Dirichlet problem for the Schrödinger operator on a cone

Abstract

In this article, a solution of the Dirichlet problem for the Schrödinger operator on a cone is constructed by the generalized Poisson integral with a slowly growing continuous boundary function. A solution of the Poisson integral for any continuous boundary function is also given explicitly by the Poisson integral with the generalized Poisson kernel depending on this boundary function.

MSC:31B05, 31B10.

1 Introduction and results

Let R and ${\mathbf{R}}_{+}$ be the set of all real numbers and the set of all positive real numbers respectively. We denote the n-dimensional Euclidean space by ${\mathbf{R}}^n$ ($n\ge 2$). A point in ${\mathbf{R}}^n$ is denoted by $P=(X,x_n)$, where $X=(x_1,x_2,\dots ,x_{n-1})$. The Euclidean distance between 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 S in ${\mathbf{R}}^n$ are denoted by ∂ S and $\overline{\mathbf{S}}$ respectively.

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

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 $(1,\mathrm{\Theta })$ on ${\mathbf{S}}^{n-1}$ and the set $\{\mathrm{\Theta };(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ for a set Ω, $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, are often identified with Θ and Ω, respectively. For two sets $\mathrm{\Xi }\subset {\mathbf{R}}_{+}$ and $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$, the set $\{(r,\mathrm{\Theta })\in {\mathbf{R}}^n;r\in \mathrm{\Xi },(1,\mathrm{\Theta })\in \mathrm{\Omega }\}$ in ${\mathbf{R}}^n$ is simply denoted by $\mathrm{\Xi }\times \mathrm{\Omega }$.

For $P\in {\mathbf{R}}^n$ and $r>0$, let $B(P,r)$ denote an open ball with a center at P and radius r in ${\mathbf{R}}^n$. $S_r=\partial B(O,r)$. By $C_n(\mathrm{\Omega })$, we denote the set ${\mathbf{R}}_{+}\times \mathrm{\Omega }$ in ${\mathbf{R}}^n$ with the domain Ω on ${\mathbf{S}}^{n-1}$. We call it a cone. We denote the sets $I\times \mathrm{\Omega }$ and $I\times \partial \mathrm{\Omega }$ with an interval on R by $C_n(\mathrm{\Omega };I)$ and $S_n(\mathrm{\Omega };I)$. By $S_n(\mathrm{\Omega };r)$ we denote $C_n(\mathrm{\Omega })\cap S_r$. By $S_n(\mathrm{\Omega })$ we denote $S_n(\mathrm{\Omega };(0,+\mathrm{\infty }))$ which is $\partial C_n(\mathrm{\Omega })-\{O\}$. We denote the $(n-1)$-dimensional volume elements induced by the Euclidean metric on $S_r$ by $d{S}_r$.

Let ${\mathcal{A}}_a$ denote the class of nonnegative radial potentials $a(P)$, i.e., $0\le a(P)=a(r)$, $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$, such that $a\in L_{\mathrm{loc}}^b(C_n(\mathrm{\Omega }))$ 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

$${Sch}_{a}u(P)=-\mathrm{\Delta }u(P)+a(P)u(P)=0,$$

(1.1)

where $P\in C_n(\mathrm{\Omega })$, Δ is the Laplace operator and $a\in {\mathcal{A}}_a$. These solutions called a-harmonic functions or generalized harmonic functions are 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 $C_0^{\mathrm{\infty }}(C_n(\mathrm{\Omega }))$ to an essentially self-adjoint operator on $L^2(C_n(\mathrm{\Omega }))$ (see [1–3]). We will denote it ${Sch}_a$ as well. This last one has a Green’s function $G(\mathrm{\Omega },a)(P,Q)$. Here $G(\mathrm{\Omega },a)(P,Q)$ is positive on $C_n(\mathrm{\Omega })$ and its inner normal derivative $\partial G(\mathrm{\Omega },a)(P,Q)/\partial n_Q\ge 0$. We denote this derivative by $\mathbb{P}(\mathrm{\Omega },a)(P,Q)$, which is called the Poisson a-kernel with respect to $C_n(\mathrm{\Omega })$. We remark that $G(\mathrm{\Omega },0)(P,Q)$ and $\mathbb{P}(\mathrm{\Omega },0)(P,Q)$ are the Green’s function and Poisson kernel of the Laplacian in $C_n(\mathrm{\Omega })$ respectively.

Given a domain $D\subset {\mathbf{R}}^n$ and a continuous function u on $\partial (D)$, we say that h is a solution of the Dirichlet problem for the Schrödinger operator on D with u if ${Sch}_{a}h=0$ in D and

$$\underset{P\in D,P\to Q}{lim}h(P)=u(Q)$$

for every $Q\in \partial (D)$. Note that h is a solution of the classical Dirichlet problem for the Laplacian in the case $a=0$.

Let ${\mathrm{\Delta }}^{\ast }$ be a Laplace-Beltrami operator (the spherical part of the Laplace) on $\mathrm{\Omega }\subset {\mathbf{S}}^{n-1}$ and ${\lambda }_j$ ($j=1,2,3,\dots ,0<{\lambda }_1<{\lambda }_2\le {\lambda }_3\le \cdots$) be the eigenvalues of the eigenvalue problem for ${\mathrm{\Delta }}^{\ast }$ on Ω (see, e.g., [4], p. 41])

Corresponding eigenfunctions are denoted by ${\phi }_{jv}$ ($1\le v\le v_j$), where $v_j$ is the multiplicity of ${\lambda }_j$. We set ${\lambda }_0=0$, norm the eigenfunctions in $L^2(\mathrm{\Omega })$ and ${\phi }_1={\phi }_{11}>0$. Then there exist two positive constants $d_1$ and $d_2$ such that

$$d_1\delta (P)\le {\phi }_1(\mathrm{\Theta })\le d_2\delta (P)$$

(1.2)

for $P=(1,\mathrm{\Theta })\in \mathrm{\Omega }$ (see Courant and Hilbert [5]), where $\delta (P)={inf}_{Q\in \partial C_n(\mathrm{\Omega })}|P-Q|$.

In order to ensure the existences of ${\lambda }_j$ ($j=1,2,3,\dots$). We put a rather strong assumption on Ω: if $n\ge 3$, then Ω 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 [6], pp. 88-89] for the definition of $C^{2,\alpha }$-domain). Then ${\phi }_{jv}\in C^2(\overline{\mathrm{\Omega }})$ ($j=1,2,3,\dots ,1\le v\le v_j$) and $\partial {\phi }_1/\partial n>0$ on ∂ Ω (here and below, $\partial /\partial n$ denotes differentiation along the interior normal).

Hence well-known estimates (see, e.g., [7], p. 14]) imply the following inequality:

$$\sum _{v=1}^{v_j}{\phi }_{jv}(\mathrm{\Theta })\frac{\partial {\phi }_{jv}(\mathrm{\Phi })}{\partial n_{\mathrm{\Phi }}}\le M(n)j^{2n-1},$$

(1.3)

where the symbol $M(n)$ denotes a constant depending only on n.

Let $V_j(r)$ and $W_j(r)$ stand, respectively, for the increasing and nonincreasing, as $r\to +\mathrm{\infty }$, solutions of the equation

$$-Q^{″}(r)-\frac{n-1}{r}{Q}^{\prime }(r)+(\frac{{\lambda }_j}{r^2}+a(r))Q(r)=0,0<r<\mathrm{\infty },$$

(1.4)

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

We shall also consider the class ${\mathcal{B}}_a$, consisting of the potentials $a\in {\mathcal{A}}_a$ such that there exists a finite limit ${lim}_{r\to \mathrm{\infty }}{r}^{2}a(r)=k\in [0,\mathrm{\infty })$; moreover, $r^{-1}|r^{2}a(r)-k|\in L(1,\mathrm{\infty })$. If $a\in {\mathcal{B}}_a$, then the solutions of Equation (1.1) are continuous (see [8]).

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

Denote

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

It is known (see [9]) that in the case under consideration the solutions to Equation (1.4) have the asymptotics

$$V_j(r)\sim d_3{r}^{{\iota }_{j,k}^{+}},W_j(r)\sim d_4{r}^{{\iota }_{j,k}^{-}},\text{as}r\to \mathrm{\infty },$$

(1.5)

where $d_3$ and $d_4$ are some positive constants.

If $a\in {\mathcal{A}}_a$, it is known that the following expansion for the Green function $G(\mathrm{\Omega },a)(P,Q)$ (see [10], Ch. 11], [1, 11])

$$G(\mathrm{\Omega },a)(P,Q)=\sum _{j=0}^{\mathrm{\infty }}\frac{1}{{\chi }^{\prime }(1)}{V}_j(min(r,t))W_j(max(r,t))(\sum _{v=1}^{v_j}{\phi }_{jv}(\mathrm{\Theta }){\phi }_{jv}(\mathrm{\Phi })),$$

where $P=(r,\mathrm{\Theta })$$Q=(t,\mathrm{\Phi })$$r\ne t$ and ${\chi }^{\prime }(s)=w(W_1(r),V_1(r)){|}_{r=s}$, is their Wronskian. The series converges uniformly if either $r\le st$ or $t\le sr$ ($0<s<1$).

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

$$K(\mathrm{\Omega },a,m)(P,Q)=\{\begin{array}{cc}0\hfill & \text{if}0<t<1,\hfill \\ \tilde{K}(\mathrm{\Omega },a,m)(P,Q)\hfill & \text{if}1\le t<\mathrm{\infty },\hfill \end{array}$$

where

$$\tilde{K}(\mathrm{\Omega },a,m)(P,Q)=\sum _{j=0}^m\frac{1}{{\chi }^{\prime }(1)}{V}_j(r)W_j(t)(\sum _{v=1}^{v_j}{\phi }_{jv}(\mathrm{\Theta }){\phi }_{jv}(\mathrm{\Phi })).$$

We introduce another function of $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$ and $Q=(t,\mathrm{\Phi })\in C_n(\mathrm{\Omega })$

$$G(\mathrm{\Omega },a,m)(P,Q)=G(\mathrm{\Omega },a)(P,Q)-K(\mathrm{\Omega },a,m)(P,Q).$$

The generalized Poisson kernel $\mathbb{P}(\mathrm{\Omega },a,m)(P,Q)$ ($P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$, $Q=(t,\mathrm{\Phi })\in S_n(\mathrm{\Omega })$) with respect to $C_n(\mathrm{\Omega })$ is defined by

$$\mathbb{P}(\mathrm{\Omega },a,m)(P,Q)=\frac{\partial G(\mathrm{\Omega },a,m)(P,Q)}{\partial n_Q}.$$

In fact,

$$\mathbb{P}(\mathrm{\Omega },a,0)(P,Q)=\mathbb{P}(\mathrm{\Omega },a)(P,Q).$$

We remark that the kernel function $\mathbb{P}(\mathrm{\Omega },0,m)(P,Q)$ coincides with the one in Yoshida and Miyamoto [12] (see [10], Ch. 11]).

Put

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

where $u(Q)$ is a continuous function on $\partial C_n(\mathrm{\Omega })$ and $d{\sigma }_Q$ is a surface area element on $S_n(\mathrm{\Omega })$.

With regard to classical solutions of the Dirichlet problem for the Laplacian, Yoshida and Miyamoto [12], Theorem 1] proved the following result.

Theorem A If u is a continuous function on $\partial C_n(\mathrm{\Omega })$ satisfying

$${\int }_{S_n(\mathrm{\Omega })}\frac{|u(t,\mathrm{\Phi })|}{1+t^{{\iota }_{m+1,0}^{+}+n-1}}d{\sigma }_Q<\mathrm{\infty },$$

then $U(\mathrm{\Omega },0,m;u)(P)$ is a classical solution of the Dirichlet problem on $C_n(\mathrm{\Omega })$ with g and satisfies

$$\underset{r\to \mathrm{\infty },P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })}{lim}{r}^{-{\iota }_{m+1,0}^{+}}U(\mathrm{\Omega },0,m;u)(P)=0.$$

Our first aim is to give growth properties at infinity for $U(\mathrm{\Omega },a,m;u)(P)$.

Theorem 1 Let$\gamma \ge 0$ (resp. $\gamma <0$), ${\iota }_{[\gamma ],k}^{+}+\{\gamma \}>-{\iota }_{1,k}^{+}+1$ (resp. $-{\iota }_{[-\gamma ],k}^{+}-\{-\gamma \}>-{\iota }_{1,k}^{+}+1$) and

If u is a measurable function on $\partial C_n(\mathrm{\Omega })$ satisfying

$${\int }_{S_n(\mathrm{\Omega })}\frac{|u(t,\mathrm{\Phi })|}{1+t^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}}}d{\sigma }_Q<\mathrm{\infty }(\text{resp.}{\int }_{S_n(\mathrm{\Omega })}|u(t,\mathrm{\Phi })|(1+t^{{\iota }_{[-\gamma ],k}^{+}+\{-\gamma \}}])d{\sigma }_Q<\mathrm{\infty }),$$

(1.6)

then

(1.7)

(1.8)

Next, we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on $C_n(\mathrm{\Omega })$.

Theorem 2 Let γ and${\iota }_{m+1,k}^{+}$be as in Theorem 1. If u is a continuous function on$\partial C_n(\mathrm{\Omega })$satisfying (1.6), then$U(\mathrm{\Omega },a,m;u)(P)$is a solution of the Dirichlet problem for the Schrödinger operator on$C_n(\mathrm{\Omega })$with u and (1.7) (resp. (1.8)) holds.

If we take ${\iota }_{[\gamma ],k}^{+}+\{\gamma \}={\iota }_{m+1,k}^{+}+n-1$, then we immediately have the following corollary, which is just Theorem A in the case $a=0$.

Corollary If u is a continuous function on $\partial C_n(\mathrm{\Omega })$ satisfying

$${\int }_{S_n(\mathrm{\Omega })}\frac{|u(t,\mathrm{\Phi })|}{1+t^{{\iota }_{m+1,k}^{+}+n-1}}d{\sigma }_Q<\mathrm{\infty },$$

(1.9)

then $U(\mathrm{\Omega },a,m;u)(P)$ is a solution of the Dirichlet problem for the Schrödinger operator on $C_n(\mathrm{\Omega })$ with u and satisfies

$$\underset{r\to \mathrm{\infty },P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })}{lim}{r}^{-{\iota }_{m+1,k}^{+}}U(\mathrm{\Omega },a,m;u)(P)=0.$$

(1.10)

By using Corollary, we can give a solution of the Dirichlet problem for any continuous function on $\partial C_n(\mathrm{\Omega })$.

Theorem 3 If u is a continuous function on$\partial C_n(\mathrm{\Omega })$satisfying (1.9) and$h(r,\mathrm{\Theta })$is a solution of the Dirichlet problem for the Schrödinger operator on$C_n(\mathrm{\Omega })$with u satisfying

$$\underset{r\to \mathrm{\infty },P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })}{lim}{r}^{-{\iota }_{m+1,k}^{+}}{h}^{+}(P)=0,$$

(1.11)

then

$$h(P)=U(\mathrm{\Omega },a,m;u)(P)+\sum _{j=0}^m(\sum _{v=1}^{v_j}{d}_{jv}{\phi }_{jv}(\mathrm{\Theta }))V_j(r),$$

where$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and$d_{jv}$are constants.

2 Lemmas

Throughout this article, let M denote various constants independent of the variables in questions, which may be different from line to line.

Lemma 1

(2.1)

(2.2)

for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and any$Q=(t,\mathrm{\Phi })\in S_n(\mathrm{\Omega })$satisfying$0<\frac{t}{r}\le \frac{4}{5}$ (resp. $0<\frac{r}{t}\le \frac{4}{5}$);

$$|\mathbb{P}(\mathrm{\Omega },0)(P,Q)|\le M\frac{1}{t^{n-1}}+M\frac{r}{{|P-Q|}^n}$$

(2.3)

for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and any$Q=(t,\mathrm{\Phi })\in S_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))$.

Proof (2.1) and (2.2) are obtained by Kheyfits (see [10], Ch. 11]). (2.3) follows from Azarin (see [13], Lemma 4 and Remark]). □

Lemma 2 (see [1])

For a nonnegative integer m, we have

$$|\mathbb{P}(\mathrm{\Omega },a,m)(P,Q)|\le M(n,m,s)V_{m+1}(r)\frac{W_{m+1}(t)}{t}{\phi }_1(\mathrm{\Theta })\frac{\partial {\phi }_1(\mathrm{\Phi })}{\partial n_{\mathrm{\Phi }}}$$

(2.4)

for any$P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$and$Q=(t,\mathrm{\Phi })\in S_n(\mathrm{\Omega })$satisfying$r\le st$ ($0<s<1$), where$M(n,m,s)$is a constant dependent of n, m and s.

Lemma 3 (see [2], Theorem 1])

If$u(r,\mathrm{\Theta })$is a solution of Equation (1.1) on$C_n(\mathrm{\Omega })$satisfying

$${\int }_{\mathrm{\Omega }}{u}^{+}(r,\mathrm{\Theta })d{S}_1=O\left(r^{{\iota }_{m,k}^{+}}\right),\text{as}r\to \mathrm{\infty },$$

(2.5)

then

$$u(r,\mathrm{\Theta })=\sum _{j=0}^m(\sum _{v=1}^{v_j}{d}_{jv}{\phi }_{jv}(\mathrm{\Theta }))V_j(r).$$

Lemma 4 Obviously, the conclusion of Lemma 3 holds true if (2.5) is replaced by

$$\underset{r\to \mathrm{\infty },(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })}{lim}{r}^{-{\iota }_{m+1,k}^{+}}{u}^{+}(r,\mathrm{\Theta })=0.$$

(2.6)

Proof Since

$$V_{m+1}(r)\sim r^{{\iota }_{m+1,k}^{+}}\text{as}r\to \mathrm{\infty }$$

from (1.5) and

$${\iota }_{m+1,k}^{+}\ge {\iota }_{m,k}^{+},$$

(2.6) gives that (2.5) holds, from which the conclusion immediately follows. □

3 Proof of Theorem 1

We only prove the case $\gamma \ge 0$, the remaining case $\gamma <0$ can be proved similarly.

For any $ϵ>0$, there exists $R_{ϵ}>1$ such that

$${\int }_{S_n(\mathrm{\Omega };(R_{ϵ},\mathrm{\infty }))}\frac{|u(Q)|}{1+t^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}}}d{\sigma }_Q<ϵ.$$

(3.1)

The relation $G(\mathrm{\Omega },a)(P,Q)\le G(\mathrm{\Omega },0)(P,Q)$ implies this inequality (see [14])

$$\mathbb{P}(\mathrm{\Omega },a)(P,Q)\le \mathbb{P}(\mathrm{\Omega },0)(P,Q).$$

(3.2)

For $0<s<\frac{4}{5}$ and any fixed point $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$ satisfying $r>\frac{5}{4}{R}_{ϵ}$, let $I_1=S_n(\mathrm{\Omega };(0,1))$, $I_2=S_n(\mathrm{\Omega };[1,R_{ϵ}])$, $I_3=S_n(\mathrm{\Omega };(R_{ϵ},\frac{4}{5}r])$, $I_4=S_n(\mathrm{\Omega };(\frac{4}{5}r,\frac{5}{4}r))$, $I_5=S_n(\mathrm{\Omega };[\frac{5}{4}r,\frac{r}{s}))$, $I_6=S_n(\mathrm{\Omega };[1,\frac{r}{s}))$ and $I_7=S_n(\mathrm{\Omega };[\frac{r}{s},\mathrm{\infty }))$, we write

$$U(\mathrm{\Omega },a,m;u)(P)\le \sum _{i=1}^7{U}_{\mathrm{\Omega },a,i}(P),$$

where

By ${\iota }_{[\gamma ],k}^{+}+\{\gamma \}>-{\iota }_{1,k}^{+}+1$, (1.6), (2.1) and (3.1), we have the following growth estimates

(3.3)

(3.4)

(3.5)

We obtain by ${\iota }_{m+1,k}^{+}\ge {\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1$, (2.2) and (3.1)

$$\begin{array}{rcl}{U}_{\mathrm{\Omega },a,5}(P)& \le & M{r}^{{\iota }_{1,k}^{+}}{\int }_{S_n(\mathrm{\Omega };[(5/4)r,\mathrm{\infty }))}{t}^{{\iota }_{1,k}^{-}-1}|u(Q)|d{\sigma }_Q\\ \le & M{r}^{{\iota }_{1,k}^{+}}{\int }_{S_n(\mathrm{\Omega };[(5/4)r,\mathrm{\infty }))}{t}^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}+{\iota }_{1,k}^{-}-1}\frac{|u(Q)|}{t^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}}}d{\sigma }_Q\\ \le & Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}.\end{array}$$

(3.6)

By (2.3) and (3.2), we consider the inequality

$$U_{\mathrm{\Omega },a,4}(P)\le U_{\mathrm{\Omega },0,4}(P)\le U_{\mathrm{\Omega },0,4}^{\prime }(P)+U_{\mathrm{\Omega },0,4}^{″}(P),$$

where

$$U_{\mathrm{\Omega },0,4}^{\prime }(P)=M{\int }_{I_4}{t}^{1-n}|u(Q)|d{\sigma }_Q,U_{\mathrm{\Omega },0,4}^{″}(P)=Mr{\int }_{I_4}\frac{|u(Q)|}{{|P-Q|}^n}d{\sigma }_Q.$$

We first have

$$\begin{array}{rcl}{U}_{\mathrm{\Omega },0,4}^{\prime }(P)& =& M{\int }_{I_4}{t}^{{\iota }_{1,k}^{+}+{\iota }_{1,k}^{-}-1}|u(Q)|d{\sigma }_Q\\ \le & M{r}^{{\iota }_{1,k}^{+}}{\int }_{S_n(\mathrm{\Omega };((4/5)r,\mathrm{\infty }))}{t}^{{\iota }_{1,k}^{-}-1}|u(Q)|d{\sigma }_Q\\ \le & Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1},\end{array}$$

(3.7)

which is similar to the estimate of $U_{\mathrm{\Omega },a,5}(P)$.

Next, we shall estimate $U_{\mathrm{\Omega },0,4}^{″}(P)$. Take a sufficiently small positive number $d_5$ such that $I_4\subset B(P,\frac{1}{2}r)$ for any $P=(r,\mathrm{\Theta })\in \mathrm{\Pi }(d_5)$, where

$$\mathrm{\Pi }(d_5)=\{P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega });\underset{z\in \partial \mathrm{\Omega }}{inf}|(1,\mathrm{\Theta })-(1,z)|<d_5,0<r<\mathrm{\infty }\}$$

and divide $C_n(\mathrm{\Omega })$ into two sets $\mathrm{\Pi }(d_5)$ and $C_n(\mathrm{\Omega })-\mathrm{\Pi }(d_5)$.

If $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })-\mathrm{\Pi }(d_5)$, then there exists a positive $d_5^{\prime }$ such that $|P-Q|\ge d_5^{\prime }r$ for any $Q\in S_n(\mathrm{\Omega })$, and hence

$$\begin{array}{rcl}{U}_{\mathrm{\Omega },0,4}^{″}(P)& \le & M{\int }_{I_4}{t}^{1-n}|u(Q)|d{\sigma }_Q\\ \le & Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1},\end{array}$$

(3.8)

which is similar to the estimate of $U_{\mathrm{\Omega },0,4}^{\prime }(P)$.

We shall consider the case $P=(r,\mathrm{\Theta })\in \mathrm{\Pi }(d_5)$. Now put

$$H_i(P)=\{Q\in I_4;2^{i-1}\delta (P)\le |P-Q|<2^i\delta (P)\}.$$

Since $S_n(\mathrm{\Omega })\cap \{Q\in {\mathbf{R}}^n:|P-Q|<\delta (P)\}=\mathrm{\varnothing }$, we have

$$U_{\mathrm{\Omega },0,4}^{″}(P)=M\sum _{i=1}^{i(P)}{\int }_{H_i(P)}r\frac{|u(Q)|}{{|P-Q|}^n}d{\sigma }_Q,$$

where $i(P)$ is a positive integer satisfying $2^{i(P)-1}\delta (P)\le \frac{r}{2}<2^{i(P)}\delta (P)$.

Since we see from (1.2)

$$r{\phi }_1(\mathrm{\Theta })\le M\delta (P)$$

for $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$. Similar to the estimate of $U_{\mathrm{\Omega },0,4}^{\prime }(P)$, we obtain

for $i=0,1,2,\dots ,i(P)$.

So

$$U_{\mathrm{\Omega },0,4}^{″}(P)\le Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}.$$

(3.9)

We only consider $U_{\mathrm{\Omega },a,6}(P)$ in the case $m\ge 1$, since $U_{\mathrm{\Omega },a,6}(P)\equiv 0$ for $m=0$. By the definition of $\tilde{K}(\mathrm{\Omega },a,m)$, (1.3) and Lemma 2, we see

$$U_{\mathrm{\Omega },a,6}(P)\le \frac{M}{{\chi }^{\prime }(1)}\sum _{j=0}^m{j}^{2n-1}{q}_j(r),$$

where

$$q_j(r)=V_j(r){\int }_{I_6}\frac{W_j(t)|u(Q)|}{t}d{\sigma }_Q.$$

To estimate $q_j(r)$, we write

$$q_j(r)\le q_j^{\prime }(r)+q_j^{″}(r),$$

where

$$q_j^{\prime }(r)=V_j(r){\int }_{I_2}\frac{W_j(t)|u(Q)|}{t}d{\sigma }_Q,q_j^{″}(r)=V_j(r){\int }_{S_n(\mathrm{\Omega };(R_{ϵ},r/s))}\frac{W_j(t)|u(Q)|}{t}d{\sigma }_Q.$$

Notice that

$$V_j(r)\frac{V_{m+1}(t)}{V_j(t)t}\le M\frac{V_{m+1}(r)}{r}\le M{r}^{{\iota }_{m+1,k}^{+}-1}(t\ge 1,R_{ϵ}<\frac{r}{s}).$$

Thus, by ${\iota }_{m+1,k}^{+}<{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+2$, (1.5) and (1.6) we conclude

$$\begin{array}{rcl}{q}_j^{\prime }(r)& =& V_j(r){\int }_{I_2}\frac{|u(Q)|}{V_j(t)t^{n-1}}d{\sigma }_Q\\ \le & M{V}_j(r){\int }_{I_2}\frac{V_{m+1}(t)}{t^{{\iota }_{m+1,k}^{+}}}\frac{|u(Q)|}{V_j(t)t^{n-1}}d{\sigma }_Q\\ \le & M{r}^{{\iota }_{m+1,k}^{+}-1}{R}_{ϵ}^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-{\iota }_{m+1,k}^{+}-n+2}.\end{array}$$

Analogous to the estimate of $q_j^{\prime }(r)$, we have

$$q_j^{″}(r)\le Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}.$$

Thus we can conclude that

$$q_j(r)\le Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1},$$

which yields

$$U_{\mathrm{\Omega },a,6}(P)\le Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}.$$

(3.10)

By ${\iota }_{m+1,k}^{+}\ge {\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1$, (1.5), (2.4) and (3.1) we have

$$\begin{array}{rcl}{U}_{\mathrm{\Omega },0,7}(P)& \le & M{V}_{m+1}(r){\int }_{I_7}\frac{|u(Q)|}{V_{m+1}(t)t^{n-1}}d{\sigma }_Q\\ \le & Mϵr^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}.\end{array}$$

(3.11)

Combining (3.3)–(3.11), we obtain that if $R_{ϵ}$ is sufficiently large and ϵ is sufficiently small, then $U(\mathrm{\Omega },a,m;u)(P)=o(r^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1})$ as $r\to \mathrm{\infty }$, where $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$. Then we complete the proof of Theorem 1.

4 Proof of Theorem 2

For any fixed $P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })$, take a number satisfying $R>max(1,\frac{r}{s})$ ($0<s<\frac{4}{5}$). By ${\iota }_{m+1,k}^{+}\ge {\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1$, (1.4), (1.6) and (2.4), we have

Thus $U(\mathrm{\Omega },a,m;u)(P)$ is finite for any $P\in C_n(\mathrm{\Omega })$. Since $\mathbb{P}(\mathrm{\Omega },a,m)(P,Q)$ is a generalized harmonic function of $P\in C_n(\mathrm{\Omega })$ for any fixed $Q\in S_n(\mathrm{\Omega })$, $U(\mathrm{\Omega },a,m;u)(P)$ is also a generalized harmonic function of $P\in C_n(\mathrm{\Omega })$. That is to say, $U(\mathrm{\Omega },a,m;u)(P)$ is a solution of Equation (1.1) on $C_n(\mathrm{\Omega })$.

Now we study the boundary behavior of $U(\mathrm{\Omega },a,m;u)(P)$. Let $Q^{\prime }=(t^{\prime },{\mathrm{\Phi }}^{\prime })\in \partial C_n(\mathrm{\Omega })$ be any fixed point and l be any positive number satisfying $l>max(t^{\prime }+1,\frac{4}{5}R)$.

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

$$U(\mathrm{\Omega },a,m;u)(P)=U^{\prime }(P)-U^{″}(P)+U^{‴}(P),$$

where

Notice that $U^{\prime }(P)$ is the Poisson a-integral of $u(Q){\chi }_{S((5/4)l)}$, we have ${lim}_{P\to Q^{\prime },P\in C_n(\mathrm{\Omega })}{U}^{\prime }(P)=u(Q^{\prime })$. Since ${lim}_{\mathrm{\Theta }\to {\mathrm{\Phi }}^{\prime }}{\phi }_{jv}(\mathrm{\Theta })=0$ ($j=1,2,3,\dots$; $1\le v\le v_j$) as $P=(r,\mathrm{\Theta })\to Q^{\prime }=(t^{\prime },{\mathrm{\Phi }}^{\prime })\in S_n(\mathrm{\Omega })$, we have ${lim}_{P\to Q^{\prime },P\in C_n(\mathrm{\Omega })}{U}^{″}(P)=0$ from the definition of the kernel function $K(\mathrm{\Omega },a,m)(P,Q)$. $U^{‴}(P)=O(r^{{\iota }_{[\gamma ],k}^{+}+\{\gamma \}-n+1}{\phi }_1(\mathrm{\Theta }))$, and therefore tends to zero.

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

$$\underset{P\to Q^{\prime },P\in C_n(\mathrm{\Omega })}{lim}U(\mathrm{\Omega },a,m;u)(P)=u\left(Q^{\prime }\right)$$

for any $Q^{\prime }=(t^{\prime },{\mathrm{\Phi }}^{\prime })\in \partial C_n(\mathrm{\Omega })$ from the arbitrariness of l. Thus we complete the proof of Theorem 2 from Theorem 1.

5 Proof of Theorem 3

From Corollary, we have the solution $U(\mathrm{\Omega },a,m;u)(P)$ of the Dirichlet problem on $C_n(\mathrm{\Omega })$ with u satisfying (1.9). Consider the function $h(P)-U(\mathrm{\Omega },a,m;u)(P)$. Then it follows that this is the solution of Equation (1.1) in $C_n(\mathrm{\Omega })$ and vanishes continuously on $\partial C_n(\mathrm{\Omega })$.

Since

$$0\le {(h-U(\mathrm{\Omega },a,m;u))}^{+}(P)\le h^{+}(P)+{(U(\mathrm{\Omega },a,m;u))}^{-}(P)$$

for any $P\in C_n(\mathrm{\Omega })$, we have

$$\underset{r\to \mathrm{\infty },P=(r,\mathrm{\Theta })\in C_n(\mathrm{\Omega })}{lim}{r}^{-{\iota }_{m+1,k}^{+}}{(h-U(\mathrm{\Omega },a,m;u))}^{+}(P)=0$$

from (1.10) and (1.11). Then the conclusions of Theorem 3 follow immediately from Lemma 4.