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RETRACTED ARTICLE: Solutions of the DirichletSchrödinger problems with continuous data admitting arbitrary growth property in the boundary
Advances in Difference Equations volume 2016, Article number: 33 (2016)
 The Retraction Note to this article has been published in Advances in Difference Equations 2020 2020:145
Abstract
By using the modified GreenSchrödinger function, we consider the Dirichlet problem with respect to the stationary Schrödinger operator with continuous data having an arbitrary growth in the boundary of the cone. As an application of the modified PoissonSchrödinger integral, the unique solution of it is also constructed.
Introduction and main theorem
We denote the ndimensional Euclidean space by \({R}^{n}\), where \(n\geq2\). The sets ∂E and E̅ denote the boundary and the closure of a set E in \({R}^{n}\). Let \(\vert VW\vert \) denote the Euclidean distance of two points V and W in \({R}^{n}\), respectively. Especially, \(\vert \mathrm{V}\vert \) denotes the distance of two points V and O in \({R}^{n}\), where O is the origin of \({R}^{n}\).
We introduce a system of spherical coordinates \((\tau,\Lambda)\), \(\Lambda=(\lambda_{1},\lambda_{2},\ldots,\lambda_{n1})\), in \({R}^{n}\) which are related to the Cartesian coordinates \((y_{1},y_{2},\ldots,y_{n1},y_{n})\) by
and if \(n\geq3\), then
where \(0\leq\tau<+\infty\), \(\frac{1}{2}\pi\leq\lambda_{n1}<\frac{3}{2}\pi\), and if \(n\geq3\), then \(0\leq\lambda_{j}\leq\pi\) (\(1\leq j\leq n2\)).
Let \(B(V,\tau)\) denote the open ball with center at V and radius r in \({R}^{n}\), where \(\tau>0\). Let \({S}^{n1}\) and \({S}_{+}^{n1}\) denote the unit sphere and the upper half unit sphere in \({R}^{n}\), respectively. The surface area \(2\pi^{n/2}\{\Gamma(n/2)\}^{1}\) of \({S}^{n1}\) is denoted by \(w_{n}\). Let \(\Xi\subset{S}^{n1}\), Λ and Ξ denote a point \((1,\Lambda)\) and the set \(\{\Lambda; (1,\Lambda)\in\Xi\}\), respectively. For two sets \(\Lambda\subset{R}_{+}\) and \(\Xi\subset\mathbf{S}^{n1}\), we denote
where \({R}_{+}\) is the set of all positive real numbers.
For the set \(\Xi\subset{S}^{n1}\), a cone \(H_{n}(\Xi)\) denote the set \({R}_{+}\times\Xi\) in \({R}^{n}\). For the set \(E\subset R\), \(C_{n}(\Xi;I)\) and \(S_{n}(\Xi;I)\) denote the sets \(E\times\Xi\) and \(E\times\partial{\Xi}\), respectively, where R is the set of all real numbers. Especially, \(S_{n}(\Xi)\) denotes the set \(S_{n}(\Xi; {R}_{+})\).
Let \(A_{a}\) denote the class of nonnegative radial potentials \(a(V)\), i.e. \(0\leq a(V)=a(\tau)\), \(V=(\tau ,\Lambda)\in H_{n}(\Xi)\), such that \(a\in L_{loc}^{b}(H_{n}(\Xi))\) with some \(b> {n}/{2}\) if \(n\geq4\) and with \(b=2\) if \(n=2\) or \(n=3\).
This article is devoted to the stationary Schrödinger equation
for \(V\in C_{n}(\Xi)\), where \(\Delta_{n}\) is the Laplace operator and \(a\in A_{a}\). These solutions are called harmonic functions with respect to \(\operatorname {SSE}_{a}\). In the case \(a=0\) we remark that they are harmonic functions. Under these assumptions the operator \(\operatorname {SSE}_{a}\) can be extended in the usual way from the space \(C_{0}^{\infty}(H_{n}(\Xi))\) to an essentially selfadjoint operator on \(L^{2}(H_{n}(\Xi))\) (see [1]). We will denote it \(\operatorname {SSE}_{a}\) as well. This last one also has a GreenSchrödinger function \(G(\Xi;a)(V,W)\). Here \(G(\Xi;a)(V,W)\) is positive on \(H_{n}(\Xi)\) and its inner normal derivative \(\partial G(\Xi;a)(V,W)/{\partial n_{W}}\geq0\). We denote this derivative by \(\mathbb{PI}(\Xi;a)(V,W)\), which is called the PoissonSchrödinger kernel with respect to \(H_{n}(\Xi)\).
Let \(\Delta'\) be the spherical part of the Laplace operator on \(\Xi\subset{S}^{n1}\) and \(\lambda_{j}\) (\(j=1,2,3\ldots\) , \(0<\lambda_{1}<\lambda_{2}\leq\lambda_{3}\leq\ldots\)) be the eigenvalues of the eigenvalue problem for \(\Delta'\) on Ξ (see, e.g., [2], p.41)
The corresponding eigenfunctions are denoted by \(\varphi_{jv}\) (\(1\leq v\leq v_{j}\)), where \(v_{j}\) is the multiplicity of \(\lambda_{j}\). We set \(\lambda_{0}=0\), norm the eigenfunctions in \(L^{2}(\Xi)\), and \(\varphi_{1}=\varphi_{11}>0\).
We wish to ensure the existence of \(\lambda_{j}\), where \(j=1,2,3\ldots \) . We put a rather strong assumption on Ξ: if \(n\geq3\), then Ξ is a \(C^{2,\alpha}\)domain (\(0<\alpha<1\)) on \(\mathbf{S}^{n1}\) surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [3], pp.8889 for the definition of a \(C^{2,\alpha}\)domain).
Given a continuous function f on \(S_{n}(\Xi)\), we say that h is a solution of the DirichletSchrödinger problem in \(H_{n}(\Xi)\) with f, if h is a harmonic function with respect to \(\operatorname {SSE}_{a}\) in \(H_{n}(\Xi )\) and
The solutions of the equation
are denoted by \(P_{j}(\tau)\) (\(j=1,2,3,\ldots\)) and \(Q_{j}(\tau )\) (\(j=1,2,3,\ldots\)), respectively, for the increasing and nonincreasing cases, as \(\tau\rightarrow+\infty\), which is normalized under the condition \(P_{j}(1)=Q_{j}(1)=1\) (see [4], Chap. 11). In the sequel, we shall write P and Q instead of \(P_{1}\) and \(Q_{1}\), respectively, for the sake of brevity.
We shall also consider the class \(B_{a}\), consisting of the potentials \(a\in A_{a}\) such that there exists a finite limit \(\lim_{\tau\rightarrow\infty}\tau^{2} a(\tau)=k\in [0,\infty)\), moreover, \(\tau^{1}\vert \tau^{2} a(\tau)k\vert \in L(1,\infty)\). If \(a\in B_{a}\), then the generalized harmonic functions are continuous (see [5]).
In the rest of this paper, we assume that \(a\in B_{a}\) and we shall suppress the explicit notation of this assumption for simplicity. Denote
for \(j=0,1,2,3\ldots\) .
It is well known (see [6]) that in the case under consideration the solutions to equation (1.1) have the asymptotics
where \(d_{1}\) and \(d_{2}\) are some positive constants.
The GreenSchrödinger function \(G(\Xi;a)(V,W)\) (see [4], Chap. 11) has the following expansion:
for \(a\in A_{a}\), where \(V=(\tau,\Lambda)\), \(W=(\iota,\Upsilon)\), \(\tau\neq\iota\), and \(\chi'(s)=w (Q_{1}(\tau),P_{1}(\tau) ) _{\tau=s}\) is their Wronskian. The series converges uniformly if either \(\tau\leq s \iota\) or \(\tau \leq s \iota\) (\(0< s<1\)).
For a nonnegative integer m and two points \(V=(\tau,\Lambda)\), \(W=(\iota,\Upsilon)\in H_{n}(\Xi)\), we put
where
The modified GreenSchrödinger function can be defined as follows (see [4], Chap. 11):
for two points \(V=(\tau,\Lambda)\), \(Q=(\iota,\Upsilon)\in H_{n}(\Xi)\), then the modified PoissonSchrödinger case on cones can be defined by
accordingly, which has the following growth estimates (see [7]):
for any \(V=(\tau,\Lambda)\in H_{n}(\Xi)\) and \(W=(\iota,\Upsilon)\in S_{n}(\Xi)\) satisfying \(\tau\leq s \iota\) (\(0< s<1\)), where \(M(n,m,s)\) is a constant dependent of n, m, and s.
We remark that
In this paper, we shall use the following modified PoissonSchrödinger integrals (see [7]):
where \(f(W)\) is a continuous function on \(\partial H_{n}(\Xi)\) and \(d\sigma_{W}\) is the surface area element on \(S_{n}(\Xi)\).
For more applications of modified GreenSchrödinger potentials and modified PoissonSchrödinger integrals, we refer the reader to the papers (see [7, 8]).
Recently, Huang and Ychussie (see [7]) gave the solutions of the DirichletSchrödinger problem with continuous data having slow growth in the boundary.
Theorem A
Iffis a continuous function on \(\partial{H_{n}(\Xi)}\)satisfying
then the modified PoissonSchrödinger integral \(\mathbb{PI}_{\Xi }^{a}(m,f)\)is a solution of the DirichletSchrödinger problem in \(H_{n}(\Xi)\)withfsatisfies
It is natural to ask if the continuous function f satisfying (1.3) can be replaced by continuous data having an arbitrary growth property in the boundary. In this paper, we shall give an affirmative answer to this question. To do this, we also construct a modified PoissonSchrödinger kernel. Let \(\phi(l)\) be a positive function of \(l\geq1\) satisfying
Denote the set
by \(\pi_{\Xi}(\phi,j)\). Then \(1\in\pi_{\Xi}(\phi,j)\). When there is an integer N such that \(\pi_{\Xi}(\phi,N)\neq\Phi\) and \(\pi_{\Xi}(\phi,N+1)= \Phi \), denote
of integers. Otherwise, denote the set of all positive integers by \(J_{\Xi}(\phi)\). Let \(l(j)=l_{\Xi}(\phi,j)\) be the minimum elements l in \(\pi_{\Xi }(\phi,j)\) for each \(j\in J_{\Xi}(\phi)\). In the former case, we put \(l(N+1)=\infty\). Then \(l(1)=1\). The kernel function \(\widetilde{K}(\Xi;a,\phi)(V,W)\) is defined by
where \(V\in H_{n}(\Xi)\) and \(W=(\iota,\Upsilon)\in S_{n}(\Xi)\).
The new modified PoissonSchrödinger kernel \(\mathbb{PI}(\Xi ;a,\phi)(V,W)\) is defined by
where \(V\in H_{n}(\Xi)\) and \(W\in S_{n}(\Xi)\).
As an application of modified PoissonSchrödinger kernel \(\mathbb {PI}(\Xi;a,\phi)(V,W)\), we have the following.
Theorem
Let \(g(V)\)be a continuous function on \(S_{n}(\Xi)\). Then there is a positive continuous function \(\phi_{g}(l)\)of \(l\geq1\)depending ongsuch that
is a solution of the DirichletSchrödinger problem in \(H_{n}(\Xi)\)withg.
Main lemmas
Lemma 1
Let \(\phi(l)\) be a positive continuous function of \(l\geq1\) satisfying
Then
for any \(V=(\tau,\Lambda)\in H_{n}(\Xi)\) and any \(W=(\iota,\Upsilon)\in S_{n}(\Xi)\) satisfying
Proof
We can choose two points \(V=(\tau,\Lambda)\in H_{n}(\Xi)\) and \(W=(\iota,\Upsilon)\in S_{n}(\Xi)\), satisfying (2.1). Moreover, we also can choose an integer \(j=j(V,W)\in J_{\Xi}(\Upsilon)\) such that
Then
Hence we have from (1.2), (2.1), and (2.2)
which is the conclusion. □
Lemma 2
(see [9])
Let \(g(V)\)be a continuous function on \(S_{n}(\Xi)\)and \(\widehat{V}(V,W)\)be a locally integrable function on \(S_{n}(\Xi)\)for any fixed \(V\in H_{n}(\Xi)\), where \(W\in S_{n}(\Xi)\). Define
for any \(V\in H_{n}(\Xi)\)and any \(W\in S_{n}(\Xi)\).
Suppose that the following two conditions are satisfied:

(I)
For any \(Q'\in S_{n}(\Xi)\)and any \(\epsilon>0\), there exists a neighborhood \(B(Q')\)of \(Q'\)such that
$$ \int_{S_{n}(\Xi;[R,\infty))}\bigl\vert \widehat{W}(V,W)\bigr\vert \bigl\vert u(W)\bigr\vert \,d\sigma _{W}< \epsilon $$(2.3)for any \(V=(\tau,\Lambda)\in H_{n}(\Xi)\cap B(W')\), whereRis a positive real number.

(II)
For any \(W'\in S_{n}(\Xi)\), we have
$$ \limsup_{V\rightarrow W', V\in H_{n}(\Xi)} \int_{S_{n}(\Xi ;(0,R))}\bigl\vert \widehat{V}(V,W)\bigr\vert \bigl\vert u(W)\bigr\vert \,d\sigma_{W}=0 $$(2.4)for any positive real numberR.
Then
for any \(W'\in S_{n}(\Xi)\).
Proof of Theorem
Take a positive continuous function \(\phi(l)\) (\(l\geq1\)) such that
and
for \(l>1\), where
For any fixed \(V=(\tau,\Lambda)\in H_{n}(\Xi)\), we can choose a number R satisfying \(R>\max\{1,4r\}\). Then we see from Lemma 1 that
Obviously, we have
which gives
To see that \(\mathbb{PI}_{\Xi}^{a}(\phi_{g},g)(V)\) is a harmonic function in \(H_{n}(\Xi)\), we remark that \(\mathbb{PI}_{\Xi}^{a}(\phi_{g},g)(V)\) satisfies the locally meanvalued property by Fubini’s theorem.
Finally we shall show that
for any \(W'=(\iota',\Upsilon')\in \partial{H_{n}(\Xi)}\). Setting
in Lemma 2, which is locally integrable on \(S_{n}(\Xi)\) for any fixed \(V\in H_{n}(\Xi)\). Then we apply Lemma 2 to \(g(V)\) and \(g(V)\).
For any \(\epsilon>0\) and a positive number δ, by (3.1) we can choose a number R (\(>\max\{1, 2(\iota'+\delta)\}\)) such that (2.2) holds, where \(V\in H_{n}(\Xi)\cap B(W',\delta)\).
Since
as \(V=(\tau,\Lambda)\rightarrow W'=(\iota',\Phi')\in S_{n}(\Xi)\), we have
where \(W\in S_{n}(\Xi)\) and \(W'\in S_{n}(\Xi)\). Then (2.3) holds.
Thus we complete the proof of the theorem.
Change history
02 April 2020
The EditorsinChief have retracted this article [1] because it significantly overlaps with an article by different authors [2] that was simultaneously under consideration at a different journal. The article also shows evidence of authorship manipulation and peer review manipulation. In addition, the identity of the third author could not be verified: the University of Mosul has confirmed that Ahmed Zama has not been affiliated with their institution. The authors have not responded to any correspondence regarding this retraction.
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Acknowledgements
We wish to express our genuine thanks to the anonymous referees for careful reading and excellent comments on this manuscript.
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All authors contributed equally to the manuscript and read and approved the final manuscript.
The EditorsinChief have retracted this article because it significantly overlaps with an article by different authors that was simultaneously under consideration at a different journal. The article also shows evidence of authorship manipulation and peer review manipulation. In addition, the identity of the third author could not be verified: the University of Mosul have confirmed that Ahmed Zama has not been affiliated with their institution. The authors have not responded to any correspondence regarding this retraction.
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Wang, J., Pu, J. & Zama, A. RETRACTED ARTICLE: Solutions of the DirichletSchrödinger problems with continuous data admitting arbitrary growth property in the boundary. Adv Differ Equ 2016, 33 (2016). https://doi.org/10.1186/s1366201607514
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Keywords
 modified GreenSchrödinger potential
 modified PoissonSchrödinger integral
 DirichletSchrödinger problem