Some remarks on Phragmén-Lindelöf theorems for weak solutions of the stationary Schrödinger operator
- Liquan Wan^{1}Email author
Received: 29 October 2015
Accepted: 8 December 2015
Published: 22 December 2015
Abstract
In this paper, we not only give the asymptotic behavior at the origin for the maximum modulus of weak solutions of the stationary Schrödinger equation in a cone but also obtain the property of the negative parts of them, which generalize the Phragmén-Lindelöf type theorems for subfunctions.
Keywords
1 Introduction and main results
Let \(\mathbf{R}^{n}\) be the n-dimensional Euclidean space, where \(n\geq2\). Let E be an open set in \(\mathbf{R}^{n}\), the boundary and the closure of it are denoted by ∂E and E̅, respectively. A point P is denoted by \((X,x_{n})\), where \(X=(x_{1},x_{2},\ldots,x_{n-1})\). For \(P\in\mathbf{R}^{n}\) and \(r>0\), let \(B(P,r)\) denote the open ball with center at P and radius r in \(\mathbf{R}^{n}\).
We introduce a system of spherical coordinates \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to the cartesian coordinates \((X,x_{n})=(x_{1},x_{2},\ldots,x_{n-1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let \(\mathbf{S}^{n-1}\) and \(\mathbf{S}_{+}^{n-1}\) denote the unit sphere and the upper half unit sphere, respectively. For \(\Omega\subset\mathbf{S}^{n-1}\), a point \((1,\Theta)\) on \(\mathbf{S}^{n-1}\), and the set \(\{\Theta; (1,\Theta)\in\Omega\}\) are simply denoted by Θ and Ω respectively. The set \(\{(r,\Theta)\in\mathbf{R}^{n}; r\in\Xi,(1,\Theta)\in\Omega\}\) in \(\mathbf{R}^{n}\) is simply denoted by \(\Xi\times\Omega\), where \(\Xi\subset\mathbf{R}_{+}\) and \(\Omega\subset \mathbf{S}^{n-1}\). Especially, the set \(\mathbf{R}_{+}\times\Omega\) by \(C_{n}(\Omega)\), where \(\mathbf{R}_{+}\) is the set of positive real number and \(\Omega\subset \mathbf{S}^{n-1}\).
Let \(C_{n}(\Omega;I)\) and \(S_{n}(\Omega;I)\) denote the sets \(I\times\Omega\) and \(I\times\partial{\Omega}\), respectively, where I is an interval on R and R is the set of real numbers. Especially, the set \(S_{n}(\Omega)\) denotes \(S_{n}(\Omega; (0,+\infty))\), which is \(\partial{C_{n}(\Omega)}-\{O\}\).
We put a rather strong assumption on Ω: if \(n\geq3\), then Ω 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 the \(C^{2,\alpha}\)-domain).
Let \(\mathscr{A}_{a}\) denote the class of nonnegative radial potentials \(a(P)\), i.e. \(0\leq a(P)=a(r)\), \(P=(r,\Theta)\in C_{n}(\Omega)\), such that \(a\in L_{\mathrm{loc}}^{b}(C_{n}(\Omega))\) with some \(b> {n}/{2}\) if \(n\geq4\) and with \(b=2\) if \(n=2\) or \(n=3\).
We will also consider the class \(\mathscr{B}_{a}\), consisting of the potentials \(a\in\mathscr{A}_{a}\) such that there exists the finite limit \(\lim _{r\rightarrow\infty}r^{2} a(r)=k\in[0,\infty)\), moreover, \(r^{-1}|r^{2} a(r)-k|\in L(1,\infty)\).
We denote by \(SbH_{a}(\Omega)\) the class of all weak solutions of the inequality (1.1) for any \(P=(r,\Theta)\in C_{n}(\Omega)\), which are continuous when \(a\in\mathscr{B}_{a}\) (see [5]). We denote by \(SpH_{a}(\Omega)\) the class of \(u(P)\) satisfying \(-u(P)\in SbH_{a}(\Omega )\). If \(u(P)\in SbH_{a}(\Omega)\) and \(u(P)\in SpH_{a}(\Omega)\), then \(u(P)\) is the solution of \(SSE_{a}u(P)=0\) for any \(P=(r,\Theta)\in C_{n}(\Omega)\). In our terminology we follow Nirenberg [6]. Other authors have under similar circumstances used various terms such as subfunctions, subsolutions, submetaharmonic function, subelliptic functions, panharmonic functions, etc.; see, for example, Duffin, Littman, Qiao et al., Topolyansky, Vekua (see [7–11]).
Throughout this paper, unless otherwise specified, we will always assume that \(u(P)\in SbH_{a}(\Omega)\) and satisfy the Phragmén-Lindelöf boundary condition on \(S_{n}(\Omega)\). Recently, about the Phragmén-Lindelöf theorems for subfunctions in a cone, Qiao and Deng (see [9], Theorem 3) proved the following result.
Theorem A
A stronger version of a Phragmén-Lindelöf type theorem is also due to Qiao and Deng (see [9], Theorem 3).
Theorem B
Theorem 1
Remark
It is obvious that \(\eta_{u}\leq L_{u}\). On the other hand, we have \(\eta_{u}\geq L_{u}\) from (1.5). Thus, if (1.3) and (1.4) are satisfied, then we have \(\eta_{u}=L_{u}\).
As an application of Theorem 1 we immediately have the following result by using Lemma 3 in Section 2.
Corollary
In [9], the authors gave the properties of the positive part of weak solutions satisfying the Phragmén-Lindelöf boundary condition on \(S_{n}(\Omega)\). Finally, we shall show one of the properties of its negative part.
Theorem 2
2 Some lemmas
Lemma 1
(see [9], Lemma 8)
- (1)
Both of the limits \(\mu_{u}\) and \(\eta_{u}\) (\(-\infty<\mu_{u},\eta_{u}\leq+\infty\)) exist.
- (2)
If \(\eta_{u}\leq0\), then \(V^{-1}(r)N_{u}(r)\) is non-decreasing on \((0,+\infty)\).
- (3)
If \(\mu_{u}\leq0\), then \(W^{-1}(r)N_{u}(r)\) is non-increasing on \((0,+\infty)\).
Lemma 2
Proof
Lemma 3
Under the conditions of the corollary, \(L_{u}>-\infty\) and \(J_{u}=L_{u}\).
Proof
Since Lemma 1 and the remark give \(\eta_{u}> -\infty\) and \(\eta_{u}= L_{u}\), respectively, we have the conclusion from (2.2) and (2.3).
Given a continuous function ψ defined on the truncated cone \(\partial C_{n}(\Omega;(R_{1},R_{2}))\), where \(R_{1}\) and \(R_{2}\) are two positive real numbers satisfying \(R_{1}< R_{2}\), then the solution of the Dirichlet-Sch problem on \(C_{n}(\Omega;(R_{1},R_{2}))\) with ψ is denoted by \(H_{\psi}(P; C_{n}(\Omega;(R_{1},R_{2})))\). □
Lemma 4
Proof
3 Proof of Theorem 1
Now we distinguish two cases.
Case 1 \(\tau=+\infty\).
These results show that (1.7) holds in this case.
Case 2 \(\tau<+\infty\).
From Theorem B, we see that \(L_{u}<+\infty\). On the other hand, we have \(L_{u}>-\infty\) from (3.1).
Subcase 2.1 \(0\leq L_{u}<+\infty\).
Thus from (3.3) we can simultaneously prove the existence of \(B_{u}\) and (1.6).
Subcase 2.2 \(-\infty\leq L_{u}<0\).
Lastly, we shall show that (1.7) holds.
Next suppose that \(\eta_{u^{+}}<+\infty\). We have \(B_{u}<+\infty\) by Lemma 4 and hence \(L_{u^{+}}=\eta_{u^{+}}\) by the remark. With (3.9), this gives (1.7).
Declarations
Acknowledgements
The author would like to thank the referees for their comments. This work was supported by the National Social Science Foundation of China (No. 13BGL047), the Humanities and Social Science Fund of Ministry of Education (No. 2013-ZD-011) and 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).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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