N}}\frac{1}{\left|\mathcal{S}x-\mathcal{S}y\right|^{\lambda}}\rho^{2N}\left(y\right)\mathrm{d}y,

which together with the stereographic projection implies that

\displaystyle\displaystyle\mathfrak{N}_{2}\left(\varphi\right)\left(x\right)=(p-1)\,2^{-(p-1)(N-2)}\rho^{4}\left(x\right)\varphi\left(x\right)\int_{\mathbb{S}^{N}}\frac{1}{\left|\mathcal{S}x-\eta\right|^{\lambda}}\mathrm{d}\eta.

(3.24)

By (2.1) and (3.24), we have

		$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left\|x-y\right\|^{N-2}}\mathfrak{N}_{2}\left(\varphi\right)\left(y\right)\mathrm{d}y$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle(p-1)\,2^{-(p-1)(N-2)}\rho^{N-2}\left(x\right)\int_{\mathbb{R}^{N}}\frac{1}{\left\|\mathcal{S}x-\mathcal{S}y\right\|^{N-2}}\int_{\mathbb{S}^{N}}\frac{1}{\left\|\mathcal{S}y-\sigma\right\|^{\lambda}}\mathrm{d}\sigma\,\rho^{2-N}\left(y\right)\varphi\left(y\right)\rho^{2N}\left(y\right)\mathrm{d}y.$		(3.25)

By the stereographic projection and (2.3), we have

		$\displaystyle\displaystyle\int_{\mathbb{R}^{N}}\frac{1}{\left\|x-y\right\|^{N-2}}\mathfrak{N}_{2}\left(\varphi\right)\left(y\right)\mathrm{d}y$
	$\displaystyle\displaystyle=$	$\displaystyle\displaystyle(p-1)\,2^{-(p-1)(N-2)}\rho^{N-2}\left(x\right)\int_{\mathbb{S}^{N}}\frac{1}{\left\|\mathcal{S}x-\eta\right\|^{N-2}}{\mathcal{S}_{\ast}\varphi}\left(\eta\right)\int_{\mathbb{S}^{N}}\frac{1}{\left\|\eta-\sigma\right\|^{\lambda}}\mathrm{d}\eta\mathrm{d}\sigma.$		(3.26)

Inserting (3.23) and (3.26) into (3.15), we obtain

\rho^{2-N}\left(x\right)\varphi\left(x\right)=\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)\mathcal{T}_{\mathbb{S}^{N}}{{\mathcal{S}_{\ast}\varphi}}\left(\mathcal{S}x\right),

which together with the fact (2.3) implies the result. ∎

Now we can use the spherical harmonic decomposition and the Funk-Hecke formula of the spherical harmonic functions in [4, 8, 51] to classify the solution of the equation (3.20) on the sphere $\displaystyle\mathbb{S}^{N}$ .

Proposition \theproposition.

Let $\displaystyle N\geqslant 3$ , $\displaystyle\lambda\in\left(0,N\right)$ , and $\displaystyle\mathcal{T}_{\mathbb{S}^{N}}$ be defined by (3.17). If $\displaystyle\Phi\in L^{2}\left(\mathbb{S}^{N}\right)\setminus\left\{0\right\}$ satisfies

\Phi\left(\xi\right)=\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)\mathcal{T}_{\mathbb{S}^{N}}\Phi\left(\xi\right),

(3.27)

then $\displaystyle\Phi\in\mathscr{H}_{1}^{N+1}$ .

Proof.

On the one hand, by (2.7), we decompose $\displaystyle\Phi\in L^{2}\left(\mathbb{S}^{N}\right)\setminus\left\{0\right\}$ as the following.

\Phi\left(\xi\right)=\sum_{k=0}^{\infty}\sum_{j=1}^{\dim{\mathscr{H}_{k}^{N+1}}}{\Phi_{k,j}}Y_{k,j}\left(\xi\right),

(3.28)

where $\displaystyle\displaystyle{\Phi_{k,j}}=\int_{\mathbb{S}^{N}}\Phi\left(\xi\right)Y_{k,j}\left(\xi\right)\mathrm{d}\xi.$

Combining (3.27), (3.28) with Section 2, we have for any $\displaystyle k\geqslant 0$ and $\displaystyle 1\leqslant j\leqslant\dim{\mathscr{H}_{k}^{N+1}}$ ,

{\Phi_{k,j}}=\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)2^{-(p-1)(N-2)}\mu_{k}\left(N-2\right)\left[{p\mu_{k}\left(\lambda\right)+\left(p-1\right)\mu_{0}\left(\lambda\right)}\right]{\Phi_{k,j}},

(3.29)

On the other hand, by (2.11) and (2.12), by a direct computation, we obtain for $\displaystyle k=1$ that

\displaystyle\displaystyle\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)2^{-(p-1)(N-2)}\mu_{1}\left(N-2\right)\left[{p\mu_{1}\left(\lambda\right)+\left(p-1\right)\mu_{0}\left(\lambda\right)}\right]=1,

(3.30)

which, together with (2.13), implies for $\displaystyle k=0$ that

\displaystyle\displaystyle\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)2^{-(p-1)(N-2)}\mu_{0}\left(N-2\right)\left[{p\mu_{0}\left(\lambda\right)+\left(p-1\right)\mu_{0}\left(\lambda\right)}\right]>1,

(3.31)

and for all $\displaystyle k\geqslant 2$ that

\displaystyle\displaystyle\mathrm{G}\left(N\right){\bm{\alpha}}\left(N,\lambda\right)2^{-(p-1)(N-2)}\mu_{k}\left(N-2\right)\left[{p\mu_{k}\left(\lambda\right)+\left(p-1\right)\mu_{0}\left(\lambda\right)}\right]<1.

(3.32)

Therefore, by inserting (3.30), (3.31) and (3.32) into (3.29), we get for any $\displaystyle k=0$ and $\displaystyle k\geqslant 2$ that

\Phi_{k,j}=0,\quad\text{for }\quad 1\leqslant j\leqslant\dim{\mathscr{H}_{k}^{N+1}}.

Hence, we obtain that $\displaystyle\Phi\in\mathscr{H}_{1}^{N+1}$ from (3.28). ∎

Proof of Theorem 1.3.

Let $\displaystyle\varphi\in L^{\infty}(\mathbb{R}^{N})$ satisfy (1.11).

Firstly, by Section 3, the function $\displaystyle\varphi$ satisfies (3.15) and the estimate $\displaystyle\left|\varphi\left(x\right)\right|\lesssim\langle x\rangle^{-(N-2)}$ .

Secondly, by Section 3, we have $\displaystyle{\mathcal{S}_{\ast}\varphi}\in L^{2}\left(\mathbb{S}^{N}\right)$ satisfies (3.20). By Proposition 3, we get $\displaystyle{\mathcal{S}_{\ast}\varphi}\in\mathscr{H}_{1}^{N+1}$ , which together with (2.8) implies that

{\mathcal{S}_{\ast}\varphi}\in\mathrm{span}\left\{\xi_{j}\,\middle|\,1\leqslant j\leqslant N+1\right\}.

(3.33)

Lastly, by Proposition 2, we have

\varphi\in\mathrm{span}\left\{\varphi_{j}\,\middle|\,1\leqslant j\leqslant N+1\right\}.

(3.34)

This completes the proof of Theorem 1.3. ∎

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