R E S E A R C H
Open Access
Some properties for a subfunction associated
with the stationary Schrödinger operator in
a cone
Pinhong Long and Guantie Deng
**Correspondence: [email protected]
School of Mathematical Science, Laboratory of Mathematics and Complex Systems, Beijing Normal University, Beijing, 100875, P.R. China
Abstract
For a subfunctionuassociated 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: Primary 31B05
Keywords: stationary Schrödinger operator; Poissona-integral; subfunction; cone
1 Introduction and main results
To begin with, let us agree to some basic conventions. As usual, letSbe an open set in Rn(n≥), whereRnis then-dimensional Euclidean space. The boundary and the closure
ofSare denoted by∂SandS, respectively. LetP= (X,xn), whereX= (x,x, . . . ,xn–), and
let|P|be the Euclidean norm ofP and|P–Q|be the Euclidean distance of two points
PandQinRn. The unit sphere and the upper half unit sphere are denoted bySn–and
Sn–
+ , respectively. For simplicity, the point (,) onSn–and the set{; (,)∈}for a
set,⊂Sn–are often identified withand, respectively. For⊂R+and⊂Sn–,
the set{(r,)∈Rn;r∈, (,)∈}inRnis simply denoted by×. In particular, the
half-spaceR+×Sn+–={(X,xn)∈Rn;xn> }will be denoted byTn. ByCn(), we denote
the setR+×inRnwith the domain onSn–and call it a cone. For an intervalI⊂
R+ and⊂Sn–, setCn(;I) =I×,Sn(;I) =I×∂andCn(;r) =Cn()∩Sr. By
Sn() we denoteSn(; (, +∞)), which is∂Cn() –{O}. Furthermore, we denote bydSr
the (n– )-dimensional volume elements induced by the Euclidean metric onSr. ForP∈
Rn andr> , letB(P,r) denote the open ball with center atP and radiusrinRn, then
Sr=∂B(O,r).
We introduce the system of spherical coordinates (r,),= (θ,θ, . . . ,θn–) forP=
(X,xn) inRnvia the following formulas:
x=r n–
j=
sinθj (n≥), xn=rcosθ
and ifn≥,
xn–k+=rcosθk k–
j=
sinθj (≤k≤n– ),
where ≤r<∞, ≤θj≤π(≤j≤n– ) and –π ≤θn–≤π.
Relative to the system of spherical coordinates, the Laplace operatormay be written
=n–
r
∂ ∂r+
∂ ∂r+
*
r,
where the explicit form of the Beltrami operator*is given by Azarin (see []).
For an arbitrary domainDinRn,ADdenotes the class of non-negative radial potentials
a(P) (i.e., ≤a(P) =a(r) forP= (r,)∈D) such thata∈Lb
loc(D) with someb>n/ ifn≥
and withb= ifn= orn= .
Ifa∈AD, the stationary Schrödinger operator with a potentiala(·),
La= –+a(·)I, (.)
can be extended in the usual way from the spaceC∞(D) to an essentially self-adjoint oper-ator onL(D), whereis the Laplace operator andIis the identical operator (see Reed and
Simon [], Chapter ). Furthermore,Lahas a Greena-functionGaD(·,·). HereGaD(·,·) is
positive onDand its inner normal derivative∂GaD(·,Q)/∂nQ≥, where∂/∂nQdenotes the
differentiation atQalong the inward normal intoD. We denote this derivative byPIDa(·,·), which is called the Poissona-kernel with respect toD. There is an inequality between the Greena-functionGa
D(·,·) and that of the Laplacian, hereafter denoted byGD(·,·). It is well
known that for any potentiala(·)≥,
GaD(·,·)≤GD(·,·). (.)
The inverse inequality is much more elaborate whenDis a bounded domain inRn. For a
bounded domainDinRn, Cranston, Fabes and Zhao (see [], the casen= is implicitly
contained in []) have proved
GaD(·,·)≥M(D)GD(·,·), (.)
where M(D) =M(D,a) is positive and independent of points in D. Ifa= , obviously
M(D)≡.
Suppose that a functionu ≡–∞is upper semi-continuous inD.u∈[–∞, +∞) is called a subfunction of the Schrödinger operatorLaif the generalized mean-value inequality
u(P)≤
S(P,ρ)
u(Q)∂G
a
B(P,ρ)(P,Q)
∂nQ
dσ(Q) (.)
is satisfied at each point P∈Dwith <ρ <infQ∈∂D|P–Q|, where S(P,ρ) =∂B(P,ρ),
GaB(P,ρ)(P,·) is the Greena-function ofLainB(P,ρ) anddσ(·) is the surface area element
Denote the class of subfunctions inDbySbH(a,D). If –u∈SbH(a,D), we callua su-perfunction and denote the class of susu-perfunctions bySpH(a,D). If a functionuis both subfunction and superfunction, clearly, it is continuous and called ana-harmonic func-tion associated with the operator La. The class ofa-harmonic functions is denoted by
H(a,D) =SbH(a,D)∩SpH(a,D). In terminology we follow Levin and Kheyfits (see [] or []). From now on, we always assumeD=Cn(). For the sake of brevity, we shall write
Ga
(·,·) instead ofGCan()(·,·),PIa(·,·) instead ofPICan()(·,·),SpH(a) (resp.SbH(a)) instead
ofSpH(a,Cn()) (resp.SbH(a,Cn())) andH(a) instead ofH(a,Cn()).
For positive functions h andh, we say thathh ifh≤Mh for some constant
M> . Ifhhandhh, we say thath≈h.
Letbe a domain onSn–with a smooth boundary, and letλbe the least positive
eigen-value for –∗on(see [], p.),
∗+λϕ() = on,
ϕ() = on∂.
(.)
The corresponding eigenfunction is denoted byϕ() satisfyingϕ()dS
= . In order
to ensure the existence ofλandϕ(), we put a rather strong assumption on: ifn≥, thenis aC,α-domain ( <α< ) onSn–surrounded by a finite number of mutually
disjoint closed hypersurfaces (e.g., see Gilbarg and Trudinger [], pp.- for the defini-tion ofC,α-domain). We denote the non-decreasing sequence of positive eigenvalues of
(.) by{λ(,k)}∞k=. In the expression, we writeλ(,k) the same number of times as the dimension of the corresponding eigenspace. When the normalized eigenfunction corre-sponding toλ(,k) is denoted byϕk(), the set of sequential eigenfunctions
correspond-ing to the same value ofλ(,k) in the sequence{ϕk()}∞k= makes an orthonormal basis
for the eigenspace of the eigenvalueλ(,k). Hence, for each⊂Sn–, there is a sequence {ki}of positive integers such thatk= ,λ(,ki) <λ(,ki+),
λ(,ki) =λ(,ki+ ) =λ(,ki+ ) =· · ·=λ(,ki+– ) (.)
and {ϕki(),ϕki+(), . . . ,ϕki+–()} is an orthonormal basis for the eigenspace of the eigenvalueλ(,ki) (i= , , , . . .). It is well known thatk= andϕ() > for any∈
(see Courant and Hilbert []). For the case=Sn–
+ (n= , , , . . .),ki=i(i= , , , . . .)
whenn= , and the situation is more complicated whenn≥ (see the Remark in [] for the detailed information). For a domainand the sequence{ki}mentioned above, byI(kl)
we denote the set of all positive integers less thankl(k= , , , . . .). In spite of the fact that
I(k) =φ, the summation overI(k) of a functionS(k) of a variablekis used by promising
k∈I(k)
S(k) = .
Ifis an (n– )-dimensional compact Riemannian manifold with its boundary to be sufficiently regular, we see that
(e.g., see Cheng and Li []) and
λ(,k)≤x
ϕk() ∼B(,n)x
n–
(x→ ∞) (.)
uniformly with respect to(e.g., see Minakshisundaram and Pleijel [] or Essén and Lewis [], p. and pp.-), whereA(,n) andB(,n) are both constants depend-ing onandn. Hence, there exist two positiveM,Msuch that
Mk
n– ≤λ(,k) (k= , , , . . .) (.)
and
ϕk()≤Mk
(∈,k= , , , . . .). (.)
Solutions of an ordinary differential equation
–Q(r) –n–
r Q
(r) +λ
r+a(r)
Q(r) = ( <r<∞) (.)
are known (see [] for more references) when the potentiala∈AD. We know equation (.) has a fundamental system of positive solutions{V,W}such thatVis non-decreasing with
≤V(+)≤V(r) asr→+∞, andWis monotonically decreasing with
+∞=W(+) >W(r) asr→+∞.
We remark that both V(r,k)ϕk() andW(r,k)ϕk() (k= , , , . . .) area-harmonic on
Cn() and vanish continuously onSn(), whereV(r,k) andW(r,k) are the solutions of
equation (.) withλ=λ(,k).
LetBDbe the class of the potentialsa∈ADsuch that
lim r→∞r
a(r) =κ∈[,∞), r–ra(r) –κ∈L(,∞).
Whena∈BD, the (super)subfunctions are continuous (e.g., see []). In the rest of paper, we assume thata∈BDand we suppress this assumption for simplicity.
Denote
ι±κ = –n±
(n– )+ (κ+λ)
.
Whena∈BD, the solutionsV(r),W(r) to equation (.) normalized byV() =W() =
have the asymptotic (see [])
and
χ=ι+κ–ικ–=(n– )+ (κ+λ), χ=ωV(r),W(r)
r=, (.)
whereχis their Wronskian atr= .
Remark . Ifa= and=Sn+–,ι+= ,ι–= –nandϕ() = (ns–
n)/cosθ, wheresnis
the surface area πn/{(n/)}–ofSn–.
Letu(r,) be a function onCn(). We introduceMu(r) =M(r,u) =sup∈u(r,),u+= max{u, }andu–=max{–u, }.
We say thatu(P) (P= (r,)) satisfies the Phragmén-Lindelöf boundary condition on
Sn(), namely
lim sup P=(r,)∈Cn(),P→Q*∈S
n()
u(P)≤. (.)
LetF(P) =F(r,) be a function onCn(). For any given positive real numberr, the
integral
F(r,)ϕ()dS
is denoted byN(F)(r), when it exists. For two non-negative integerspandq, the finite or infinite limit
lim r→∞
N(F)(r)
V(r,kp+)
resp. lim r→
N(F)(r)
W(r,kq+)
is denoted byVP(F) (resp.Wq(F)), when it exists.
Iff(l) is a real finite-valued function defined on an interval (, +∞), for any givenl,l
( <l<l<∞) andl∈(, +∞), we have
E(l;f,V,W,l,l) =
f(l) V(l) W(l)
f(l) V(l) W(l)
f(l) V(l) W(l)
≥
if and only if
f(l)≤F(l;f,V,W,l,l),
whereF(l;f,V,W,l,l) has the following expression:
W(l)
W(l)
f(l)
V(l)
W(l)
– V(l)
W(l)
+ W(l)
W(l)
f(l)
V(l)
W(l)–
V(l)
W(l)
V(l)
W(l)
– V(l)
W(l)
–
.
We say thatf(l) is (V,W)-convex on (, +∞) ifE(l;f,V,W,l,l)≥ (l≤l≤l) for any
Remark . A functionf(l) is (V,W)-convex on (, +∞) if and only ifW–(l)f(l) is a
con-vex function ofW–(l)V(l) on (, +∞) or, equivalently, if and only ifV–(l)f(l) is a convex
function ofV–(l)W(l) on (, +∞); refer to Dinghas [] for the relevant properties of a convex function with respect to an ODE.
The Poissona-integralPIa[g](P) ofgrelative toCn() is defined by
PIa[g](P) =c–n
Sn()
PIa(P,Q)g(Q)dσQ, (.)
where
PIa(P,Q) =
∂Ga(P,Q) ∂nQ
, cn=
⎧ ⎨ ⎩
π, n= , (n– )sn, n≥, ∂
∂nQ denotes the differentiation atQalong the inward normal intoCn() anddσQis the surface area element onSn().
For two non-negative integersl,mand two pointsP= (r,)∈Cn() andQ= (t,)∈
Sn(), we put
VCn(),l
(P,Q)
=
k∈I(kl+)
χ()t
–V(t,k)W(r,k)ϕ k()
∂ϕk() ∂n
(.)
and
VCn(),m
(P,Q)
=
k∈I(km+)
χ()t
–W(t,k)V(r,k)ϕ k()
∂ϕk() ∂n
. (.)
We introduce two functions ofP∈Cn() andQ= (t,)∈Sn() as follows:
WCn(),l
(P,Q) =
⎧ ⎨ ⎩
V(Cn(),l)(P,Q) (≤t<∞),
( <t< ) and
WCn(),m
(P,Q) =
⎧ ⎨ ⎩
V(Cn(),m)(P,Q) ( <t< ),
(≤t<∞).
The kernelK(Cn(),l,m)(P,Q) with respect toCn() is defined by
KCn(),l,m
(P,Q) =c–n ∂G
a Cn()
∂n
(P,Q) –WCn(),l
(P,Q) –WCn(),m
In fact
KCn(),l,
(P,Q) =c–n ∂G
a Cn()
∂n
(P,Q) –WCn(),l
(P,Q) (l≥) (.)
and
KCn(), ,
(P,Q) =c–n ∂G
a Cn()
∂n
(P,Q). (.)
Based on the elaborate research, Yoshida ([] and []) 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 [] 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 [] or [] Yoshida and Miyamoto generalized some theorems (from []) to the conical case and extended the results (from [] and []) 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 [] extended Yoshida’s results (from []) to the situa-tion for the stasitua-tionary Schrödinger operator; for the relevant research on the stasitua-tionary Schrödinger operator, we may refer to Bramanti [], Kheyfits [–] and Levinet al.
[, ]. However, we find a falsehood in [] and have to make a correction. In [] or [] 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 operatorLaonCn() and with their
growth properties. Furthermore, we note the existence of solutions of the Dirichlet prob-lem for the stationary Schrödinger operatorLain 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 Sn()satisfying
∞
t–V(t)–
∂
g(t,)dσ
dt<∞ (.)
and
t–W(t)–
∂
g(t,)dσ
dt<∞. (.)
Then the function PIa
[g](P) (P= (r,))satisfies
PIa[g]∈CCn()
∩CCn()
,
LaPIa[g] = in Cn(),
PIa[g] =g on∂Cn(), lim
r→∞,P=(r,)∈Cn()V(r)
–ϕ()–PIa
and
lim
r→,P=(r,)∈Cn()W(r)
–ϕ()–PIa
[g](P) = . (.)
Remark . As to Theorem in the paper [], the factorϕn–can be replaced withϕ–
such that it is true, that is, Theorem A corrects Theorem (from []) which is a general-ization for a result from Siegel and Talvila (see []). Moreover, as to Theorem A we may follow the proof procedure of Theorem in [].
Next, we state our main results as follows.
Theorem . Let l,m be two non-negative integers and g(Q) =g(t,)be a continuous function on∂Cn()satisfying
∞
t–V(t,kl+)–
∂
g(t,)dσ
dt<∞ (.)
and
t–W(t,km+)–
∂
g(t,)dσ
dt<∞. (.)
Then
HCn(),l,m;g
(P) =
Sn()
g(Q)KCn(),l,m
(P,Q)dσQ (.)
is a solution of the Dirichlet problem for the stationary Schrödinger operator on Cn()with
g satisfying
VlH
Cn(),l,m;g=WmH
Cn(),l,m;g= . (.)
Theorem . Let l be a non-negative integer and g(Q) =g(t,)be a continuous function on∂Cn()satisfying
∞
t–V(t,kl+)–
∂
g(t,)dσ
dt<∞.
Then
HCn(),l, ;g
(P) =
Sn()
g(Q)KCn(),l,
(P,Q)dσQ (.)
is a solution of the Dirichlet problem for the stationary Schrödinger operator on Cn()with
g satisfying
VlH
Cn(),l, ;g= . (.)
It is natural to ask if in (.) can be replaced with a general functiong(Q*) onS n().
The following Theorem . gives an affirmative answer to this question. For related re-sults, we refer the readers to the paper by Levin and Kheyfits (see [], Section or [], Chapter ).
Theorem . Let p,q be two positive integers satisfying p,q≥.Let g(Q*)be a continuous
function on Sn()satisfying(.)and(.)and u(P)be a subfunction on Cn()such that lim sup
P∈Cn(),P→Q*∈Sn()
u(P)≤gQ*. (.)
Then all of the limitsVp(u),Wq(u),Vp(u+)andWq(u+) (–∞<Vp(u),Wq(u)≤+∞, ≤
Vp(u+),Wq(u+)≤+∞)exist.Moreover,when
Vp
u+< +∞ and Wq
u+< +∞, (.)
u(P)≤PIa[g](P) +
k∈I(kp+)
Au(k)V(r,k)ϕk()
+
k∈I(kq+)
Bu(k)W(r,k)ϕk() (.)
for any P= (r,)∈Cn(),whereAu(k) (k= , , . . . ,kp+– )andBu(k) (k= , , . . . ,kq+– )
are all constants.
As an application of Theorems A and ., we obtain the following result.
Theorem . Let p, q be two positive integers satisfying p,q ≥. Let g(Q) be defined as in Theorem .and h(P)be any solution of the Dirichlet problem for the stationary Schrödinger operatorLaon Cn()with g.Then all of the limitsVp(h),Wq(h),Vp(|h|)and
Wq(|h|) (–∞<Vp(h),Wq(h)≤+∞, ≤Vp(|h|),Wq(|h|)≤+∞)exist.Moreover,when
Vp
|h|< +∞ and Wq
|h|< +∞, (.)
h(P) =PIa[g](P) +
k∈I(kp+)
Ah(k)V(r,k)ϕk()
+
k∈I(kq+)
Bh(k)W(r,k)ϕk() (.)
for any P= (r,)∈Cn(),whereAh(k) (k= , , . . . ,kp+– )andBh(k) (k= , , . . . ,kq+– )
are all constants.
Remark . Forp=q= , Theorems . and . come from Qiao and Deng []. Fur-thermore, whena= andp=q= , Theorems . and . are due to Yoshida (see [], Theorems and (II)). In fact, fork∈I(kp+) we knowAu(k),Bu(k) (orAh(k),Bh(k)) are
equal to the correspondingV(u),W(u) (orV(h),W(h)), respectively. Without the potential function, we may refer to Yoshida (see []).
m,respectively.If h(P)is any solution of the Dirichlet problem for the stationary Schrödinger operatorLaon Cn()with g satisfying
Vp
h+= and Wq
h+= , (.)
h(P) =HCn(),l,m;g
(P) +
k∈I(kp+)
Ah(k)V(r,k)ϕk()
+
k∈I(kq+)
Bh(k)W(r,k)ϕk() (.)
for any P= (r,)∈Cn(),whereAh(k) (k= , , . . . ,kp+– )andBh(k) (k= , , . . . ,kq+– )
are all constants.
Theorem . Let l be a non-negative integer and p be a positive integer satisfying p≥.
If h(r,)is a generalized harmonic function on Cn()and continuous on Cn()such that
the restriction h=h|∂Cn()of h to∂Cn()satisfies
∞
t–V(t,kl+)–
∂n
h(t,)dσ
dt<∞
for some non-negative integer l,and for a positive integerp
lim sup r→∞
logN(h+)(r)
logV(r,kp+)
<∞,
then,for some positive integer p=max{l,p},
h(P) =HCn(),l;g
(P) +
k∈I(kp+)
Ah(k)V(r,k)ϕk()
for any P= (r,)∈Cn(),whereAh(k) (k= , , . . . ,kp+– )are all constants.
Remark . If we takea= , Theorems . and . are similar to Theorems and in [], respectively. In [] Yoshida and Miyamoto considered the case whenq= ,m= anda= about Theorem . and gave the proof. In addition, with Theorem . we easily get the conclusion of Theorem ., then we do not have to prove it.
2 Some lemmas
In our arguments, we need some important results and techniques, which result from [, , , ], and [] (Lemma and Remark).
Lemma .
PIa(P,Q)≈t–V(t)W(r)ϕ() ∂ϕ()
∂n
(.)
resp. PIa(P,Q)≈V(r)t–W(t)ϕ()∂ϕ()
∂n
for any P= (r,)∈Cn()and any Q= (t,)∈Sn()satisfying <rt ≤(resp. <rt ≤).
In addition,
PI(P,Q) ϕ()
tn–
∂ϕ()
∂n
+ rϕ()
|P–Q|n ∂ϕ()
∂n
(.)
for any P= (r,)∈Cn()and any Q= (t,)∈Sn(; (r,r)).
Lemma . Let a∈BD.For a non-negative integer kl+,we have
c–n ∂G
a Cn()
∂n
(P,Q) –WCn(),l
(P,Q)
≤M(kl+,n,s)V(r,kl+)t–W(t,kl+)ϕk() ∂ϕk()
∂n
(.)
for any P = (r,)∈Cn() and Q= (t,)∈Sn() satisfying r≤st ( <s< ), where
M(kl+,n,s)is a constant dependent on n,kl+and s.
Lemma . Let g(Q)be a locally integrable and upper semicontinuous function on∂Cn().
Let W(P,Q)be a function of P∈Cn()and Q∈∂Cn()such that for any fixed P∈Cn()
the function W(P,Q)of Q∈∂Cn()is a locally integrable function on∂Cn().Put
K(P,Q) =c–n ∂G
a Cn()
∂n
(P,Q) –W(P,Q) P∈Cn(),Q∈∂Cn()
. (.)
Suppose that the following(I)and(II)are satisfied. (I) For anyQ*∈∂C
n()and anyε> ,there exist a neighborhoodU(Q*)ofQ*inRn and a numberR( <R<∞)such that
Sn(;[R,∞))
g(Q)K(P,Q)dσ<ε (.)
for anyp= (r,)∈∂Cn()∩U(Q*)
(II) For anyQ*∈∂C
n()and anyR( <R<∞),
lim sup P→Q*,P∈Cn()
Sn(;(,R))
g(Q)K(P,Q)dσ= . (.)
Then
lim sup P→Q*,P∈C
n()
Sn()
g(Q)K(P,Q)dσ≤g
Q* (.)
for anyQ*∈∂C n().
Lemma . If h(r,) is an a-harmonic function on Cn()vanishing continuously on
Sn(),
Er;N(h),V,W,r,r
=
for any r,r( <r<r< +∞)and every r( <r< +∞).
Lemma . If f(l)is(V,W)-convex on(,d) ( <d≤+∞),then
β=lim l→
f(l)
W(l) (–∞<α≤+∞)
exists.Further,ifβ≤,V–(l)f(l)is non-decreasing on(,d).
It is known thatCn() is regular, the Dirichlet problem forandLais solvable in it (see
[]). Based on this fact, Lemmas ., . and . could be derived from (.), (.), (.), Remark ., Lemmas . and . with their means of proof essentially due to Yoshida (see [], Theorems . and ., and [], Lemma ).
Lemma . If u(r,)is a subfunction on Cn()satisfying the Phragmén-Lindelöf
bound-ary condition on Sn(),then
N(u)(r) > –∞
for <r< +∞and N(u)(r)is(V,W)-convex on(, +∞).If there are three numbers r,r
and rsatisfying <r<r<r< +∞such that
Er;N(u),V,W,r,r
= ,
we have that
() E(r;N(u),V,W,r,r) = (r≤r≤r);
() u(r,)is ana-harmonic function onCn(; (r,r))and vanishes continuously on
Sn(; (r,r)).
Lemma . Let g(Q)be defined as in Theorem..Then PIa
[g](P) (resp.PIa[|g|](P))is an
a-harmonic function on Cn()such that both of the limitsV(PIa[g])andW(PIa[g]) (resp.
V(PIa
[|g|])andW(PIa[|g|]))exist,and
VPIa[g]
=WPIa[g]
= resp.VPIa
|g|=WPIa
|g|= .
Lemma . Let u(P)be a subfunction on Cn()satisfying the Phragmén-Lindelöf
bound-ary condition on Sn().If(.)is satisfied for p= and q= ,
u(P)≤V(u)V(r) +W(u)W(r)ϕ()
By the Kelvin transformation (see [], p.), Lemmas . and ., we immediately have the following result, which is due to Yoshida in the casea= (see [], Theorem .). Lemma . Let u(P)be defined as in Lemma..Then
() Both of the limitsV(u)andW(u)(–∞<V(u),W(u)≤+∞)exist. () IfW(u)≤,thenV–(r)N(u)(r)is non-decreasing on(, +∞).
() IfV(u)≤,thenW–(r)N(u)(r)is non-increasing on(, +∞).
Lemma . Let H(r,)be an a-harmonic function in Cn()vanishing continuously on ∂Cn(),and p,q be two positive integers.h satisfies
Vp
h+= and Wq
h+= ,
then
h(P) =
k∈I(kp+)
Ah(k)V(r,k)ϕk() +
k∈I(kq+)
Bh(k)W(r,k)ϕk()
for any P= (r,)∈Cn(),whereAu(k) (k= , , . . . ,kp+– )andBu(k) (k= , , . . . ,kq+– )
are all constants.
Remark . Whenq= and a= , Yoshida states the result in []. Later Qiao [] proves Lemma . when q= . Similarly, we may complete the proof of Lemma .. Herein we leave out the detailed information for the proof.
3 Proofs of the theorems
Proof of Theorem. For any fixedP∈Cn(), we take a numberRsatisfyingsR>max{,r}
( <s< ). Then from Lemma . and (.)
Sn(;[R,∞))
g(Q)KCn(),l,
(P,Q)dσ
=
Sn(;[R,∞))
g(Q)C–n ∂G
a Cn()
∂n
(P,Q) –WCn(),l
(P,Q)dσ
V(r,kl+)ϕk()
∞
R
t–V(t,kl+)–
∂
g(t,)dσ
dt<∞. (.)
ThenH(Cn(),l, ;g)(P) is absolutely convergent and finite for anyP∈Cn(). We remark
that K(Cn(),l, )(P,Q) is a harmonic function ofP∈Cn() for anyQ∈Cn(). Thus,
H(Cn(),l, ;g)(P) is a generalized harmonic function ofP∈Cn().
Next, we consider the boundary behavior ofH(Cn(),l, ;g)(P). To prove that lim sup
P→Q*,P∈C n()
HCn(),l, ;g
(P) =gQ*
for anyQ*∈∂C
n(), we may apply Lemma . tog(Q) and –g(Q) by putting
W(P,Q) =WCn(),l,
which is locally integrable onCn() for any fixedP∈Cn(). Letδbe a positive number
and takeQ*= (t*,*)∈∂C
n() and anyε> . Then from (.) and (.) we can choose a
numberR(sR>max{,r}) ( <s< ) such that for anyP∈Cn()∩Uδ(Q*), whereUδ(Q*) = {X∈Rn:|X–Q*|<δ}andδis a positive number
Sn(;[R,∞))
g(Q)KCn(),l,
(P,Q)dσ<ε,
which is (I) in Lemma .. Because
lim
→*ϕk() = (k= , , . . .) asP= (r,)→Q*= (t*,*)∈∂C
n(), we know that for anyQ*∈∂Cn() andQ∈∂Sn(), lim
P→Q*,P∈C n()W
Cn(),l,
(P,Q) = .
According to Lemma ., we get the required results. Next, we note the inequality
NHCn(),l, ;g+(r)≤I(r) +I(r), (.)
where
I(r) =
Sn(;[r,∞))
g+(Q)KCn(),l,
(P,Q)dσQ
ϕk()dσ
and
I(r) =
Sn(;[,r))
g+(Q)KCn(),l,
(P,Q)dσQ
ϕk()dσ
forP= (r,), <r<∞. For any positive numberε, from (.) we can take a sufficiently large numberrsuch that
∞
r
t–V(t,kl+)–
∂
g(t,)dσ
dt< ε
LM (r>r),
whereMis the constant in Lemma . and
L=
ϕk()dσ.
Then from Lemma . we have
≤I(r)≤L(kl+)M(n,kl+,s)V(r,kl+)
∞
r
t–V(t,kl+)–
∂
g+(t,)dσ
dt
≤εV(r,kl+) (r>r) (.)
Following this, we see the inequality
I(r)≤I(r) +I(r), (.)
where
I(r) =
Sn(;[r,∞))
g+(Q)c–n ∂G
a Cn()
∂n
(P,Q)dσQ
ϕk()dσ
and
I(r) =
Sn(;[,r))
g+(Q)WCn(),l
(P,Q)dσQ
ϕk()dσ
forP= (r,) andr> . First, we know from (.) and (.) that ifl≥
I(r)
k∈I(km+)
kV(r,k)k(r) (r> ), (.)
where
k(r) =
r
t–V(t,k)–
∂
g+(t,)dσ
dt r> ,k∈I(km+)
.
We claim that
k(r) =O
V(r,kl+)V(r,k)– r> ,k∈I(km+)
. (.)
First we note increasingk(r) and Lemma C. in [], Chapter or [], then by (.) we
get
k(r)V(r,kl+)–V(r,k)
r
t–V(t,k+)–
∂
g+(t,)dσ
dt<∞ (r> ).
Hence, we can conclude that ifl≥, then
Vl(I) = . (.)
Next, we seeIand note that
<I=N
HCn(),l, ;g+
(r) –I*(r) +I* (r), (.) where
I*(r) =
Sn(;[r,∞))
g+(Q)KCn(),l,
(P,Q)dσQ
ϕk()dσ
and
I* (r) =
Sn(;[,r))
g+(Q)KCn(),l,
(P,Q)dσQ
Since
I*(r)≤I(r) and I* (r)≤I(r) (r> ),
we see from (.) and (.) that
VlI*=VlI* = . (.)
If we can show that
lim sup r→∞
V–(t,kl+)N
HCn(),l, ;g+
(r)≤, (.)
we can finally conclude from (.) and (.) that
lim sup r→∞
V–(t,kl+)I(r)≤,
which gives the required result. To prove (.), we recall that –H(Cn(),l, ;g+)(P) is an
a-harmonic function onCn() satisfying
lim P→Q*,P∈Cn()–H
Cn(),l, ;g+
(P) =g+Q*≤
for anyQ*∈Cn(). Hence, we know
–∞<Vl
–HCn(),l, ;g+
≤ ∞
and so
–∞<Vl
HCn(),l, ;g+
≤ ∞.
Thus we obtain that ifl≥, then
lim sup
r→∞ V(t,kl+)
–NHC
n(),l, ;g+
(r)≤. (.)
Therefore,
Vl
HCn(),l, ;g+
= ,
and so we conclude that
VlH
Cn(),l, ;g+= .
Similarly, we apply the method tog–, then we have
VlH
Since
HCn(),l, ;g≤H
Cn(),l, ;g++H
Cn(),l, ;g–,
we get the desired conclusion.
Proof of Theorem. We see from Theorem A that
lim sup P∈Cn(),P→Q*∈Sn()
PIa[g](P) =g
Q* and
lim sup P∈Cn(),P→Q*∈Sn()
PIa
|g|(P) =gQ*.
(.)
Set the two subfunctions onCn() as follows:
U(P) =u(P) –PIa[g](P) and U(P) =u(P) –PIa
|g|(P).
We have from (.) and (.)
lim sup P∈Cn(),P→Q*∈Sn()
U(P)≤ and lim sup
P∈Cn(),P→Q*∈Sn()
U(P)≤.
Hence, it follows from Lemma . that all of the limits Vp(U), Wq(U), Vp(U) and
Wq(U) (–∞<Vp(U),Wq(U)≤+∞, ≤Vp(U),Wq(U)≤+∞) for anyp,q∈N∪ {}
exist. Since
N(U)(r) =N(u)(r) –N
PIa[g]
(r) and N(U)(r) =N(u)(r) –N
PIa
|g|(r),
according to Lemma ., we know all of the limitsVp(u),Wq(u),Vp(u+) andWq(u+) exist
and that
Vp(U) =Vp(u), Wq(U) =Wq(u),
Vp(U) =Vp
u+, Wq(U) =Wq
u+
(.)
for anyp,q∈N∪ {}. Since
U+(P)≤u+(P) +PIa[g]–(P),
we have from Lemma . and (.) that
Vp
U+≤Vp
u+<∞ and Wq
U+≤Wq
u+<∞. Applying Lemma . toU, we can obtain from (.)
u(P)≤PIa[g](P) +
k∈I(kp+)
Au(k)V(r,k)ϕk() +
k∈I(kq+)
Bu(k)W(r,k)ϕk()
Proof of Theorem. We putu(P) =h(P) and –h(P) in Theorem ., respectively. Then Theorem . gives the existence of all limitsVp(h),Wq(h),Vp(h+),Wq(h+)
Vp
(–h)+=Vp
h– and Wq
(–h)+=Wq
h– (.)
for anyp,q∈N∪ {}. Since Vp
|h|=Vp
h++Vp
h– and Wq
|h|=Wq
h++Wq
h–, (.)
it follows that both limitsVp(|h|) andWq(|h|) exist. If
Vp
|h|<∞ and Wq
|h|<∞,
then we see from (.), (.) and (.) that
Vp
h+<∞, Vp
(–h)+<∞, Wp
h+<∞ and Wp
(–h)+<∞.
Hence, by applying Theorem . tou(P) =h(P) and –h(P) again, we obtain from (.) that
h(P)≤PIa[g](P) +
k∈I(kp+)
Ah(k)V(r,k)ϕk() +
k∈I(kq+)
Bh(k)W(r,k)ϕk()
and
h(P)≥PIa[g](P) +
k∈I(kp+)
Ah(k)V(r,k)ϕk() +
k∈I(kq+)
Bh(k)W(r,k)ϕk(),
respectively, which gives the required result.
Proof of Theorem . From Theorem . we have the solution H(Cn(),l,m;g)(P) of
the Dirichlet problem for the stationary Schrödinger operator onCn() withg
satisfy-ing (.). We consider the function h–H(Cn(),l,m;g)(P). Then we see that it is an
a-harmonic function inCn() and vanishes continuously on∂Cn(). Since
≤h–HCn(),l,m;g +
(P)≤h+(P) +HCn(),l,m;g –
(P) for anyP∈Cn(),
Vl
HCn(),l,m;g
–
=
and
Wl
HCn(),l,m;g
–
=
from (.), (.) gives that
Vp
h–HCn(),l,m;g +
and
Wq
h–HCn(),l,m;g +
= .
From Lemma . we see that
h(P) –HCn(),l,m;g
(P) =
k∈I(kp+)
Ah(k)V(r,k)ϕk() +
k∈I(kq+)
Bh(k)W(r,k)ϕk()
for anyP= (r,)∈Cn(), whereAu(k) (k= , , . . . ,kp+– ) andBu(k) (k= , , . . . ,kq+– )
are all constants. Thus, we obtain the conclusion of Theorem ..
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
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.
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).
Received: 9 February 2012 Accepted: 21 November 2012 Published: 12 December 2012 References
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doi:10.1186/1029-242X-2012-295
