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Some properties for a subfunction associated with the stationary Schrödinger operator in a cone

Abstract

For a subfunction u associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we correct Theorem 1 in (Qiao and Deng in Glasg. Math. J. 53(3):599-610, 2011). Then by the theorem we generalize some theorems of Phragmén-Lindelöf type for a subfunction in a cone. Meanwhile, we obtain some results about the existence of solutions of the Dirichlet problem associated with the stationary Schrödinger operator in a cone and about the type of their uniqueness.

MSC:31B05.

1 Introduction and main results

To begin with, let us agree to some basic conventions. As usual, let S be an open set in R n (n2), where R n is the n-dimensional Euclidean space. The boundary and the closure of S are denoted by S and S ¯ , respectively. Let P=(X, x n ), where X=( x 1 , x 2 ,, x n 1 ), and let |P| be the Euclidean norm of P and |PQ| be the Euclidean distance of two points P and Q in R n . The unit sphere and the upper half unit sphere are denoted by S n 1 and S + n 1 , respectively. For simplicity, the point (1,Θ) on S n 1 and the set {Θ;(1,Θ)Ω} for a set Ω, Ω S n 1 are often identified with Θ and Ω, respectively. For Ξ R + and Ω S n 1 , the set {(r,Θ) R n ;rΞ,(1,Θ)Ω} in R n is simply denoted by Ξ×Ω. In particular, the half-space R + × S + n 1 ={(X, x n ) R n ; x n >0} will be denoted by T n . By C n (Ω), we denote the set R + ×Ω in R n with the domain Ω on S n 1 and call it a cone. For an interval I R + and Ω S n 1 , set C n (Ω;I)=I×Ω, S n (Ω;I)=I×Ω and C n (Ω;r)= C n (Ω) S r . By S n (Ω) we denote S n (Ω;(0,+)), which is C n (Ω){O}. Furthermore, we denote by d S r the (n1)-dimensional volume elements induced by the Euclidean metric on S r . For P R n and r>0, let B(P,r) denote the open ball with center at P and radius r in R n , then S r =B(O,r).

We introduce the system of spherical coordinates (r,Θ), Θ=( θ 1 , θ 2 ,, θ n 1 ) for P=(X, x n ) in R n via the following formulas:

x 1 =r j = 1 n 1 sin θ j (n2), x n =rcos θ 1

and if n3,

x n k + 1 =rcos θ k j = 1 k 1 sin θ j (2kn1),

where 0r<, 0 θ j π (1jn2) and π 2 θ n 1 3 π 2 .

Relative to the system of spherical coordinates, the Laplace operator Δ may be written

Δ= n 1 r r + 2 r 2 + Δ r 2 ,

where the explicit form of the Beltrami operator Δ is given by Azarin (see [1]).

For an arbitrary domain D in R n , A D denotes the class of non-negative radial potentials a(P) (i.e., 0a(P)=a(r) for P=(r,Θ)D) such that a L loc b (D) with some b>n/2 if n4 and with b=2 if n=2 or n=3.

If a A D , the stationary Schrödinger operator with a potential a(),

L a =Δ+a()I,
(1.1)

can be extended in the usual way from the space C 0 (D) to an essentially self-adjoint operator on L 2 (D), where Δ is the Laplace operator and I is the identical operator (see Reed and Simon [2], Chapter 13). Furthermore, L a has a Green a-function G D a (,). Here G D a (,) is positive on D and its inner normal derivative G D a (,Q)/ n Q 0, where / n Q denotes the differentiation at Q along the inward normal into D. We denote this derivative by P I D a (,), which is called the Poisson a-kernel with respect to D. There is an inequality between the Green a-function G D a (,) and that of the Laplacian, hereafter denoted by G D 0 (,). It is well known that for any potential a()0,

G D a (,) G D 0 (,).
(1.2)

The inverse inequality is much more elaborate when D is a bounded domain in R n . For a bounded domain D in R n , Cranston, Fabes and Zhao (see [3], the case n=2 is implicitly contained in [4]) have proved

G D a (,)M(D) G D 0 (,),
(1.3)

where M(D)=M(D,a) is positive and independent of points in D. If a=0, obviously M(D)1.

Suppose that a function u is upper semi-continuous in D. u[,+) is called a subfunction of the Schrödinger operator L a if the generalized mean-value inequality

u(P) S ( P , ρ ) u(Q) G B ( P , ρ ) a ( P , Q ) n Q dσ(Q)
(1.4)

is satisfied at each point PD with 0<ρ< inf Q D |PQ|, where S(P,ρ)=B(P,ρ), G B ( P , ρ ) a (P,) is the Green a-function of L a in B(P,ρ) and dσ() is the surface area element on S(P,ρ).

Denote the class of subfunctions in D by SbH(a,D). If uSbH(a,D), we call u a superfunction and denote the class of superfunctions by SpH(a,D). If a function u is both subfunction and superfunction, clearly, it is continuous and called an a-harmonic function associated with the operator L a . The class of a-harmonic functions is denoted by H(a,D)=SbH(a,D)SpH(a,D). In terminology we follow Levin and Kheyfits (see [5] or [6]). From now on, we always assume D= C n (Ω). For the sake of brevity, we shall write G Ω a (,) instead of G C n ( Ω ) a (,), P I Ω a (,) instead of P I C n ( Ω ) a (,), SpH(a) (resp. SbH(a)) instead of SpH(a, C n (Ω)) (resp. SbH(a, C n (Ω))) and H(a) instead of H(a, C n (Ω)).

For positive functions h 1 and h 2 , we say that h 1 h 2 if h 1 M h 2 for some constant M>0. If h 1 h 2 and h 2 h 1 , we say that h 1 h 2 .

Let Ω be a domain on S n 1 with a smooth boundary, and let λ be the least positive eigenvalue for Δ on Ω (see [7], p.41),

( Δ + λ ) φ ( Θ ) = 0 on  Ω , φ ( Θ ) = 0 on  Ω .
(1.5)

The corresponding eigenfunction is denoted by φ(Θ) satisfying Ω φ 2 (Θ)d S 1 =1. In order to ensure the existence of λ and φ(Θ), we put a rather strong assumption on Ω: if n3, then Ω is a C 2 , α -domain (0<α<1) on S n 1 surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g., see Gilbarg and Trudinger [8], pp.88-89 for the definition of C 2 , α -domain). We denote the non-decreasing sequence of positive eigenvalues of (1.5) by { λ ( Ω , k ) } k = 1 . In the expression, we write λ(Ω,k) the same number of times as the dimension of the corresponding eigenspace. When the normalized eigenfunction corresponding to λ(Ω,k) is denoted by φ k (Θ), the set of sequential eigenfunctions corresponding to the same value of λ(Ω,k) in the sequence { φ k ( Θ ) } k = 1 makes an orthonormal basis for the eigenspace of the eigenvalue λ(Ω,k). Hence, for each Ω S n 1 , there is a sequence { k i } of positive integers such that k 1 =1, λ(Ω, k i )<λ(Ω, k i + 1 ),

λ(Ω, k i )=λ(Ω, k i +1)=λ(Ω, k i +2)==λ(Ω, k i + 1 1)
(1.6)

and { φ k i (Θ), φ k i + 1 (Θ),, φ k i + 1 1 (Θ)} is an orthonormal basis for the eigenspace of the eigenvalue λ(Ω, k i ) (i=1,2,3,). It is well known that k 2 =2 and φ 1 (Θ)>0 for any ΘΩ (see Courant and Hilbert [9]). For the case Ω= S + n 1 (n=2,3,4,), k i =i (i=1,2,3,) when n=2, and the situation is more complicated when n3 (see the Remark in [10] for the detailed information). For a domain Ω and the sequence { k i } mentioned above, by I( k l ) we denote the set of all positive integers less than k l (k=1,2,3,). In spite of the fact that I( k 1 )=ϕ, the summation over I( k 1 ) of a function S(k) of a variable k is used by promising

k I ( k 1 ) S(k)=0.

If Ω is an (n1)-dimensional compact Riemannian manifold with its boundary to be sufficiently regular, we see that

λ(Ω,k)A(Ω,n) k 2 n 1 (k)
(1.7)

(e.g., see Cheng and Li [11]) and

λ ( Ω , k ) x { φ k ( Θ ) } 2 B(Ω,n) x n 1 2 (x)
(1.8)

uniformly with respect to Θ (e.g., see Minakshisundaram and Pleijel [12] or Essén and Lewis [13], p.120 and pp.126-128), where A(Ω,n) and B(Ω,n) are both constants depending on Ω and n. Hence, there exist two positive M 1 , M 2 such that

M 1 k 2 n 1 λ(Ω,k)(k=1,2,3,)
(1.9)

and

| φ k ( Θ ) | M 2 k 1 2 (ΘΩ,k=1,2,3,).
(1.10)

Solutions of an ordinary differential equation

Q (r) n 1 r Q (r)+ ( λ r 2 + a ( r ) ) Q(r)=0(0<r<)
(1.11)

are known (see [14] for more references) when the potential a A D . We know equation (1.11) has a fundamental system of positive solutions {V,W} such that V is non-decreasing with

0V(0+)V(r)as r+,

and W is monotonically decreasing with

+=W(0+)>W(r)0as r+.

We remark that both V(r,k) φ k (Θ) and W(r,k) φ k (Θ) (k=1,2,3,) are a-harmonic on C n (Ω) and vanish continuously on S n (Ω), where V(r,k) and W(r,k) are the solutions of equation (1.11) with λ=λ(Ω,k).

Let B D be the class of the potentials a A D such that

lim r r 2 a(r)=κ[0,), r 1 | r 2 a ( r ) κ | L(1,).

When a B D , the (super)subfunctions are continuous (e.g., see [15]). In the rest of paper, we assume that a B D and we suppress this assumption for simplicity.

Denote

ι κ ± = 2 n ± ( n 2 ) 2 + 4 ( κ + λ ) 2 .

When a B D , the solutions V(r), W(r) to equation (1.11) normalized by V(1)=W(1)=1 have the asymptotic (see [8])

V(r) r ι κ + ,W(r) r ι κ ,as r
(1.12)

and

χ= ι κ + ι κ = ( n 2 ) 2 + 4 ( κ + λ ) , χ =ω ( V ( r ) , W ( r ) ) | r = 1 ,
(1.13)

where χ is their Wronskian at r=1.

Remark 1.1 If a=0 and Ω= S + n 1 , ι 0 + =1, ι 0 =1n and φ(Θ)= ( 2 n s n 1 ) 1 / 2 cos θ 1 , where s n is the surface area 2 π n / 2 { Γ ( n / 2 ) } 1 of S n 1 .

Let u(r,Θ) be a function on C n (Ω). We introduce M u (r)=M(r,u)= sup Θ Ω u(r,Θ), u + =max{u,0} and u =max{u,0}.

We say that u(P) (P=(r,Θ)) satisfies the Phragmén-Lindelöf boundary condition on S n (Ω), namely

lim sup P = ( r , Θ ) C n ( Ω ) , P Q S n ( Ω ) u(P)0.
(1.14)

Let F(P)=F(r,Θ) be a function on C n (Ω). For any given positive real number r, the integral

Ω F(r,Θ)φ(Θ)d S 1

is denoted by N(F)(r), when it exists. For two non-negative integers p and q, the finite or infinite limit

lim r N ( F ) ( r ) V ( r , k p + 1 ) ( resp.  lim r 0 N ( F ) ( r ) W ( r , k q + 1 ) )

is denoted by V P (F) (resp. W q (F)), when it exists.

If f(l) is a real finite-valued function defined on an interval (0,+), for any given l 1 , l 2 (0< l 1 < l 2 <) and l(0,+), we have

E(l;f,V,W, l 1 , l 2 )=| f ( l ) V ( l ) W ( l ) f ( l 1 ) V ( l 1 ) W ( l 1 ) f ( l 2 ) V ( l 2 ) W ( l 2 ) |0

if and only if

f(l)F(l;f,V,W, l 1 , l 2 ),

where F(l;f,V,W, l 1 , l 2 ) has the following expression:

{ W ( l ) W ( l 1 ) f ( l 1 ) ( V ( l 2 ) W ( l 2 ) V ( l ) W ( l ) ) + W ( l ) W ( l 2 ) f ( l 2 ) ( V ( l ) W ( l ) V ( l 1 ) W ( l 1 ) ) } { V ( l 2 ) W ( l 2 ) V ( l 1 ) W ( l 1 ) } 1 .

We say that f(l) is (V,W)-convex on (0,+) if E(l;f,V,W, l 1 , l 2 )0 ( l 1 l l 2 ) for any l 1 , l 2 (0< l 1 < l 2 <+).

Remark 1.2 A function f(l) is (V,W)-convex on (0,+) if and only if W 1 (l)f(l) is a convex function of W 1 (l)V(l) on (0,+) or, equivalently, if and only if V 1 (l)f(l) is a convex function of V 1 (l)W(l) on (0,+); refer to Dinghas [16] for the relevant properties of a convex function with respect to an ODE.

The Poisson a-integral P I Ω a [g](P) of g relative to C n (Ω) is defined by

P I Ω a [g](P)= c n 1 S n ( Ω ) P I Ω a (P,Q)g(Q)d σ Q ,
(1.15)

where

P I Ω a (P,Q)= G Ω a ( P , Q ) n Q , c n ={ 2 π , n = 2 , ( n 2 ) s n , n 3 ,

n Q denotes the differentiation at Q along the inward normal into C n (Ω) and d σ Q is the surface area element on S n (Ω).

For two non-negative integers l, m and two points P=(r,Θ) C n (Ω) and Q=(t,Φ) S n (Ω), we put

(1.16)

and

(1.17)

We introduce two functions of P C n (Ω) and Q=(t,Φ) S n (Ω) as follows:

W ¯ ( C n ( Ω ) , l ) (P,Q)={ V ¯ ( C n ( Ω ) , l ) ( P , Q ) ( 1 t < ) , 0 ( 0 < t < 1 )

and

W ̲ ( C n ( Ω ) , m ) (P,Q)={ V ̲ ( C n ( Ω ) , m ) ( P , Q ) ( 0 < t < 1 ) , 0 ( 1 t < ) .

The kernel K( C n (Ω),l,m)(P,Q) with respect to C n (Ω) is defined by

K ( C n ( Ω ) , l , m ) ( P , Q ) = c n 1 G C n ( Ω ) a n Φ ( P , Q ) W ¯ ( C n ( Ω ) , l ) ( P , Q ) W ̲ ( C n ( Ω ) , m ) ( P , Q ) .
(1.18)

In fact

K ( C n ( Ω ) , l , 0 ) (P,Q)= c n 1 G C n ( Ω ) a n Φ (P,Q) W ¯ ( C n ( Ω ) , l ) (P,Q)(l1)
(1.19)

and

K ( C n ( Ω ) , 0 , 0 ) (P,Q)= c n 1 G C n ( Ω ) a n Φ (P,Q).
(1.20)

Based on the elaborate research, Yoshida ([17] and [18]) has considered the subharmonic function defined on a cone or a cylinder which is dominated on the boundary by a certain function and generalized the classical Phragmén-Lindelöf theorem by making a harmonic majorant. Later Yoshida [19] proved the property of a harmonic function defined on a half-space which is represented by the generalized Poisson integral with a slowly growing continuous function on the boundary. In [20] or [10] Yoshida and Miyamoto generalized some theorems (from [19]) to the conical case and extended the results (from [17] and [18]) given particular solutions and a type of general solutions of the Dirichlet problem on a cone by introducing conical generalized Poisson kernels and Poisson integrals. On the other hand, Qiao and Deng [21] extended Yoshida’s results (from [18]) to the situation for the stationary Schrödinger operator; for the relevant research on the stationary Schrödinger operator, we may refer to Bramanti [22], Kheyfits [2330] and Levin et al.[6, 31]. However, we find a falsehood in [21] and have to make a correction. In [5] or [6] we also know the Green function associated with the stationary Schrödinger operator. Dependent on the related statement above, we are to be concerned with the solutions of the Dirichlet problem for the stationary Schrödinger operator L a on C n (Ω) and with their growth properties. Furthermore, we note the existence of solutions of the Dirichlet problem for the stationary Schrödinger operator L a in a cone and the type of their uniqueness. First of all, we start with the following result.

Theorem A Let g(Q) be a continuous function on S n (Ω) satisfying

t 1 V ( t ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt<
(1.21)

and

0 t 1 W ( t ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt<.
(1.22)

Then the functionP I Ω a [g](P) (P=(r,Θ)) satisfies

(1.23)

and

lim r 0 , P = ( r , Θ ) C n ( Ω ) W ( r ) 1 φ ( Θ ) 1 P I Ω a [g](P)=0.
(1.24)

Remark 1.3 As to Theorem 1 in the paper [21], the factor φ Ω n 1 can be replaced with φ Ω 1 such that it is true, that is, Theorem A corrects Theorem 1 (from [21]) which is a generalization for a result from Siegel and Talvila (see [32]). Moreover, as to Theorem A we may follow the proof procedure of Theorem 1 in [21].

Next, we state our main results as follows.

Theorem 1.4 Let l, m be two non-negative integers andg(Q)=g(t,Φ)be a continuous function on C n (Ω)satisfying

t 1 V ( t , k l + 1 ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt<
(1.25)

and

0 t 1 W ( t , k m + 1 ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt<.
(1.26)

Then

H ( C n ( Ω ) , l , m ; g ) (P)= S n ( Ω ) g(Q)K ( C n ( Ω ) , l , m ) (P,Q)d σ Q
(1.27)

is a solution of the Dirichlet problem for the stationary Schrödinger operator on C n (Ω) with g satisfying

V l ( | H ( C n ( Ω ) , l , m ; g ) | ) = W m ( | H ( C n ( Ω ) , l , m ; g ) | ) =0.
(1.28)

Theorem 1.5 Let l be a non-negative integer andg(Q)=g(t,Φ)be a continuous function on C n (Ω)satisfying

t 1 V ( t , k l + 1 ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt<.

Then

H ( C n ( Ω ) , l , 0 ; g ) (P)= S n ( Ω ) g(Q)K ( C n ( Ω ) , l , 0 ) (P,Q)d σ Q
(1.29)

is a solution of the Dirichlet problem for the stationary Schrödinger operator on C n (Ω) with g satisfying

V l ( | H ( C n ( Ω ) , l , 0 ; g ) | ) =0.
(1.30)

Remark 1.6 When m=0, Theorem 1.4 is equal to Theorem 1.5. Since Theorem 1.4 may follow the proof for Theorem 1.5, for convenience, we only prove the latter.

It is natural to ask if 0 in (1.14) can be replaced with a general function g( Q ) on S n (Ω). The following Theorem 1.7 gives an affirmative answer to this question. For related results, we refer the readers to the paper by Levin and Kheyfits (see [6], Section 3 or [5], Chapter 11).

Theorem 1.7 Let p, q be two positive integers satisfyingp,q1. Letg( Q )be a continuous function on S n (Ω)satisfying (1.25) and (1.26) andu(P)be a subfunction on C n (Ω)such that

lim sup P C n ( Ω ) , P Q S n ( Ω ) u(P)g ( Q ) .
(1.31)

Then all of the limits V p (u), W q (u), V p ( u + )and W q ( u + ) (< V p (u), W q (u)+, 0 V p ( u + ), W q ( u + )+) exist. Moreover, when

(1.32)
(1.33)

for anyP=(r,Θ) C n (Ω), where A u (k) (k=1,2,, k p + 1 1) and B u (k) (k=1,2,, k q + 1 1) are all constants.

As an application of Theorems A and 1.7, we obtain the following result.

Theorem 1.8 Let p, q be two positive integers satisfyingp,q1. Letg(Q)be defined as in Theorem  1.7 andh(P)be any solution of the Dirichlet problem for the stationary Schrödinger operator L a on C n (Ω)with g. Then all of the limits V p (h), W q (h), V p (|h|)and W q (|h|) (< V p (h), W q (h)+, 0 V p (|h|), W q (|h|)+) exist. Moreover, when

(1.34)
(1.35)

for anyP=(r,Θ) C n (Ω), where A h (k) (k=1,2,, k p + 1 1) and B h (k) (k=1,2,, k q + 1 1) are all constants.

Remark 1.9 For p=q=1, Theorems 1.7 and 1.8 come from Qiao and Deng [21]. Furthermore, when a=0 and p=q=1, Theorems 1.7 and 1.8 are due to Yoshida (see [18], Theorems 2 and 3(II)). In fact, for kI( k p + 1 ) we know A u (k), B u (k) (or A h (k), B h (k)) are equal to the corresponding V(u),W(u) (or V(h), W(h)), respectively. Without the potential function, we may refer to Yoshida (see [33]).

Theorem 1.10 Let l, m be two non-negative integers and p, q be two positive integers satisfyingp,q1. Letg(Q)be defined as in Theorem  1.7 satisfying (1.25) with l and (1.26) with m, respectively. Ifh(P)is any solution of the Dirichlet problem for the stationary Schrödinger operator L a on C n (Ω)with g satisfying

(1.36)
(1.37)

for anyP=(r,Θ) C n (Ω), where A h (k) (k=1,2,, k p + 1 1) and B h (k) (k=1,2,, k q + 1 1) are all constants.

Theorem 1.11 Let l be a non-negative integer and p be a positive integer satisfyingp1. Ifh(r,Θ)is a generalized harmonic function on C n (Ω)and continuous on C n ( Ω ) ¯ such that the restrictionh=h | C n ( Ω ) of h to C n (Ω)satisfies

1 t 1 V ( t , k l + 1 ) 1 ( Ω n | h ( t , Φ ) | d σ Φ ) dt<

for some non-negative integer l, and for a positive integer p ˜

lim sup r log N ( h + ) ( r ) log V ( r , k p ˜ + 1 ) <,

then, for some positive integerp=max{l, p ˜ },

h(P)=H ( C n ( Ω ) , l ; g ) (P)+ k I ( k p + 1 ) A h (k)V(r,k) φ k (Θ)

for anyP=(r,Θ) C n (Ω), where A h (k) (k=1,2,, k p + 1 1) are all constants.

Remark 1.12 If we take a=0, Theorems 1.10 and 1.11 are similar to Theorems 7 and 9 in [20], respectively. In [33] Yoshida and Miyamoto considered the case when q=0, m=0 and a=0 about Theorem 1.10 and gave the proof. In addition, with Theorem 1.10 we easily get the conclusion of Theorem 1.11, then we do not have to prove it.

2 Some lemmas

In our arguments, we need some important results and techniques, which result from [5, 6, 27, 34], and [1] (Lemma 4 and Remark).

Lemma 2.1

(2.1)
(2.2)

for anyP=(r,Θ) C n (Ω)and anyQ=(t,Φ) S n (Ω)satisfying0< t r 4 5 (resp. 0< r t 4 5 ). In addition,

P I Ω 0 (P,Q) φ ( Θ ) t n 1 φ ( Φ ) n Φ + r φ ( Θ ) | P Q | n φ ( Φ ) n Φ
(2.3)

for anyP=(r,Θ) C n (Ω)and anyQ=(t,Φ) S n (Ω;( 4 5 r, 5 4 r)).

Lemma 2.2 Leta B D . For a non-negative integer k l + 1 , we have

(2.4)

for anyP=(r,Θ) C n (Ω)andQ=(t,Φ) S n (Ω)satisfyingrst (0<s<1), whereM( k l + 1 ,n,s)is a constant dependent on n, k l + 1 and s.

Lemma 2.3 Letg(Q)be a locally integrable and upper semicontinuous function on C n (Ω). LetW(P,Q)be a function ofP C n (Ω)andQ C n (Ω)such that for any fixedP C n (Ω)the functionW(P,Q)ofQ C n (Ω)is a locally integrable function on C n (Ω). Put

K(P,Q)= c n 1 G C n ( Ω ) a n Φ (P,Q)W(P,Q) ( P C n ( Ω ) , Q C n ( Ω ) ) .
(2.5)

Suppose that the following (I) and (II) are satisfied.

  1. (I)

    For any Q C n (Ω) and any ε>0, there exist a neighborhood U( Q ) of Q in R n and a number R (0<R<) such that

    S n ( Ω ; [ R , ) ) | g ( Q ) K ( P , Q ) | d σ Φ <ε
    (2.6)

for any p=(r,Θ) C n (Ω)U( Q )

  1. (II)

    For any Q C n (Ω) and any R (0<R<),

    lim sup P Q , P C n ( Ω ) S n ( Ω ; ( 0 , R ) ) | g ( Q ) K ( P , Q ) | d σ Φ =0.
    (2.7)

Then

lim sup P Q , P C n ( Ω ) S n ( Ω ) g(Q)K(P,Q)d σ Φ g ( Q )
(2.8)

for any Q C n (Ω).

Remark 2.4 When a=0, Lemma 2.3 is due to Yoshida (see [10], Lemma 5). By (1.2), obviously we reduce to Yoshida’s results. Therefore, we may omit the proof.

Lemma 2.5 Ifh(r,Θ)is an a-harmonic function on C n (Ω)vanishing continuously on S n (Ω),

E ( r ; N ( h ) , V , W , r 1 , r 2 ) =0

for any r 1 , r 2 (0< r 1 < r 2 <+) and every r (0<r<+).

Lemma 2.6 Iff(l)is(V,W)-convex on(0, d 1 ) (0< d 1 +), then

β= lim l 0 f ( l ) W ( l ) (<α+)

exists. Further, ifβ0, V 1 (l)f(l)is non-decreasing on(0, d 1 ).

It is known that C n (Ω) is regular, the Dirichlet problem for Δ and L a is solvable in it (see [6]). Based on this fact, Lemmas 2.7, 2.8 and 2.9 could be derived from (1.2), (1.3), (1.12), Remark 1.2, Lemmas 2.3 and 2.5 with their means of proof essentially due to Yoshida (see [17], Theorems 3.1 and 5.1, and [18], Lemma 3).

Lemma 2.7 Ifu(r,Θ)is a subfunction on C n (Ω)satisfying the Phragmén-Lindelöf boundary condition on S n (Ω), then

N(u)(r)>

for0<r<+andN(u)(r)is(V,W)-convex on(0,+). If there are three numbers r 1 , r 2 and r 0 satisfying0< r 1 < r 0 < r 2 <+such that

E ( r 0 ; N ( u ) , V , W , r 1 , r 2 ) =0,

we have that

  1. (1)

    E(r;N(u),V,W, r 1 , r 2 )=0 ( r 1 r r 2 );

  2. (2)

    u(r,Θ) is an a-harmonic function on C n (Ω;( r 1 , r 2 )) and vanishes continuously on S n (Ω;( r 1 , r 2 )).

Lemma 2.8 Letg(Q)be defined as in Theorem  1.7. ThenP I Ω a [g](P) (resp. P I Ω a [|g|](P)) is an a-harmonic function on C n (Ω)such that both of the limitsV(P I Ω a [g])andW(P I Ω a [g]) (resp. V(P I Ω a [|g|])andW(P I Ω a [|g|])) exist, and

V ( P I Ω a [ g ] ) =W ( P I Ω a [ g ] ) =0 ( resp. V ( P I Ω a [ | g | ] ) = W ( P I Ω a [ | g | ] ) = 0 ) .

Lemma 2.9 Letu(P)be a subfunction on C n (Ω)satisfying the Phragmén-Lindelöf boundary condition on S n (Ω). If (1.32) is satisfied forp=1andq=1,

u(P) ( V ( u ) V ( r ) + W ( u ) W ( r ) ) φ(Θ)

for anyP=(r,Θ) C n (Ω).

By the Kelvin transformation (see [35], p.59), Lemmas 2.6 and 2.7, we immediately have the following result, which is due to Yoshida in the case a=0 (see [17], Theorem 3.3).

Lemma 2.10 Letu(P)be defined as in Lemma  2.9. Then

  1. (1)

    Both of the limits V(u) and W(u) (<V(u), W(u)+) exist.

  2. (2)

    If W(u)0, then V 1 (r)N(u)(r) is non-decreasing on (0,+).

  3. (3)

    If V(u)0, then W 1 (r)N(u)(r) is non-increasing on (0,+).

Lemma 2.11 LetH(r,Θ)be an a-harmonic function in C n (Ω)vanishing continuously on C n (Ω), and p, q be two positive integers. h satisfies

V p ( h + ) =0and W q ( h + ) =0,

then

h(P)= k I ( k p + 1 ) A h (k)V(r,k) φ k (Θ)+ k I ( k q + 1 ) B h (k)W(r,k) φ k (Θ)

for anyP=(r,Θ) C n (Ω), where A u (k) (k=1,2,, k p + 1 1) and B u (k) (k=1,2,, k q + 1 1) are all constants.

Remark 2.12 When q=0 and a=0, Yoshida states the result in [10]. Later Qiao [34] proves Lemma 2.11 when q=0. Similarly, we may complete the proof of Lemma 2.11. Herein we leave out the detailed information for the proof.

3 Proofs of the theorems

Proof of Theorem 1.5 For any fixed P C n (Ω), we take a number R satisfying sR>max{1,r} (0<s<1). Then from Lemma 2.2 and (1.25)

(3.1)

Then H( C n (Ω),l,0;g)(P) is absolutely convergent and finite for any P C n (Ω). We remark that K( C n (Ω),l,0)(P,Q) is a harmonic function of P C n (Ω) for any Q C n (Ω). Thus, H( C n (Ω),l,0;g)(P) is a generalized harmonic function of P C n (Ω).

Next, we consider the boundary behavior of H( C n (Ω),l,0;g)(P). To prove that

lim sup P Q , P C n ( Ω ) H ( C n ( Ω ) , l , 0 ; g ) (P)=g ( Q )

for any Q C n (Ω), we may apply Lemma 2.3 to g(Q) and g(Q) by putting

W(P,Q)= W ¯ ( C n ( Ω ) , l , 0 ) (P,Q)

which is locally integrable on C n (Ω) for any fixed P C n (Ω). Let δ be a positive number and take Q =( t , Φ ) C n (Ω) and any ε>0. Then from (1.25) and (3.1) we can choose a number R (sR>max{1,r}) (0<s<1) such that for any P C n (Ω) U δ ( Q ), where U δ ( Q )={X R n :|X Q |<δ} and δ is a positive number

S n ( Ω ; [ R , ) ) | g ( Q ) K ( C n ( Ω ) , l , 0 ) ( P , Q ) | d σ Φ <ε,

which is (I) in Lemma 2.3.

Because

lim Θ Φ φ k (Θ)=0(k=1,2,)

as P=(r,Θ) Q =( t , Φ ) C n (Ω), we know that for any Q C n (Ω) and Q S n (Ω),

lim P Q , P C n ( Ω ) W ¯ ( C n ( Ω ) , l , 0 ) (P,Q)=0.

According to Lemma 2.3, we get the required results.

Next, we note the inequality

N ( | H ( C n ( Ω ) , l , 0 ; g + ) | ) (r) I 1 (r)+ I 2 (r),
(3.2)

where

I 1 (r)= Ω ( S n ( Ω ; [ r , ) ) g + ( Q ) | K ( C n ( Ω ) , l , 0 ) ( P , Q ) | d σ Q ) φ k (Θ)d σ Φ

and

I 2 (r)= Ω ( S n ( Ω ; [ 0 , r ) ) g + ( Q ) | K ( C n ( Ω ) , l , 0 ) ( P , Q ) | d σ Q ) φ k (Θ)d σ Φ

for P=(r,Θ), 0<r<. For any positive number ε, from (1.25) we can take a sufficiently large number r 0 such that

r t 1 V ( t , k l + 1 ) 1 ( Ω | g ( t , Φ ) | d σ Φ ) dt< ε L M (r> r 0 ),

where M is the constant in Lemma 2.2 and

L= Ω φ k (Θ)d σ Θ .

Then from Lemma 2.2 we have

0 I 1 ( r ) L ( k l + 1 ) M ( n , k l + 1 , s ) V ( r , k l + 1 ) r t 1 V ( t , k l + 1 ) 1 ( Ω g + ( t , Φ ) d σ Φ ) d t ε V ( r , k l + 1 ) ( r > r 0 )
(3.3)

which gives V l ( I 1 )=0.

Following this, we see the inequality

I 2 (r) I 21 (r)+ I 22 (r),
(3.4)

where

I 21 (r)= Ω ( S n ( Ω ; [ r , ) ) g + ( Q ) c n 1 G C n ( Ω ) a n Φ ( P , Q ) d σ Q ) φ k (Θ)d σ Φ

and

I 22 (r)= Ω ( S n ( Ω ; [ 0 , r ) ) g + ( Q ) W ¯ ( C n ( Ω ) , l ) ( P , Q ) d σ Q ) φ k (Θ)d σ Φ

for P=(r,Θ) and r>1. First, we know from (1.10) and (1.16) that if l1

I 22 (r) k I ( k m + 1 ) kV(r,k) Ψ k (r)(r>1),
(3.5)

where

Ψ k (r)= 1 r t 1 V ( t , k ) 1 ( Ω g + ( t , Φ ) d σ Φ ) dt ( r > 1 , k I ( k m + 1 ) ) .

We claim that

Ψ k (r)=O ( V ( r , k l + 1 ) V ( r , k ) 1 ) ( r > 1 , k I ( k m + 1 ) ) .
(3.6)

First we note increasing Ψ k (r) and Lemma C.1 in [5], Chapter 13 or [6], then by (1.12) we get

Ψ k (r)V ( r , k l + 1 ) 1 V(r,k) 1 r t 1 V ( t , k + 1 ) 1 ( Ω g + ( t , Φ ) d σ Φ ) dt<(r>1).

Hence, we can conclude that if l1, then

V l ( I 22 )=0.
(3.7)

Next, we see I 21 and note that

0< I 21 =N ( H ( C n ( Ω ) , l , 0 ; g + ) ) (r) I 1 (r)+ I 22 (r),
(3.8)

where

I 1 (r)= Ω ( S n ( Ω ; [ r , ) ) g + ( Q ) K ( C n ( Ω ) , l , 0 ) ( P , Q ) d σ Q ) φ k (Θ)d σ Φ

and

I 22 (r)= Ω ( S n ( Ω ; [ 0 , r ) ) g + ( Q ) K ( C n ( Ω ) , l , 0 ) ( P , Q ) d σ Q ) φ k (Θ)d σ Φ .

Since

| I 1 ( r ) | I 1 (r)and | I 22 ( r ) | I 22 (r)(r>1),

we see from (3.3) and (3.7) that

V l ( | I 1 | ) = V l ( | I 22 | ) =0.
(3.9)

If we can show that

lim sup r V 1 (t, k l + 1 )N ( H ( C n ( Ω ) , l , 0 ; g + ) ) (r)0,
(3.10)

we can finally conclude from (3.8) and (3.9) that

lim sup r V 1 (t, k l + 1 ) I 21 (r)0,

which gives the required result. To prove (3.10), we recall that H( C n (Ω),l,0; g + )(P) is an a-harmonic function on C n (Ω) satisfying

lim P Q , P C n ( Ω ) H ( C n ( Ω ) , l , 0 ; g + ) (P)= g + ( Q ) 0

for any Q C n (Ω). Hence, we know

< V l ( H ( C n ( Ω ) , l , 0 ; g + ) )

and so

< V l ( H ( C n ( Ω ) , l , 0 ; g + ) ) .

Thus we obtain that if l1, then

lim sup r V ( t , k l + 1 ) 1 N ( H ( C n ( Ω ) , l , 0 ; g + ) ) (r)0.
(3.11)

Therefore,

V l ( H ( C n ( Ω ) , l , 0 ; g + ) ) =0,

and so we conclude that

V l ( | H ( C n ( Ω ) , l , 0 ; g + ) | ) =0.

Similarly, we apply the method to g , then we have

V l ( | H ( C n ( Ω ) , l , 0 ; g ) | ) =0.

Since

| H ( C n ( Ω ) , l , 0 ; g ) | | H ( C n ( Ω ) , l , 0 ; g + ) | + | H ( C n ( Ω ) , l , 0 ; g ) | ,

we get the desired conclusion. □

Proof of Theorem 1.7 We see from Theorem A that

lim sup P C n ( Ω ) , P Q S n ( Ω ) P I Ω a [ g ] ( P ) = g ( Q ) and lim sup P C n ( Ω ) , P Q S n ( Ω ) P I Ω a [ | g | ] ( P ) = | g ( Q ) | .
(3.12)

Set the two subfunctions on C n (Ω) as follows:

U 1 (P)=u(P)P I Ω a [g](P)and U 2 (P)=u(P)P I Ω a [ | g | ] (P).

We have from (1.31) and (3.12)

lim sup P C n ( Ω ) , P Q S n ( Ω ) U 1 (P)0and lim sup P C n ( Ω ) , P Q S n ( Ω ) U 2 (P)0.

Hence, it follows from Lemma 2.10 that all of the limits V p ( U 1 ), W q ( U 1 ), V p ( U 2 ) and W q ( U 2 ) (< V p ( U 1 ), W q ( U 1 )+, 0 V p ( U 2 ), W q ( U 2 )+) for any p,qN{0} exist. Since

N( U 1 )(r)=N(u)(r)N ( P I Ω a [ g ] ) (r)andN( U 2 )(r)=N(u)(r)N ( P I Ω a [ | g | ] ) (r),

according to Lemma 2.8, we know all of the limits V p (u), W q (u), V p ( u + ) and W q ( u + ) exist and that

V p ( U 1 ) = V p ( u ) , W q ( U 1 ) = W q ( u ) , V p ( U 2 ) = V p ( u + ) , W q ( U 2 ) = W q ( u + )
(3.13)

for any p,qN{0}. Since

U 1 + (P) u + (P)+P I Ω a [ g ] (P),

we have from Lemma 2.8 and (1.32) that

V p ( U 1 + ) V p ( u + ) <and W q ( U 1 + ) W q ( u + ) <.

Applying Lemma 2.9 to U, we can obtain from (3.13)

u(P)P I Ω a [g](P)+ k I ( k p + 1 ) A u (k)V(r,k) φ k (Θ)+ k I ( k q + 1 ) B u (k)W(r,k) φ k (Θ)

for P=(r,Θ) C n (Ω), which is required. □

Proof of Theorem 1.8 We put u(P)=h(P) and h(P) in Theorem 1.7, respectively. Then Theorem 1.7 gives the existence of all limits V p (h), W q (h), V p ( h + ), W q ( h + )

V p ( ( h ) + ) = V p ( h ) and W q ( ( h ) + ) = W q ( h )
(3.14)

for any p,qN{0}. Since

V p ( | h | ) = V p ( h + ) + V p ( h ) and W q ( | h | ) = W q ( h + ) + W q ( h ) ,
(3.15)

it follows that both limits V p (|h|) and W q (|h|) exist. If

V p ( | h | ) <and W q ( | h | ) <,

then we see from (3.14), (3.15) and (1.33) that

V p ( h + ) <, V p ( ( h ) + ) <, W p ( h + ) <and W p ( ( h ) + ) <.

Hence, by applying Theorem 1.7 to u(P)=h(P) and h(P) again, we obtain from (1.34) that

h(P)P I Ω a [g](P)+ k I ( k p + 1 ) A h (k)V(r,k) φ k (Θ)+ k I ( k q + 1 ) B h (k)W(r,k) φ k (Θ)

and

h(P)P I Ω a [g](P)+ k I ( k p + 1 ) A h (k)V(r,k) φ k (Θ)+ k I ( k q + 1 ) B h (k)W(r,k) φ k (Θ),

respectively, which gives the required result. □

Proof of Theorem 1.10 From Theorem 1.4 we have the solution H( C n (Ω),l,m;g)(P) of the Dirichlet problem for the stationary Schrödinger operator on C n (Ω) with g satisfying (1.28). We consider the function hH( C n (Ω),l,m;g)(P). Then we see that it is an a-harmonic function in C n (Ω) and vanishes continuously on C n (Ω). Since

0 { h H ( C n ( Ω ) , l , m ; g ) } + (P) h + (P)+ { H ( C n ( Ω ) , l , m ; g ) } (P)

for any P C n (Ω),

V l ( H ( C n ( Ω ) , l , m ; g ) ) =0

and

W l ( H ( C n ( Ω ) , l , m ; g ) ) =0

from (1.28), (1.36) gives that

V p ( { h H ( C n ( Ω ) , l , m ; g ) } + ) =0

and

W q ( { h H ( C n ( Ω ) , l , m ; g ) } + ) =0.

From Lemma 2.11 we see that

h(P)H ( C n ( Ω ) , l , m ; g ) (P)= k I ( k p + 1 ) A h (k)V(r,k) φ k (Θ)+ k I ( k q + 1 ) B h (k)W(r,k) φ k (Θ)

for any P=(r,Θ) C n (Ω), where A u (k) (k=1,2,, k p + 1 1) and B u (k) (k=1,2,, k q + 1 1) are all constants. Thus, we obtain the conclusion of Theorem 1.10. □

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Acknowledgements

We wish to express our appreciation to the referee for their careful reading and some useful suggestions which led to an improvement of our original manuscript. Supported by SRFDP (No. 20100003110004) and NSF of China (No. 11071020 and No. 11271045).

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PL carried out the study, participated in the design and drafted the manuscript, GD conceived the study and participated in the design. All authors read and approve the final manuscript.

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Long, P., Deng, G. Some properties for a subfunction associated with the stationary Schrödinger operator in a cone. J Inequal Appl 2012, 295 (2012). https://doi.org/10.1186/1029-242X-2012-295

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Keywords

  • stationary Schrödinger operator
  • Poisson a-integral
  • subfunction
  • cone