Further result on Dirichlet Sch type inequality and its application

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R E S E A R C H

Open Access

Further result on Dirichlet-Sch type

inequality and its application

Liquan Wan

*

*_{Correspondence:} [email protected] School of Accounting, Henan University of Economics and Law, Zhengzhou, 450046, China

Abstract

In this paper we deal with a theoretical question raised in connection with the application of Dirichlet-Sch type inequality, obtained by Huang (Int. Math. J. 27(02):1650009, 2016), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers. We also discuss a particular case of it in more detail. As an application, we deduce the least harmonic majorant and log-concavity of extended subharmonic functions.

Keywords: Dirichlet-Sch type inequality; harmonic function; Schrödinger PWB solution

1 Introduction

Letbe the subset of the upper half unit sphere. The set R+× in Rnis called a cone. We denote it byCn(), where⊂S. The setsI×andI×∂with an interval on R are denoted byCn(;I) andSn(;I), respectively. We denoteCn()∩SRandSn(; (, +∞))

bySn(;R) andSn(), respectively.

Furthermore, we denote bydσ(resp.dSR) the (n– )-dimensional volume elements

in-duced by the Euclidean metric on∂Cn() (resp.SR) and bydwthe elements of the

Eu-clidean volume in Rn_.

It is well known (see,e.g., [], p.) that

∗ϕ() +λϕ() =  in,

(.) ϕ() =  on∂,

where∗is the Laplace-Beltrami operator. We denote the least positive eigenvalue of this boundary value problem (.) byλand the normalized positive eigenfunction correspond-ing toλbyϕ(),ϕ₍₎_dS_{= .}

We remark that the function rℵ±ϕ() is harmonic in Cn(), belongs to the class

C(Cn()\{O}) and vanishes onSn(), where

ℵ±= –n+ ±(n– )_{+ }_λ_.

For simplicity we shall writeχinstead ofℵ+_–_ℵ–_.

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For simplicity we shall assume that the boundary of the domainis twice continuously diﬀerentiable,ϕ∈C₍_{) and} ∂ϕ

∂n>  on∂. Then (see [], p.-)

dist(,∂)≈ϕ(), (.)

where∈.

Letδ(P) =dist(P,∂Cn()), we have

ϕ()≈δ(P), (.)

for anyP= (,)∈(see []).

Letu(r,) be a function onCn(). For any givenr∈R+, the integral

u(r,)ϕ()dS,

is denoted byNu(r), when it exists. The ﬁnite or inﬁnite limits

lim r→∞r

–ℵ+_N

u(r) and lim r→r

–ℵ–_N

u(r)

are denoted byUuandVu, respectively, when they exist.

Remark  A functiong(t) on (,∞) isAd,d-convex if and only ifg(t)td is a convex func-tion oftd_(d₌_d₊_{d) on (,}_∞_{), or, equivalently, if and only if}_g(t)t–d_{is a convex function} oft–d_{on (,}_∞_).

Remark  Nu(r) is aAℵ+_,_γ–-convex on (,∞), where uis a subharmonic function on

Cn() such that

lim sup P∈Cn(),P→Q∈∂Cn()

u(P)≤c, (.)

wherecis a nonnegative number (see []). The function

PCn()(P,Q) =

∂GCn()(P,Q)

∂nQ

is called the ordinary Poisson kernel, whereGCn()is the Green function. The Poisson integral ofgrelative toCn() is deﬁned by

PICn()[g](P) =

 cn

Sn()P

Cn()(P,Q)g(Q)dσ,

wheregis a continuous function on∂Cn() and∂nQ∂ denotes the diﬀerentiation atQalong

the inward normal intoCn().

We set functionsf satisfying

Sn()

|f(t,)|p

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where – <p< +∞and –ℵ+_–_n_{+ }

p <γ <

–ℵ+_–_n_{+ } p +n– .

Let – <p< +∞. we denoteAthe class of all measurable functionsg(t,) (Q= (t,) =

(Y,yn)∈Cn()) satisfying the following inequality:

Cn()

|g(t,)|p–_ϕ

 +tγ– dw<∞

and the classB, consists of all measurable functionsh(t,) ((t,) = (Y,yn)∈Sn())

sat-isfying

Sn()

|h(t,)|q

 +tγ

∂ϕ

∂ndσ<∞, whereq> .

We will also consider the class of all continuous functionsu(t,) ((t,)∈Cn())

har-monic inCn() withu+(t,)∈A((t,)∈Cn()) andu+(t,)∈B ((t,)∈Sn()) is

denoted byC(see []).

In , Jiang, Hou and Peixoto-de-Büyükkurt (see []) obtained the following result.

Theorem A Let g be a measurable function on∂Tnsuch that

∂Tn

 +|Q|–ng(Q)dQ<∞.

Then the harmonic functionPITn[g]satisﬁesPITn[g](P) =o(rsecn–θ)as r→ ∞in Tn.

Recently,Wang,Huang and N.Yamini(see[])generalized TheoremAto the conical case.

Theorem B Let g be a continuous function on∂Cn()satisfying(.)with p=q= and

γ =ℵ+_{+  –}_ℵ–_._Then

UPICn()[g]=UPICn()[|g|]= .

The remainder of the paper is organized as follows:in Section,we shall give our main theorem;in Section,some necessary lemmas are given;in Section,we shall prove the main result.

2 Main result

In this section, we give the main result of this paper.

Our main aim is to give a least harmonic majorant of a nonnegative subharmonic func-tion onCn().

Theorem  Let u be a function subharmonic inCn()and u be the restriction of u to

∂Cn().If usatisfy(.)and–≤Uu≤then

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for any P= (r,)∈Cn(),where hu(P)is the least harmonic majorant of u onCn()and

has the following expression:

hu(P) =PICn()

u (P) +Vurℵ

–

ϕ() +Uurℵ

+ ϕ().

Remark  Theorem  solves a theoretical question raised in connection with the applica-tion of Dirichlet-Sch type inequality, obtained by Huang (see []), which has been already applied to obtain multiplicity results for boundary value problems in several recent papers.

3 Main lemmas

In order to prove our main result, we need the following lemmas.

Lemma (see []) Let u be a function subharmonic onCn()satisfying(.).Then the

limitUu(– <Uu≤)exists.

Lemma  Let u be a function subharmonic onCn()satisfying(.)and

Uu+≤ and U_u+< +∞. (.)

Then

u(r,)≤Vu+rℵ –

ϕ() +Uu+rℵ +

ϕ(). (.)

Proof Take any (r,)∈Cn() and any pair of numbersτ,τ( <τ<r<τ< +∞). We deﬁne a boundary function on∂Cn(; (τ,τ)) by

ν(r,) = ⎧ ⎨ ⎩

u(τi,) on{τi} ×(i= , ),

 on [τ,τ]×∂.

If we denote Schrödinger PWB solution of the Dirichlet-Sch problem onCn(; (τ,τ))

withνbyHν((r,);Cn(; (τ,τ))), then we have

u(r,)≤Hν

(r,);Cn

; (τ,τ)

≤

u+(τ,)∂GCn(;(τ,τ))((τ,), (r,))

∂R τ

n–  dS

–

u+(τ,)∂GCn(;(τ,τ))((τ,), (r,))

∂R τ

n–  dS,

which shows that (.) holds from (.).

Lemma  Let g be a locally integrable function on∂Cn()satisfying(.)and u be a

sub-harmonic function onCn()satisfying

–≤ lim inf P∈Cn(),P→Q∈∂Cn()

u(P) –PICn()[g](P)

≤ (.)

and

lim inf P∈Cn(),P→Q∈∂Cn()

u+(P) –PICn()

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Then the limitsUuandVu+(–∞<U_u≤, ≤U_u+≤+∞)exist,and if(.)is satisﬁed, then

u(P)≤PICn()[g](P) +Vu+rℵ –

ϕ() +Uu+rℵ +

ϕ() (.)

for any P= (r,)∈Cn().

Proof Put

U(P) =u(P) –PICn()[g](P) and U(P) =u+(P) –PICn()

|g| (P)

onCn(). From (.) and (.) we have

lim inf

P∈Cn(),P→QU(P)≤– and P∈Clim infn(),P→QU

_(P)_≤_–.

Hence it follows from Lemma  that the limitsUUandVU (– <UU≤, ≤VU≤)

exist. So Theorem B gives the existence of the limitsUu,Vu+,

UU=Vu and UU=Vu+. (.)

Since ≤U+(P)≤u+(P) + (PICn()[g])

–_{(P) on} _C

n(), it also follows from Theorem B

and (.) that

VU+≤V_u+<∞,

which together with Lemma  gives the conclusion.

Lemma  Let g be a lower semi-continuous function on∂Cn()satisfying(.)and u be a

superharmonic function onCn()such that

lim inf

P∈Cn(),P→Qu(P)≤g(Q) +c (.)

for any Q∈∂Cn()and c is a positive number.Then the limitUu(–≤Uu≤+)exists,

and ifUu< +∞,then

u(P)≤PICn()[g](P) +Vurℵ

–

ϕ() +Uurℵ

+ ϕ()

for any P= (r,)∈Cn().

Proof Since –gis upper semi-continuous function in∂Cn(), it follows from [], p., that

lim inf

P∈Cn(),P→QPICn()[g](P)≥g(Q) –c. (.)

We see from (.) and (.) that

–≤ lim sup P∈Cn(),P→Q

u(P) –PICn()[g](P)

≤,

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Lemma  Let u be a subharmonic function in Cn()such that u=u|∂Cn() satisﬁes

(.).Then PICn()[u](P)≤h(P)onCn(),where h(P)is the any harmonic majorant of

u onCn().

Proof Take anyP= (r,)∈Cn(). Let be any positive number. In the same way as in

the proof of Lemma , we can chooseRsuch that

Sn(;(R,∞))

PCn()

P,Qu(Q)dσ<

. (.)

Further, take an integerj(j>R) such that (see [])

Sn(;(,R))

∂j(P,Q)

∂nQ

u(Q)dσ<

. (.)

Since

Sn(;(,R))

∂GCn(;(,j))(P,Q)

∂nQ

u(Q)dσ≤Hu

P;Cn

; (,j)

for anyP∈Cn(; (,j)), we have from (.) and (.)

PICn()

u P–Hu

P;Cn

; (,j)

≤

Sn(;(,R))

∂j(P,Q)

∂nQ

u(Q)dσ

+  cn

Sn(;(R,∞))

PCn()

P,Qu(Q)dσ

<. (.)

Here note that Hu(P;Cn(; (,j))) is the least harmonic majorant of u onCn(; (,j))

(see [], Theorem .). Ifhis a harmonic majorant ofuonCn(), then

Hu

P;Cn

; (,j)≤hP.

Thus we obtain from (.)

PICn()

u P<hP+,

which gives the conclusion of Lemma .

4 Proof of Theorem 1

LetP be any point ofCn() and be any positive number. By the Vitali-Carathéodory

theorem with respect to the Schrödinger operator (see [], p.), there exists a lower semi-continuous functiong(Q) on∂Cn() such that

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and

PICn()

g (P) <PICn()

u (P) +. (.)

Since

lim

P∈Cn(),P→Qu(P)≤u

_(Q)_≤_g_(Q)

for anyQ∈∂Cn() from (.), it follows from [], Lemma ., that the limitsUuandUu

exist, and if –≤Uu<  and –≤Vu< , then

u(P)≤PICn()

g (P) +Vurℵ

–

ϕ() +Uurℵ

+

ϕ(). (.)

Hence we see from (.) and (.) that (.) holds.

Next we call the least harmonic majorant ofuonCn():hu(P). Seth(P) is a Schrödinger

harmonic function inCn() such that (see [])

u(P)≤h(P) +. (.)

Put

h∗(P) =hu(P) –h(P) onCn().

It is easy to see that

h∗(P)≤hu(P).

It follows from Theorem B thatVh∗+< +∞. Further, from Lemma  we see that lim sup

P∈Cn(),P→Q

h∗(P) = lim inf P∈Cn(),P→Q

PICn()

u (P) –h(P)≤–.

From Theorem B and (.) we know

Vh∗=Vhu–Vh=Vu–Uh≤Uu–Uu= .

We see from Lemma  that –≤h∗(P)≤onCn(), which shows thathu(P) is the least

harmonic majorant inCn(). Theorem  is proved.

5 Conclusion

In this article, we dealt with a theoretical question raised in connection with the applica-tion of Dirichlet-Sch type inequality. Addiapplica-tionally, we discussed a particular case of it in more detail. As applications, we deduced the least harmonic majorant and log-concavity of extended subharmonic functions.

Competing interests

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Acknowledgements

I would like to thank the referees for their constructive suggestions and useful comments which resulted in an improved version of this paper. This work was supported by the 2015 Universities Philosophy Social Sciences Innovation Team of Henan Province: the Institutional Arrangements for Mixed Ownership Reform of Stateowned Enterprises in Henan Province (No. 2015-CXTD-09).

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Received: 4 March 2017 Accepted: 21 April 2017

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