Integral representations for the solutions of infinite order of the stationary Schrödinger equation in a cone

Abstract

Each solution of infinite order of the stationary Schrödinger equation defined in a smooth cone and continuous in the closure can be represented in terms of the modified Poisson integral and an infinite series vanishing continuously on the boundary.

1 Introduction and results

Let $\mathbf R$ and $\mathbf{R}_+$ be the set of all real numbers and the set of all positive real numbers, respectively. We denote by $\mathbf{R}^{n} (n\ge 2)$ the $n$-dimensional Euclidean space. A point in $\mathbf{R}^{n}$ is denoted by $P=(X,x_n),$ $X=(x_1,x_2,\ldots ,x_{n-1}).$ The Euclidean distance of two points $P$ and $Q$ in $\mathbf{R}^{n}$ is denoted by $|P-Q|$. Also $|P-O|$ with the origin $O$ of $\mathbf{R}^{n}$ is simply denoted by $|P|$. The boundary and the closure of a set $\mathbf S$ in $\mathbf{R}^{n}$ are denoted by $\partial \mathbf{S}$ and $\overline{\mathbf{S}}$, respectively.

We introduce a system of spherical coordinates $(r,\Theta ),$ $\Theta =(\theta _1,\theta _2,\ldots ,\theta _{n-1}),$ in $\mathbf{R}^{n}$ which are related to cartesian coordinates $(X,x_n)=(x_1,x_2,\ldots ,x_{n-1},x_n)$ by $x_n=r\cos \theta _1$.

For $P\in \mathbf{R}^{n}$ and $r>0$, let $B(P,r)$ denote the open ball with center at $P$ and radius $r$ in $\mathbf{R}^{n}$. $S_{r}=\partial {B(O,r)}$. The unit sphere and the upper half unit sphere in $\mathbf{R}^{n}$ are denoted by $\mathbf{S}^{n-1}$ and $\mathbf{S}_{+}^{n-1}$, respectively. For simplicity, a point $(1,\Theta )$ on $\mathbf{S}^{n-1}$ and the set $\{\Theta ; (1,\Theta )\in \Omega \}$ for a set $\Omega$, $\Omega \subset \mathbf{S}^{n-1},$ are often identified with $\Theta$ and $\Omega$, respectively. For two sets $\Lambda \subset \mathbf{R}_+$ and $\Omega \subset \mathbf{S}^{n-1},$ the set $\{(r,\Theta )\in \mathbf{R}^{n}; r\in \Lambda ,(1,\Theta )\in \Omega \}$ in $\mathbf{R}^{n}$ is simply denoted by $\Lambda \times \Omega .$ In particular, the half space $\mathbf{R}_{+}\times \mathbf{S}_{+}^{n-1}=\{(X,x_n)\in \mathbf{R}^{n}; x_n>0\}$ will be denoted by $\mathbf{T}_n$.

By $C_n(\Omega )$, we denote the set $\mathbf{R}_+\times \Omega$ in $\mathbf{R}^{n}$ with the domain $\Omega$ on $\mathbf{S}^{n-1}.$ We call it a cone. We denote the sets $I\times \Omega$ and $I\times \partial {\Omega }$ with an interval on $\mathbf R$ by $C_n(\Omega ;I)$ and $S_n(\Omega ;I)$. By $S_n(\Omega ; r)$ we denote $C_n(\Omega )\cap S_{r}$. By $S_n(\Omega )$ we denote $S_n(\Omega ; (0,+\infty ))$ which is $\partial {C_n(\Omega )}-\{O\}.$

Furthermore, we denote by $d\sigma _{Q}$ (resp. $dS_{r}$) the $(n-1)$-dimensional volume elements induced by the Euclidean metric on $\partial {C_n(\Omega )}$ (resp. $S_{r}$) and by $dw$ the elements of the Euclidean volume in $\mathbf{R}^{n}$.

Let $\fancyscript{A}_a$ denote the class of nonnegative radial potentials $a(P)$, i.e. $0\le a(P)=a(r)$, $P=(r,\Theta )\in C_n(\Omega )$, such that $a\in L_{loc}^{b}(C_n(\Omega ))$ with some $b> {n}/{2}$ if $n\ge 4$ and with $b=2$ if $n=2$ or $n=3$.

This article is devoted to the stationary Schrödinger equation

$$\begin{aligned} Sch_au(P)=-\Delta u(P)+a(P)u(P)=0\quad \mathrm{{for}}\quad P\in C_n(\Omega ), \end{aligned}$$

where $\Delta$ is the Laplace operator and $a\in \fancyscript{A}_a$. These solutions are called $a$-harmonic functions or generalized harmonic functions (g.h.f.s) associated with the operator $Sch_a$. Note that they are classical harmonic functions in the case $a=0$. Under these assumptions the operator $Sch_a$ can be extended in the usual way from the space $C_0^{\infty }(C_n(\Omega ))$ to an essentially self-adjoint operator on $L^{2}(C_n(\Omega ))$ (see [11, Ch. 13]). We will denote it $Sch_a$ as well. This last one has a Green function $G(\Omega ,a)(P,Q)$ which is positive on $C_n(\Omega )$ and its inner normal derivative $\partial G(\Omega ,a)(P,Q)/{\partial n_Q}\ge 0$, where ${\partial }/{\partial n_{Q}}$ denotes the differentiation at $Q$ along the inward normal into $C_n(\Omega )$. We denote this derivative $P(\Omega ,a)(P,Q)$, which is called the Poisson $a$-kernel with respect to $C_n(\Omega )$.

Let $\Delta ^{*}$ be a Laplace-Beltrami operator (spherical part of the Laplace) on $\Omega \subset \mathbf{S}^{n-1}$ and $\lambda _j~(j=1,2,3\ldots , 0<\lambda _1<\lambda _2\le \lambda _3\le \ldots )$ be the eigenvalues of the eigenvalue problem for $\Delta ^{*}$ on $\Omega$ (see, e.g., [12, p. 41])

$$\begin{aligned}&\Delta ^{*}\varphi (\Theta )+\lambda \varphi (\Theta )=0 \quad \mathrm{{in}} \quad \Omega ,\\&\varphi (\Theta )=0 \quad \mathrm{{on}} \quad \partial {\Omega }. \end{aligned}$$

Corresponding eigenfunctions are denoted by $\varphi _{j}(\Theta )$. We set $\lambda _0=0$, norm the eigenfunctions in $L^2(\Omega )$ and $\varphi _1(\Theta )>0$.

In order to ensure the existences of $\lambda _j~(j=1,2,3,\ldots )$. We put a rather strong assumption on $\Omega$: if $n\ge 3,$ then $\Omega$ is a $C^{2,\alpha }$-domain $(0<\alpha <1)$ on $\mathbf{S}^{n-1}$ surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [5, p. 88–89] for the definition of $C^{2,\alpha }$-domain), $\varphi _j\in C^2(\overline{\Omega })$ $(j=1,2,3,\ldots )$ and ${\partial \varphi _1}/{\partial n}>0$ on $\partial {\Omega }$ (here and below, ${\partial }/{\partial n}$ denotes differentiation along the interior normal).

Here well-known estimates (see, e.g., [2], and also [4, p. 120 and p. 126–128] imply the following inequalities:

$$\begin{aligned} M_1j^{\frac{2}{n-1}}\le \lambda _j ~~(j=1,2,3,\ldots ) \end{aligned}$$

(1.1)

and

$$\begin{aligned} |\varphi _j(\Theta )|\le M_2j^{\frac{1}{2}}~~~~(\Theta \in \Omega , j=1,2,3,\ldots ), \end{aligned}$$

(1.2)

where $M_1$ and $M_2$ are two positive constants.

Let $V_j(r)$ and $W_j(r)$ stand, respectively, for the increasing and non-increasing, as $r\rightarrow +\infty$, solutions of the equation

$$\begin{aligned} -Q^{\prime \prime }(r)-\frac{n-1}{r}Q^{\prime }(r)+\left( \frac{\lambda _j}{r^2}+a(r)\right) Q(r)=0, \quad 0<r<\infty , \end{aligned}$$

(1.3)

normalized under the condition $V_j(1)=W_j(1)=1$.

We will also consider the class $\fancyscript{B}_a$, consisting of the potentials $a\in \fancyscript{A}_a$ such that there exists the finite limit $\lim \nolimits _{r\rightarrow \infty }r^2 a(r)=k\in [0,\infty )$, moreover, $r^{-1}|r^2 a(r)-k|\in L(1,\infty )$. If $a\in \fancyscript{B}_a$, then the g.h.f.s are continuous (see [13]).

In the rest of paper, we assume that $a\in \fancyscript{B}_a$ and we shall suppress this assumption for simplicity. Meanwhile, we use the standard notations $u^{+}=\max \{u,0\}$, $u^{-}=-\min \{u,0\}$ and $[d]$ is the integer part of $d$, where $d$ is a positive real number.

Denote

$$\begin{aligned} \iota _{j,k}^{\pm }=\frac{2-n\pm \sqrt{(n-2)^2+4(k+\lambda _j)}}{2}\quad (j=0,1,2,3\ldots ). \end{aligned}$$

The solutions to Eq. (1.3) have the asymptotic (see [6])

$$\begin{aligned} V_j(r)\sim M_3 r^{\iota _{j,k}^{+}}, W_j(r)\sim M_4 r^{\iota _{j,k}^{-}}, \mathrm{as}\,\, r\rightarrow \infty , \end{aligned}$$

(1.4)

where $M_3$ and $M_4$ are some positive constants.

Further, we have

$$\begin{aligned} \iota _{j,k}^{+}\ge \iota _{j,0}^{+}> M_5 j^{\frac{1}{n-1}}\quad (j=1,2,3\ldots ) \end{aligned}$$

(1.5)

from (1.1), where $M_5$ is a positive constant independent of $j$.

If $a\in \fancyscript{A}_a$, it is known that the following expansion for the Green function $G(\Omega ,a)(P,Q)$ (see [3, Ch. 11])

$$\begin{aligned} G(\Omega ,a)(P,Q)=\sum _{j=0}^{\infty }\frac{1}{\chi ^{\prime }(1)}V_j(\min \{r,t\}) W_j(\max \{r,t\})\varphi _{j}(\Theta )\varphi _{j}(\Phi ), \end{aligned}$$

where $P=(r,\Theta )$, $Q=(t,\Phi )$, $r\ne t$ and $\chi ^{\prime }(t)=w\left( W_1(r),V_1(r)\right) |_{r=1}$ is their Wronskian. This series converges uniformly if either $r\le s t$ or $t\le s r$ $(0<s<1)$. In the case $a=0$, this expansion coincides with the well-known result by Lelong-Ferrand (see [9]).

For a nonnegative integer $m$ and two points $P=(r,\Theta ), Q=(t,\Phi )\in C_n(\Omega )$, we put

$$\begin{aligned} K(\Omega ,a,m)(P,Q)=\left\{ \begin{array}{ll} 0 &{} \quad \text{ if }\ \ 0<t<1, \\ \widetilde{K}(\Omega ,a,m)(P,Q) &{}\quad {\text{ if }} \ \ 1\le t<\infty , \end{array}\right. \end{aligned}$$

where

$$\begin{aligned} \widetilde{K}(\Omega ,a,m)(P,Q)=\sum _{j=0}^{m}\frac{1}{\chi ^{\prime }(1)}V_j(r)W_j(t)\varphi _{j} (\Theta )\varphi _{j}(\Phi ). \end{aligned}$$

To obtain the modified Poisson integral representation for the Schrödinger operator in a cone, we use the following modified kernel function defined by

$$\begin{aligned} G(\Omega ,a,m)(P,Q)=G(\Omega ,a)(P,Q)-K(\Omega ,a,m)(P,Q) \end{aligned}$$

for two points $P=(r,\Theta ), Q=(t,\Phi )\in C_n(\Omega ).$

Write

$$\begin{aligned} U(\Omega ,a,m;u)(P)=\int _{S_n(\Omega )}P(\Omega ,a,m)(P,Q)u(Q)d\sigma _Q, \end{aligned}$$

where

$$\begin{aligned} P(\Omega ,a,m)(P,Q)=\frac{\partial G(\Omega ,a,m)(P,Q)}{\partial n_Q},\quad P(\Omega ,a,0)(P,Q)=P(\Omega ,a)(P,Q) \end{aligned}$$

and $u(Q)$ is a continuous function on $\partial C_n(\Omega )$.

Now we define the function $\rho (R)$ under consideration. Hereafter, the function $\rho (R)~(\ge 1)$ is always supposed to be nondecreasing and continuously differentiable on the interval $[0,+\infty )$. We assume further that

$$\begin{aligned} \epsilon _{0}=\limsup _{R\rightarrow \infty } \frac{(\iota _{{[\rho (R)]+1},k}^{+})^{\prime }R\ln R}{\iota _{{[\rho (R)]+1},k}^{+}}<1. \end{aligned}$$

(1.6)

Remark

$\iota _{{[\rho (R)]+1},k}^{+}$ in (1.6) is not the function $V_j(R)$. For any $\epsilon$ $(0<\epsilon <1-\epsilon _0)$, there exists a sufficiently large positive number $R_{\epsilon }$ such that $R>R_{\epsilon }$, by (1.5) and (1.6) we have

$$\begin{aligned} M(\rho (R))^{\frac{1}{n-1}}<\iota _{{[\rho (R)]+1},k}^{+}< \iota _{{[\rho (e)]+1,k}}^{+}(\ln R)^{\epsilon _0+\epsilon }, \end{aligned}$$

where $M$ is a positive constant.

For positive real numbers $\beta$, we denote $\mathcal C _{\Omega ,\beta ,a}$ the class of all measurable functions $f(t,\Phi )$ $(Q=(t,\Phi )\in C_n(\Omega ))$ satisfying the following inequality

$$\begin{aligned} \int _{C_n(\Omega )}\frac{|f(t,\Phi )|\varphi _1}{1+V_{[\rho (t)]+1} (t)t^{n+\beta -1}}dw<\infty \end{aligned}$$

(1.7)

and the class $\mathcal D _{\Omega ,\beta ,a}$, consists of all measurable functions $g(t,\Phi )\,(Q=(t,\Phi )\in S_n(\Omega ))$ satisfying

$$\begin{aligned} \int _{S_n(\Omega )}\frac{|g(t,\Phi )|V_1(t)W_1(t)}{1+\chi ^{\prime }(t)V_{[\rho (t)]+1}(t)t^{n+\beta -2}} \frac{\partial \varphi _1}{\partial n}d\sigma _Q<\infty , \end{aligned}$$

(1.8)

where $\chi ^{\prime }(t)=w\left( W_1(r),V_1(r)\right) |_{r=t}$ is their Wronskian.

We will also consider the class of all continuous functions $u(t,\Phi )$ $((t,\Phi )\in \overline{C_n(\Omega )})$ generalized harmonic in $C_n(\Omega )$ with $u^{+}(t,\Phi )\in \mathcal C _{\Omega ,\beta ,a}\,((t,\Phi )\in C_n(\Omega ))$ and $u^{+}(t,\Phi )\in \mathcal D _{\Omega ,\beta ,a}\,((t,\Phi )\in S_n(\Omega ))$ is denoted by $\mathcal E _{\Omega ,\beta ,a}$.

Next we define the order of g.h.f, which is similar to the F. Riesz’ definition for the order of classical harmonic function (see [7, Definition 4.1]). We shall say that a g.h.f.-$u(P)\,(P=(r,\Theta )\in C_n(\Omega ))$ is of order $\lambda$ if

$$\begin{aligned} \lambda =\limsup _{r\rightarrow \infty } \frac{\log \left( \sup _{C_n(\Omega )\cap S_{r}}|u|\right) }{\log r}. \end{aligned}$$

If $\lambda <\infty$, then $u$ is said to be of finite order.

In case $\lambda <\infty$, about the solutions of the Dirichlet problem for the Schrödinger operator with continuous data in $\mathbf{T}_n$, we refer the readers to the paper by Kheyfits (see [8]).

Motivated by Kheyfits’s conclusions, we prove the following results for the g.h.f.s of infinite order. In the case $a=0$, we refer readers to the paper by Qiao (see [10]).

Theorem 1

If $u\in \mathcal E _{\Omega ,\beta ,a}$, then $u\in \mathcal D _{\Omega ,\beta ,a}$.

Theorem 2

If $u\in \mathcal E _{\Omega ,\beta ,a}$, then the following properties hold:

• (I) $U(\Omega , a,[\rho (t)];u)(P)$ is a g.h.f. on $C_n(\Omega )$ and can be continuously extended to $\overline{C_n(\Omega )}$ such that $U(\Omega , a,[\rho (t)];u)(P)=u(P)$ for $P=(r,\Theta )\in S_n(\Omega )$.

• (II) There exists an infinite series $h(P)=\sum \nolimits _{j=1}^{\infty }A_jV_j(r)\varphi _j(\Theta )$ vanishing continuously on $\partial {C_n(\Omega )}$ such that

  $$\begin{aligned} u(P)=U(\Omega , a,[\rho (t)];u)(P)+h(P) \end{aligned}$$

  for $P=(r,\Theta )\in C_n(\Omega )$, where $A_j$ $(j=1,2,3,\ldots )$ is a constant.

2 Lemmas

The following Lemma generalizes the Carleman’s formula (referring to the holomorphic functions in the half space) (see [1]) to the g.h.f.s in a cone, which is due to Levin and Kheyfits (see [3, Ch. 11]).

Lemma 1

If $u(t,\Phi )$ is a g.h.f. on a domain containing $C_n(\Omega ;(1,R))$, then

$$\begin{aligned}&m_{+}(R)+\int _{S_n(\Omega ;(1,R))}u^{+}\Psi (t) \frac{\partial \varphi _1}{\partial n} d\sigma _Q+M_6+\frac{W_1(R)}{V_1(R)}M_7=m_{-}(R)\\&\quad + \int _{S_n(\Omega ;(1,R))}u^{-}\Psi (t) \frac{\partial \varphi _1}{\partial n} d\sigma _Q, \end{aligned}$$

where

$$\begin{aligned} \Psi (t)=W_1(t)-\frac{W_1(R)}{V_1(R)}V_1(t),~~~~m_{\pm }(R)= \int _{S_n(\Omega ;R)}\frac{\chi ^{\prime }(R)}{V_1(R)}u^{\pm }\varphi _1 d S_{R}, \end{aligned}$$

$$\begin{aligned} M_6=\int _{S_n(\Omega ;1)}u\varphi _1W^{\prime }_1(1)-W_1(1)\varphi _1 \frac{\partial u}{\partial n}dS_1\quad \mathrm{{and}} \quad M_7=\int _{S_n(\Omega ;1)}V_1(1)\varphi _1\frac{\partial u}{\partial n}-u\varphi _1V^{\prime }_1(1) dS_1. \end{aligned}$$

Lemma 2

(see [3, Ch. 11]) For a non-negative integer $m$, we have

$$\begin{aligned} |P(\Omega ,a,m)(P,Q)|\le M_8V_{m+1}(2r)\frac{W_{m+1}(t)}{t}\varphi _1(\Theta )\frac{\partial \varphi _1(\Phi )}{\partial n_\Phi } \end{aligned}$$

(2.1)

for any $P=(r,\Theta )\in C_n(\Omega )$ and $Q=(t,\Phi )\in S_n(\Omega )$ satisfying $2r\le t$, where $M_8$ is a constant depending only $n$.

Lemma 3

If $h(r,\Theta )$ is a g.h.f. in $C_n(\Omega )$ vanishing continuously on $\partial {C_n(\Omega )}$, then

$$\begin{aligned} h(r,\Theta )=\sum _{j=1}^{\infty }B_jV_j(r)\varphi _j(\Theta ), \end{aligned}$$

(2.2)

where the series converges uniformly and absolutely in any compact set of $\overline{C_n(\Omega )}$, and $B_j$ $(j=1,2,3,\ldots )$ is a constant satisfying

$$\begin{aligned} B_jV_j(r)=\int _\Omega h(r,\Theta )\varphi _j(\Theta )d S_1 \end{aligned}$$

(2.3)

for every $r~(0<r<\infty )$.

Proof

Set

$$\begin{aligned} y_{j}(r)=\int _\Omega h(r,\Theta )\varphi _j(\Theta )d S_1~~~~~(j=1,2,3,\ldots ). \end{aligned}$$

Making use of the assumptions on $h$ and self-adjoint of the Laplace-Beltrami operator $\Delta ^{*}$, one can check directly (by differentiating under the integral sign) that the functions $y_j~(j=1,2,3,\ldots )$ satisfy the Eq. (1.3). This equation has a general solution $y_j(r)=B_jV_j(r)+D_jW_j(r)$, where $B_j$ and $D_j$ are constants independent of $r~(j=1,2,3,\ldots )$. We note that $h(r,\Theta )$ converges uniformly to zero as $r\rightarrow 0$ and hence $\lim \nolimits _{r\rightarrow 0}y_j(r)=0~(j=1,2,3,\ldots ).$ Thus we see that $D_j=0~(j=1,2,3,\ldots )$. Since $y_j(r)$ takes the value $y_j(r_1)$ at $r=r_1$, we have

$$\begin{aligned} y_j(r)=\frac{V_j(r)}{V_j(r_1)}y_j(r_1) \end{aligned}$$

for any $r$ and $r_1$ $(0<r,r_1<R_1)$, where $0<R_1\le +\infty$. In particular, if $R_1=\infty$, then

$$\begin{aligned} \lim _{r\rightarrow \infty }\frac{y_j(r)}{V_j(r)}=B_{j}~~~(j=1,2,3,\ldots ) \end{aligned}$$

(2.4)

exists.

Since $y_j(r_1)\rightarrow y_j(R_1)$ as $r_1\rightarrow R_1$, we see that

$$\begin{aligned} y_j(r)=\frac{V_j(r)}{V_j(R_1)}y_j(R_1)~~~(j=1,2,3,\ldots ), \end{aligned}$$

(2.5)

which gives

$$\begin{aligned} |y_j(r)|\le M_2w_n\left( \frac{r}{R_1}\right) ^{\iota _{j,k}^{+}}j^{\frac{1}{2}}\sup _{\Theta \in \Omega }|h(R_1,\Theta )| \end{aligned}$$

from (1.2) and (1.4), where $w_n$ is the surface area $2\pi ^{n/2}\{\Gamma (n/2)\}^{-1}$ of $\mathbf{S}^{n-1}$.

Now we set

$$\begin{aligned} J=\max \left\{ j;~\iota _{j,k}^{+}-\iota _{{l+1},k}^{+} <\frac{1}{2}M_5j^{\frac{1}{n-1}}\right\} ~~(l=1,2,3,\ldots ). \end{aligned}$$

The existence of this $J$ is known from (1.5). Hence if we put

$$\begin{aligned} M_{9}=w_nM_2^2\left\{ \sum _{j=l+1}^{J}j\left( \frac{1}{2}\right) ^{\iota _{j,k}^{+} -\iota _{{m+1},k}^{+}}+\sum _{j=J+1}^{\infty } j\left( \frac{1}{2}\right) ^{2^{-1}M_5j^{\frac{1}{n-1}}}\right\} \end{aligned}$$

and use (1.2), then from the completeness of $\{\varphi _j(\Theta )\}$ we can expand $h(r,\Theta )$ into the Fourier series

$$\begin{aligned} h(r,\Theta )=\sum _{j=1}^{\infty }y_j(r)\varphi _j(\Theta ) \end{aligned}$$

satisfying

$$\begin{aligned} \sum _{j=l+1}^{\infty }|y_j(r)||\varphi _j(\Theta )|\le M_{9}\left( \frac{r}{R_1}\right) ^{\iota _{{l+1},k}^{+}}\sup _{\Theta \in \Omega }|h(R_1,\Theta )|~~(l=1,2,3,\ldots ) \qquad \end{aligned}$$

(2.6)

on $C_n(\Omega ;(0,\frac{R_1}{2}))$, where $M_{9}$ is a positive constant independent of $r$ and $R_1$.

Take any compact $H$, $H\subset \overline{C_n(\Omega )}$ and a number $R_1$ satisfying $R_1>2\max \{r;(r,\Theta )\in H\}$. So we can represent $h(r,\Theta )$ as

$$\begin{aligned} h(r,\Theta )=\sum _{j=1}^{\infty }y_j(r)\varphi _j(\Theta ), \end{aligned}$$

(2.7)

where $(r,\Theta )$ is a point in $H$. Hence we observe in (2.5) that $y_j(r)$ is a number independent of $R_1$. Hence as $R_1\rightarrow \infty$, we see from (2.4) that $y_j(r)=B_jV_j(r)$, which is (2.3). This and (2.7) give (2.2).

To prove the absolute and uniform convergence of (2.7) on $H$, see from (2.6) that

$$\begin{aligned} \sum _{j=l+1}^{\infty }|y_j(r)||\varphi _j(\Theta )|\le M_{9}2^{-\iota _{{l+1},k}^{+}}\sup _{\Theta \in \Omega }|h(R_1,\Theta )|, \end{aligned}$$

which converges to 0 as $l\rightarrow \infty$. Then Lemma 3 is proved. $\square$

3 Proof of Theorem 1

Since $u\in \mathcal E _{\Omega ,\beta ,a}$, we obtain by (1.7)

$$\begin{aligned} \int _{1}^{\infty }\frac{m_{+}(R)V_1(R)}{\chi ^{\prime }(R)V_{[\rho (R)]+1}(R)R^{n+\beta -1}}dR \!\le \! 2\int _{C_n(\Omega )}\frac{u^{+}\varphi _1}{1+V_{[\rho (t)]+1}(t)t^{n+\beta -1}}dw\!<\!\infty . \qquad \quad \end{aligned}$$

(3.1)

From (1.4) and (1.8), we conclude that

$$\begin{aligned}&\int _{1}^{\infty }\frac{V_1(R)}{\chi ^{\prime }(R)V_{[\rho (R)]+1} (R)R^{n+\beta -1}}\int _{S_n(\Omega ;(1,R))}u^{+}\Psi (t) \frac{\partial \varphi _1}{\partial n} d\sigma _Q d R\nonumber \\&\quad \le \frac{2\chi _{1,k}}{(\chi _{1,k}+\beta )\beta } \int _{S_n(\Omega )}\frac{u^{+}V_1(t)W_1(t)}{1+\chi ^{\prime }(t)V_{[\rho (t)]+1} (t)t^{n+\beta -2}}\frac{\partial \varphi _1}{\partial n}d\sigma _Q \nonumber \\&\quad <\infty , \end{aligned}$$

(3.2)

where $\chi _{1,k}=\iota _{1,k}^{+}-\iota _{1,k}^{-}.$

It follows from (1.4), Remark and the L’hospital’s rule

$$\begin{aligned} \lim _{t\rightarrow \infty }\frac{\chi ^{\prime }(t)V_{[\rho (t)]+1} (t)t^{n+\beta -2}}{W_1(t)}\int _{t}^{\infty }\frac{V_1(R)}{\chi ^{\prime }(R)V_{[\rho (R)]+1}(R)R^{n+\frac{\beta }{2}-1}} \left( \frac{W_1(t)}{V_1(t)}-\frac{W_1(R)}{V_1(R)}\right) d R=+\infty , \end{aligned}$$

which yields that there exists a positive constant $M_{10}$ such that for any $t\ge 1$,

$$\begin{aligned} \int _{t}^{\infty }\frac{V_1(R)}{\chi ^{\prime }(R)V_{[\rho (t)]+1} (R)R^{n+\frac{\beta }{2}-1}}\Psi (t)dR\ge \frac{M_{10}V_1(t)W_1(t)}{\chi ^{\prime }(t)V_{[\rho (t)]+1}(t)t^{n+\beta -2}}. \end{aligned}$$

From (3.1), (3.2) and Lemma 1 we see that

$$\begin{aligned}&M_{10}\int _{S_n(\Omega ;(1,\infty ))} \frac{u^{-}V_1(t)W_1(t)}{\chi ^{\prime }(t)V_{[\rho (t)]+1}(t)t^{n+\beta -2}} \frac{\partial \varphi _1}{\partial n}d\sigma _Q\\&\quad \le \int _{S_n(\Omega ;(1,\infty ))}u^{-}\int _{t}^{\infty } \frac{V_1(R)}{\chi ^{\prime }(R)V_{[\rho (t)]+1}(R)R^{n+\frac{\beta }{2}-1}} \Psi (t) dR \frac{\partial \varphi _1}{\partial n} d\sigma _Q\\&\quad <\infty . \end{aligned}$$

Then Theorem 1 is proved from $|u|=u^{+}+u^{-}$.

4 Proof of Theorem 2

Let $l_1$ be any positive number such that $l_1\ge 2\beta$. For any fixed $P=(r,\Theta )\in C_n(\Omega )$, take a number $\sigma$ satisfying $\sigma >\sigma _r=\max \{[2r]+1,\vartheta _r \}$, where $\vartheta _r=\exp (\frac{l_1}{\beta }\iota _{{[\rho (e)]+1,k}}^{+} 2^{1+\epsilon _0+\epsilon }\ln 2r)^{\frac{1}{1-\epsilon _0-\epsilon }}$.

From the Remark we see that there exists a constant $M(r)$ dependent only on $r$ such that $M(r)\ge (2r)^{\iota _{{[\rho (i+1)]+1,k}}^{+}}i^{-\frac{\beta }{l_1}}$ from $\sigma \ge \vartheta _r$.

By (1.4), (1.8), (2.1) and Theorem 1, we have

$$\begin{aligned}&\int _{S_n(\Omega ;(\sigma ,\infty ))}|P(\Omega ,a,[\rho (t)]) (P,Q)||u(Q)|d\sigma _{Q}\\&\quad \le M_8\varphi _1(\Theta )\sum _{i=\sigma _r}^{\infty } \int _{S_n(\Omega ;[i,i+1))}\frac{(2r)^{\iota _{{[\rho (t)]+1},k}^{+}}}{t^{\frac{\beta }{l_1}}}\frac{|u(t,\Phi )|}{V_{[\rho (t)]+1} (t)t^{n-2+\frac{\beta }{l_1}}}d\sigma _{Q} \\&\quad \le M_8\sum _{i=\sigma _r}^{\infty } \frac{(2r)^{\iota _{{[\rho (i+1)]+1,k}}^{+}}}{i^{\frac{\beta }{l_1}}} \int _{S_n(\Omega ;[i,i+1))}\frac{|u(t,\Phi )|}{V_{[\rho (t)]+1} (t)t^{n-2+\frac{\beta }{l_1}}}d\sigma _{Q}\\&\quad \le M_8M(r)\varphi _1(\Theta ) \int _{S_n(\Omega ;[\sigma _r,\infty ))}\frac{|u(t,\Phi )|}{V_{[\rho (t)]+1}(t)t^{n-2+\frac{\beta }{l_1}}}d\sigma _{Q}\\&\quad <\infty . \end{aligned}$$

Hence $U(\Omega ,a,[\rho (t)];u)(P)$ is absolutely convergent and finite for any $P\in C_n(\Omega )$. Thus $U(\Omega ,a,[\rho (t)];u)(P)$ is generalized harmonic on $C_n(\Omega )$.

Now we study the boundary behavior of $U(\Omega ,a,[\rho (t)];u)(P)$. Let $Q^{\prime }=(t^{\prime },\Phi ^{\prime })\in \partial {C_n(\Omega )}$ be any fixed point and $l_2$ be any positive number such that $l_2>t^{\prime }+1$.

Set $\chi _{S(l_2)}$ is the characteristic function of $S(l_2)=\{Q=(t,\Phi )\in \partial {C_n(\Omega )},t\le l_2\}$ and write

$$\begin{aligned} U(\Omega ,a,[\rho (t)];u)(P)=U^{\prime }(P)-U^{\prime \prime }(P)+U^{\prime \prime \prime }(P), \end{aligned}$$

where

$$\begin{aligned} U^{\prime }(P)&= \int _{S_n(\Omega ;(0,2l_2])}P(\Omega ,a)(P,Q)u(Q)d\sigma _Q,\\ U^{\prime \prime }(P)&= \int _{S_n(\Omega ;(1,2l_2])}\frac{\partial { K(\Omega ,a,[\rho (t)])(P,Q)}}{\partial n_Q}u(Q)d\sigma _Q \end{aligned}$$

and

$$\begin{aligned} U^{\prime \prime \prime }(P)=\int _{S_n(\Omega ;(2l_2,\infty ))}P (\Omega ,a,[\rho (t)])(P,Q)u(Q)d\sigma _Q. \end{aligned}$$

Notice that $U^{\prime }(P)$ is the Poisson integral of $u(Q)\chi _{S(2l_2)}$, we have $\lim \nolimits _{P\in C_n(\Omega ),P\rightarrow Q^{\prime }} U^{\prime }(P)=u(Q^{\prime })$. Since $\lim \nolimits _{\Theta \rightarrow \Phi ^{\prime }}\varphi _{j}(\Theta )=0~(j=1,2,3\ldots )$ as $P=(r,\Theta )\rightarrow Q^{\prime }=(t^{\prime },\Phi ^{\prime })\in S_n(\Omega )$, we have $\lim \nolimits _{P\in C_n(\Omega ),P\rightarrow Q^{\prime }}U^{\prime \prime }(P)=0$ from the definition of the kernel function $K(\Omega ,a,[\rho (t)])(P,Q)$. $U^{\prime \prime \prime }(P)=O(M(r)\varphi _1(\Theta ))$ and therefore tends to zero.

So the function $U(\Omega ,a,[\rho (t)];u)(P)$ can be continuously extended to $\overline{C_n(\Omega )}$ such that

$$\begin{aligned} \lim \limits _{P\in C_n(\Omega ),P\rightarrow Q^{\prime }}U(\Omega ,a,[\rho (t)];u)(P)=u(Q^{\prime }) \end{aligned}$$

for any $Q^{\prime }=(t^{\prime },\Phi ^{\prime })\in \partial {C_n(\Omega )}$ from the arbitrariness of $l_2$.

So (I) is proved. Finally (I) and Lemma 3 give the conclusion of (II). Then we complete the proof of Theorem 2.