AAA Abstract and Applied Analysis 1687-0409 1085-3375 Hindawi Publishing Corporation 578197 10.1155/2012/578197 578197 Research Article Dirichlet Problem for the Schrödinger Operator in a Half Space Su Baiyun Gossez Jean Pierre Department of Mathematics and Information Science Henan University of Economics and Law Zhengzhou 450002 China 2012 28 8 2012 2012 22 04 2012 12 07 2012 2012 Copyright © 2012 Baiyun Su. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

For continuous boundary data, the modified Poisson integral is used to write solutions to the half space Dirichlet problem for the Schrödinger operator. Meanwhile, 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.

1. Introduction and Results

Let R and R+ be the sets of all real numbers and of all positive real numbers, respectively. Let Rn(n2) denote the n-dimensional Euclidean space with points x=(x',xn), where x'=(x1,x2,,xn-1)Rn-1 and xnR. The unit sphere and the upper half unit sphere in Rn are denoted by Sn-1 and S+n-1, respectively. The boundary and closure of an open set D of Rn are denoted by D and D¯, respectively. The upper half space is the set H={(x',xn)Rn:  xn>0}, whose boundary is H.

For a set E, ER+{0}, we denote {xH;|x|E} and {xH;|x|E} by HE and HE, respectively. We identify Rn with Rn-1×R and Rn-1 with Rn-1×{0}, writing typical points x,yRn as x=(x',xn),y=(y',yn), where y=(y1,y2,,yn-1)Rn-1, and putting (1.1)x·y=j=1nxjyj,|x|=x·x,Θ=x|x|,Φ=y|y|.

For xRn and r>0, let B(x,r) denote the open ball with center at x and radius r(>0) in Rn. We will say that a set EH has a covering {rj,Rj} if there exists a sequence of balls {Bj} with centers in H such that Ej=1Bj, where rj is the radius of Bj and Rj is the distance between the origin and the center of Bj.

Let 𝒜a denote the class of nonnegative radial potentials a(x), that is, 0a(x)=a(|x|), xH, such that aLlocb(H) with some b>n/2 if n4 and with b=2 if n=2 or n=3.

This paper is devoted to the stationary Schrödinger equation (1.2)SSEu(x)=-Δu(x)+a(x)u(x)=0, where xH, Δ is the Laplace operator and a𝒜a. These solutions are called a-harmonic functions or generalized harmonic functions associated with the operator SSE. Note that they are (classical) harmonic functions in the case a=0. Under these assumptions the operator SSE can be extended in the usual way from the space C0(H) to an essentially self-adjoint operator on L2(H) (see ). We will denote it by SSE as well. This last one has a Green function Ga(x,y). Here, Ga(x,y) is positive on H and its inner normal derivative Ga(x,y)/n(y')0. We denote this derivative by Pa(x,y'), which is called the Poisson a-kernel with respect to H. We remark that G(x,y) and P(x,y') are the Green function and Poisson kernel of the Laplacian in H, respectively.

Let Δ* be a Laplace-Beltrami operator (spherical part of the Laplace) on the unit sphere. It is known (see, e.g., [4, page 41]) that the eigenvalue problem (1.3)Δ*φ(Θ)+λφ(Θ)=0,ΘS+n-1,φ(Θ)=0,ΘS+n-1, has the eigenvalues λj=j(j+n-2)(j=0,1,2). Corresponding eigenfunctions are denoted by φjv(1vvj), where vj is the multiplicity of λj. We norm the eigenfunctions in L2(S+n-1) and φ1=φ11>0.

Hence, well-known estimates (see, e.g., [5, page 14]) imply the following inequality: (1.4)v=1vjφjv(Θ)φjv(Φ)nΦM(n)j2n-1, where the symbol M(n) denotes a constant depending only on n.

Let Vj(r) and Wj(r) stand, respectively, for the increasing and nonincreasing, as r+, solutions of the equation (1.5)-y′′(r)-n-1ry(r)+(λjr2+a(r))y(r)=0,0<r<, normalized under the condition Vj(1)=Wj(1)=1.

We will also consider the class a, consisting of the potentials a𝒜a such that there exists a finite limit limrr2a(r)=k[0,). Moreover, r-1|r2a(r)-k|L(1,). If aa, then solutions of (1.2) are continuous (see ).

In the rest of paper, we assume that aa, and we will 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 (1.6)ιj,k±=2-n±(n-2)2+4(k+λj)2(j=0,1,2,3).

Remark 1.1.

ι j , 0 + = j ( j = 0,1 , 2,3 , ) in the case a=0.

It is known (see ) that in the case under consideration the solutions to (1.5) have the asymptotics (1.7)Vj(r)~d1rιj,k+,Wj(r)~d2rιj,k-,asr, where d1 and d2 are some positive constants.

If a𝒜a, it is known that the following expansion for the Green function Ga(x,y) (see [8, Chapter 11], [1, 9]) (1.8)Ga(x,y)=j=01χ(1)Vj(min(|x|,|y|))Wj(max(|x|,|y|))(v=1vjφjv(Θ)φjv(Φ)), where |x||y| and χ(1)=w(W1(r),V1(r))|r=1 is its Wronskian. The series converges uniformly if either |x|s|y| or |y|s|x|(0<s<1).

For a nonnegative integer m and two points x,yH, we put (1.9)K(a,m)(x,y)={0if|y|<1,K~(a,m)(x,y)if1|y|<, where (1.10)K~(a,m)(x,y)=j=0m1χ(1)Vj(|x|)Wj(|y|)(v=1vjφjv(Θ)φjv(Φ)).

We introduce another function of x,yH(1.11)G(a,m)(x,y)=Ga(x,y)-K(a,m)(x,y).

The generalized Poisson kernel P(a,m)(x,y') with respect to H is defined by (1.12)P(a,m)(x,y)=G(a,m)(x,y)n(y).

In fact (1.13)P(a,0)(x,y)=Pa(x,y).

We remark that the kernel function P(0,m)(x,y') coincides with ones in Finkelstein and Scheinberg  and Siegel and Talvila  (see [8, Chapter 11]).

Put (1.14)U(a,m;u)(x)=HP(a,m)(x,y)u(y)dy', where u(y) is a continuous function on H.

If γ is a real number and γ0,(resp.,γ<0), ι[γ],k++{γ}>-ι1,k++1(resp.,-ι[-γ],k+-{-γ}>-ι1,k++1) and (1.15)ι[γ],k++{γ}-n+1ιm+1,k+<ι[γ],k++{γ}-n+2(resp.,-ι[-γ],k+-{-γ}-n+1ιm+1,k+<-ι[-γ],k+-{-γ}-n+2).

If these conditions all hold, we write γ𝒞(k,m,n)(resp.,γ𝒟(k,m,n)).

Let γ𝒞(k,m,n)(resp.,γ𝒟(k,m,n)) and u be functions on H satisfying (1.16)H|u(y)|1+|y|ι[γ],k++{γ}dy<(resp.H|u(y)|(1+|y|ι[-γ],k++{-γ})dy<).

For γ and u, we define the positive measure μ (resp., ν) on Rn by (1.17)dμ(y)={|u(y)||y|-ι[γ],k+-{γ}dy,y'H(1,+),0,y'Rn-H(1,+)(resp.dν(y)={|u(y)||y|ι[-γ],k++{-γ}dy,y'H(1,+),0,y'Rn-H(1,+)).

We remark that the total mass of μ and ν is finite.

Let ϵ>0 and ξ0, and let μ be any positive measure on Rn having finite mass. For each x=(x',xn)Rn, the maximal function is defined by (1.18)M(x;μ,ξ)=sup0<ρ<|x|/2μ(B(x,ρ))ρξ. The set {x=(x',xn)Rn;M(x;μ,ξ)|x|ξ>ϵ} is denoted by E(ϵ;μ,ξ).

About classical solutions of the Dirichlet problem for the Laplacian, Siegel and Talvila (cf. [11, Corollary 2.1]) proved the following result.

Theorem A.

If u is a continuous function on H satisfying (1.19)H|u(y)|1+|y|n+mdy'<, then, the function U(0,m;u)(x) satisfies (1.20)U(0,m;u)C2(H)C0(H¯),ΔU(0,m;u)=0 in H,U(0,m;u)=u on H,lim|x|,xHU(0,m;u)(x)=o(xn1-n|x|n+m).

Our first aim is to give the growth properties at infinity for U(a,m;u)(x).

Theorem 1.2.

If 0ζn, γ𝒞(k,m,n)(resp.,γ𝒟(k,m,n)) and u is a measurable function on H satisfying (1.16), then there exists a covering {rj,Rj} of E(ϵ;μ,n-ζ)(resp.,E(ϵ;ν,n-ζ))(H) satisfying (1.21)j=0(rjRj)2-ζVj(Rjrj)Wj(Rjrj)< such that (1.22)lim|x|,xH-E(ϵ;μ,n-ζ)|x|-ι[γ],k+-{γ}+n-1φ1ζ-1(Θ)U(a,m;u)(x)=0(1.23)(resp.,lim|x|,xH-E(ϵ;ν,n-ζ)|x|ι[-γ],k++{-γ}+n-1φ1ζ-1(Θ)U(a,m;u)(x)=0).

If u is a measurable function on H satisfying (1.24)H|u(y)|1+|y|γdy'<, where γ is a real number, for this γ and u, we define (1.25)dμ(y)={|u(y)||y|-γdy,y'H(1,+),0,y'Rn-H(1,+).

Obviously, the total mass of μ' is also finite.

If we take a=0 in Theorem 1.2, then we immediately have the following growth property based on (1.5) and Remark 1.1.

Corollary 1.3.

Let 0ζn, γ>-(n-1)(p-1) and γ-nm<γ-n+1. If u is defined as previously, then the function U(0,m;u)(x) is a harmonic function on H and there exists a covering {rj,Rj} of E(ϵ;μ',n-ζ)(H) satisfying (1.26)j=0(rjRj)n-ζ< such that (1.27)lim|x|,xH-E(ϵ;μ,n-ζ)|x|n-γ-1φ1ζ-1(Θ)U(a,m;u)(x)=0.

Remark 1.4.

In the case ζ=n, (1.26) is a finite sum, and the set E(ϵ;μ',0) is a bounded set and (1.27) holds in H.

Next we are concerned with solutions of the Dirichlet problem for the Schrödinger operator on H. For related results, we refer the readers to the paper by Kheyfits .

Theorem 1.5.

If γ𝒞(k,m,n)(resp.,γ𝒟(k,m,n)) and u is a continuous function on H satisfying (1.16), then (1.28)U(a,m;u)C2(H)C0(H¯), SSE U(a,m;u)=0 in H,U(a,m;u)=u on H,(1.29)lim|x|,xH|x|-ι[γ],k+-{γ}+n-1φ1n-1(Θ)U(a,m;u)(x)=0(1.30)(resp.,lim|x|,xH|x|ι[-γ],k++{-γ}+n-1φ1n-1(Θ)U(a,m;u)(x)=0).

If we take ι[γ],k++{γ}=ιm+1,k++n-1, then we immediately have the following corollary, which is just Theorem A in the case a=0.

Corollary 1.6.

If u is a continuous function on H satisfying (1.31)H|u(y)|1+|y|ιm+1,k++n-1dy'<, then (1.28) hold and (1.32)lim|x|,xH|x|-ιm+1,k+φ1n-1(Θ)U(a,m;u)(x)=0.

As an application of Corollary 1.6, we can give a solution of the Dirichlet problem for any continuous function on H.

Theorem 1.7.

If u is a continuous function on H satisfying (1.31) and h(x) is a solution of the Dirichlet problem for the Schrödinger operator on H with u satisfying (1.33)lim|x|,xH|x|-ιm+1,k+h+(x)=0, then (1.34)h(x)=U(a,m;u)(x)+j=0m(v=1vjdjvφjv(Θ))Vj(|x|), where xH and djv are constants.

2. Lemmas

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

Lemma 2.1.

If 1|y'|<(1/2)|x|, then (2.1)|Pa(x,y)|M|x|ι1,k-|y|ι1,k+-1φ1(Θ). If |y'|1 and |y'|2|x|, then (2.2)|P(a,m)(x,y)|MVm+1(|x|)Wm+1(|y|)|y|φ1(Θ)φ1(Φ)nΦ. If (1/2)|x|<|y'|<2|x|, then (2.3)|P(x,y)|M|x-y|-n|x|φ1(Θ).

Proof.

Equations (2.1) and (2.2) are obtained by Kheyfits (see [8, Chapter 11] or [1, Lemma 1]). Equation (2.3) follows from Hayman and Kennedy (see [12, Lemma 4.2]).

Lemma 2.2 (see [<xref ref-type="bibr" rid="B8">2</xref>, Theorem 1]).

If u(x) is a solution of (1.2) on H satisfying (2.4)lim|x|,xH|x|-ιm+1,k+u+(x)=0, then (2.5)u(x)=j=0m(v=1vjdjvφjv(Θ))Vj(|x|).

Lemma 2.3.

Let ϵ>0 and ξ0, and let μ be any positive measure on Rn having finite total mass. Then, E(ϵ;μ,ξ) has a covering {rj,Rj}(j=1,2,) satisfying (2.6)j=1(rjRj)2-n+ξVj(Rjrj)Wj(Rjrj)<.

Proof.

Set (2.7)Ej(ϵ;μ,ξ)={xE(ϵ;μ,ξ):2j|x|<2j+1}(j=2,3,4,).

If xEj(ϵ;μ,ξ), then there exists a positive number ρ(x) such that (2.8)(ρ(x)|x|)2-n+ξVj(|x|ρ(x))Wj(|x|ρ(x))~(ρ(x)|x|)ξμ(B(x,ρ(x)))ϵ.

Here, Ej(ϵ;μ,ξ) can be covered by the union of a family of balls {B(xj,i,ρj,i):xj,iEj(ϵ;μ,ξ)}(ρj,i=ρ(xj,i)). By the Vitali lemma (see ), there exists ΛjEj(ϵ;μ,ξ), which is at most countable, such that {B(xj,i,ρj,i):xj,iΛj} are disjoint and Ej(ϵ;μ,ξ)xj,iΛjB(xj,i,5ρj,i).

So (2.9)j=2Ej(ϵ;μ,ξ)j=2xj,iΛjB(xj,i,5ρj,i).

On the other hand, note that xj,iΛjB(xj,i,ρj,i){x:2j-1|x|<2j+2}, so that (2.10)Pj,iΛj(5ρj,i|xj,i|)2-n+ξVj(|xj,i|5ρj,i)Wj(|xj,i|5ρj,i)~xj,iΛj(5ρj,i|xj,i|)ξ5ξxj,iΛjμ(B(xj,i,ρj,i))ϵ5ξϵμ(H[2j-1,2j+2)).

Hence, we obtain (2.11)j=1xj,iΛj(ρj,i|xj,i|)2-n+ξVj(|xj,i|ρj,i)Wj(|xj,i|ρj,i)~j=1xj,iΛj(ρj,i|xj,i|)ξj=1μ(H[2j-1,2j+2))ϵ3μ(Rn)ϵ.

Since E(ϵ;μ,ξ){xRn;|x|4}=j=2Ej(ϵ;μ,ξ), then E(ϵ;μ,ξ) is finally covered by a sequence of balls (B(xj,i,ρj,i),B(x1,6))(j=2,3,;i=1,2,) satisfying (2.12)j,i(ρj,i|xj,i|)2-n+ξVj(|xj,i|ρj,i)Wj(|xj,i|ρj,i)~j,i(ρj,i|xj,i|)ξ3μ(Rn)ϵ+6ξ<+, where B(x1,6)(x1=(1,0,,0)Rn) is the ball that covers {xRn;|x|<4}.

3. Proof of Theorem <xref ref-type="statement" rid="thm1">1.2</xref>

We only prove the case γ0, the remaining case γ<0 can be proved similarly.

For any ϵ>0, there exists Rϵ>1 such that (3.1)H(Rϵ,)|u(y)|1+|y|ι[γ],k++{γ}dy'<ϵ.

The relation Ga(x,y)G(x,y) implies this inequality (see ) (3.2)Pa(x,y)P(x,y).

For any fixed point xH(Rϵ,+)-E(ϵ;μ,n-ζ) satisfying |x|>2Rϵ, letting I1=H[0,1), I2=H[1,Rϵ], I3=H(Rϵ,(1/2)|x|], I4=H((1/2)|x|,2|x|), I5=H[2|x|,) and I6=H[1,2|x|), we write (3.3)|U(a,m;u)(x)|i=16Ua,i(x), where (3.4)Ua,i(x)=Ii|Pa(x,y)||u(y)|dy(i=1,2,3,4),Ua,5(x)=I5|P(a,m)(x,y)||u(y)|dy',Ua,6(x)=I6|K~(Ω,a,m)(x,y)n(y)||u(y)|dy'.

By ι[γ],k++{γ}>-ι1,k++1, (1.16), (2.1), and (3.1), we have the following growth estimates (3.5)Ua,2(x)M|x|ι1,k-φ1(Θ)I2|y|ι1,k+-1|u(y)|dy'M|x|ι1,k-Rϵι[γ],k++{γ}+ι1,k+-1φ1(Θ),Ua,1(x)M|x|ι1,k-φ1(Θ),Ua,3(x)Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ).

Next, we will estimate Ua,4(x).

Take a sufficiently small positive number d3 such that I4B(x,(1/2)|x|) for any xΠ(d3), where (3.6)Π(d3)={xH;infzS+n-1|x|x|-z|z||<d3,0<|x|<}, and divide H into two sets Π(d3) and H-Π(d3).

If xH-Π(d3), then there exists a positive d3 such that |x-y'|d3|x| for any y'H, and hence (3.7)Ua,4(x)M|x|1-nφ1(Θ)I4|u(y)|dy'Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ).

We will consider the case xΠ(d3). Now put (3.8)Ξi(x)={yI4;2i-1δ(x)|x-y|<2iδ(x)}, where δ(x)=infy'H|x-y'|.

Since H{yRn:|x-y|<δ(x)}=, we have (3.9)Ua,4(x)=Mi=1i(x)Ξi(x)|x|φ1(Θ)|u(y)||x-y|ndy', where i(x) is a positive integer satisfying 2i(x)-1δ(x)|x|/2<2i(x)δ(x).

Since |x|φ1(Θ)Mδ(x)(xH), we obtain (3.10)Ξi(x)|x|φ1(Θ)|u(y)||x-y|ndy2(1-i)nφ1(Θ)δ(x)ζ-nΞi(x)|x|δ(x)-ζ|u(y)|dy'Mφ11-ζ(Θ)δ(x)ζ-nΞi(x)|x|1-ζ|u(y)|dy'M|x|n-ζφ11-ζ(Θ)δ(x)ζ-nΞi(x)|y|1-n|u(y)|dy'Mϵ|x|ι[γ],k++{γ}-ζ+1φ11-ζ(Θ)(μ(Ξi(x))(2iδ(x))n-ζ) for i=0,1,2,,i(x).

Since xE(ϵ;μ,n-ζ), we have (3.11)μ(Ξi(x))(2iδ(x))n-ζμ(B(x,2iδ(x)))(2iδ(x))n-ζM(x;μ,n-ζ)ϵ|x|ζ-n(i=0,1,2,,i(x)-1),μ(Λi(x)(x))(2iδ(x))n-ζμ(B(x,|x|/2))(|x|/2)n-ζϵ|x|ζ-n.

So (3.12)Ua,4(x)Mϵ|x|ι[γ],k++{γ}-n+1φ11-ζ(Θ).

By ιm+1,k+ι[γ],k++{γ}-n+1, (1.7), (2.2), and (3.1), we have (3.13)Ua,5(x)MVm+1(|x|)I5|u(y)|Vm+1(|y|)|y|n-1dy'Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ).

We only consider Ua,6(x) in the case m1, since Ua,6(x)0 for m=0. By the definition of K~(a,m), (1.4), and (2.2), we see that (3.14)Ua,6(x)Mχ(1)j=0mj2n-1qj(|x|), where (3.15)qj(|x|)=Vj(|x|)I6Wj(|y|)|u(y)||y|dy'.

To estimate qj(|x|), we write (3.16)qj(|x|)qj(|x|)+qj′′(|x|), where (3.17)qj(|x|)=Vj(|x|)φ1(Θ)I2Wj(|y|)|u(y)||y|dy',qj′′(|x|)=Vj(|x|)φ1(Θ){yH:Rϵ<|y|<2|x|}Wj(|y|)|u(y)||y|dy.

Notice that (3.18)Vj(|x|)Vm+1(|y|)Vj(|y|)|y|MVm+1(|x|)|x|M|x|ιm+1,k+-1(|y|1,Rϵ<2|x|). Thus, by ιm+1,k+<ι[γ],k++{γ}-n+2, (1.7), and (1.16), we conclude (3.19)qj(|x|)=Vj(|x|)φ1(Θ)I2|u(y)|Vj(|y|)|y|n-1dy'MVj(|x|)φ1(Θ)I2Vm+1(|y|)|y|ιm+1,k+|u(y)|Vj(|y|)|y|n-1dy'M|x|ιm+1,k+-1Rϵι[γ],k++{γ}-ιm+1,k+-n+2φ1(Θ).

Analogous to the estimate of qj(|x|), we have (3.20)qj′′(|x|)Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ).

Thus, we can conclude that (3.21)qj(|x|)Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ), which yields (3.22)Ua,6(x)Mϵ|x|ι[γ],k++{γ}-n+1φ1(Θ).

Combining (3.5)–(3.22), we obtain that if Rϵ is sufficiently large and ϵ is sufficiently small, then U(a,m;u)(x)=o(|x|ι[γ],k++{γ}-n+1φ11-ζ(Θ)) as |x|, where xH(Rϵ,+)-E(ϵ;μ,n-ζ). Finally, there exists an additional finite ball B0 covering H[0,Rϵ], which together with Lemma 2.3 gives the conclusion of Theorem 1.2.

4. Proof of Theorem <xref ref-type="statement" rid="thm2">1.5</xref>

For any fixed xH, take a number satisfying R>max{1,2|x|}. By ιm+1,k+ι[γ],k++{γ}-n+1, (1.5), (1.16), and (2.2), we have (4.1)H(R,)|P(a,m)(x,y)||u(y)|dyMVm+1(|x|)φ1(Θ)H(R,)|u(y)||y|ιm+1,k++n-1dy'M|x|ιm+1,k+φ1(Θ)H(2|x|,)|y|ι[γ],k++{γ}-ιm+1,k+-n+1dy'M|x|ι[γ],k++{γ}-n+1φ1(Θ)<. Then, U(a,m;u)(x) is absolutely convergent and finite for any xH. Thus U(a,m;u)(x) is a solution of (1.2) on H.

Now we study the boundary behavior of U(a,m;u)(x). Let y'H be any fixed point and l any positive number satisfying l>max{|y'|+1,(1/2)R}.

Set χS(l) as the characteristic function of S(l)={yH,|y|l}, and write (4.2)U(a,m;u)(x)=U(x)-U′′(x)+U′′′(x), where (4.3)U(x)=H[0,2l]Pa(x,y)u(y)dy',U′′(x)=H(1,2l]K(a,m)(x,y)n(y')u(y')dy',U′′′(x)=H(2l,)P(a,m)(x,y)u(y)dy'.

Notice that U'(x) is the Poisson a-integral of u(y)χS(2l), We have limxy',xHU(x)=u(y'). Since limΘΦφjv(Θ)=0(j=1,2,3;1vvj) as xyH, we have limxy,xHU′′(x)=0 from the definition of the kernel function K(a,m)(x,y). U′′′(x)=O(|x|ι[γ],k++{γ}-n+1φ1(Θ)) and therefore tends to zero.

So the function U(a,m;u)(x) can be continuously extended to H¯ such that (4.4)limxy,xHU(a,m;u)(x)=u(y) for any yH from the arbitrariness of l.

Finally, (1.29) and (1.30) follow from (1.22) and (1.23), respectively, in the case ζ=n. Thus, we complete the proof of Theorem 1.5.

5. Proof of Theorem <xref ref-type="statement" rid="thm3">1.7</xref>

From Corollary 1.6, we have the solution U(a,m;u)(x) of the Dirichlet problem on H with u satisfying (1.31). Consider the function h(x)-U(a,m;u)(x). Then, it follows that this is a solution of (1.2) in H and vanishes continuously on H.

Since (5.1)0(h-U(Ω,a,m;u))+(x)h+(x)+(U(a,m;u))-(x) for any xH, we have (5.2)lim|x|,xH|x|-ιm+1,k+(h-U(Ω,a,m;u))+(x)  =0 from (1.32) and (1.33). Then, the conclusion of Theorem 1.7 follows immediately from Lemma 2.2.

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