Solutions of the Dirichlet-Schrödinger problems with continuous data admitting arbitrary growth property in the boundary
- Jianjie Wang^{1}Email author,
- Jun Pu^{2} and
- Ahmed Zama^{3}
https://doi.org/10.1186/s13662-016-0751-4
© Wang et al. 2016
Received: 15 October 2015
Accepted: 11 January 2016
Published: 1 February 2016
Abstract
By using the modified Green-Schrödinger 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 Poisson-Schrödinger integral, the unique solution of it is also constructed.
Keywords
1 Introduction and main theorem
We denote the n-dimensional 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 \(\vert V-W\vert \) 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 \({R}^{n}\).
For the set \(\Xi\subset{S}^{n-1}\), 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 \(n=3\).
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 \(\mathbf{S}^{n-1}\) surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [3], pp.88-89 for the definition of a \(C^{2,\alpha}\)-domain).
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]).
For more applications of modified Green-Schrödinger potentials and modified Poisson-Schrödinger integrals, we refer the reader to the papers (see [7, 8]).
Recently, Huang and Ychussie (see [7]) gave the solutions of the Dirichlet-Schrödinger problem with continuous data having slow growth in the boundary.
Theorem A
As an application of modified Poisson-Schrödinger kernel \(\mathbb {PI}(\Xi;a,\phi)(V,W)\), we have the following.
Theorem
2 Main lemmas
Lemma 1
Proof
Lemma 2
(see [9])
- (I)For any \(Q'\in S_{n}(\Xi)\) and any \(\epsilon>0\), there exists a neighborhood \(B(Q')\) of \(Q'\) such thatfor any \(V=(\tau,\Lambda)\in H_{n}(\Xi)\cap B(W')\), where R is a positive real number.$$ \int_{S_{n}(\Xi;[R,\infty))}\bigl\vert \widehat{W}(V,W)\bigr\vert \bigl\vert u(W)\bigr\vert \,d\sigma _{W}< \epsilon $$(2.3)
- (II)For any \(W'\in S_{n}(\Xi)\), we havefor any positive real number R.$$ \limsup_{V\rightarrow W', V\in H_{n}(\Xi)} \int_{S_{n}(\Xi ;(0,R))}\bigl\vert \widehat{V}(V,W)\bigr\vert \bigl\vert u(W)\bigr\vert \,d\sigma_{W}=0 $$(2.4)
3 Proof of Theorem
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 mean-valued property by Fubini’s theorem.
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 \(V\in H_{n}(\Xi)\cap B(W',\delta)\).
Thus we complete the proof of the theorem.
Declarations
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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Authors’ Affiliations
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