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Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator
Boundary Value Problems volume 2015, Article number: 239 (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 PhragménLindelöf type theorems for subfunctions.
1 Introduction and main results
Let \(\mathbf{R}^{n}\) be the ndimensional 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 \(X=(x_{1},x_{2},\ldots,x_{n1})\). 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 \((r,\Theta)\), \(\Theta=(\theta_{1},\theta_{2},\ldots,\theta_{n1})\), in \(\mathbf{R}^{n}\) which are related to the cartesian coordinates \((X,x_{n})=(x_{1},x_{2},\ldots,x_{n1},x_{n})\) by \(x_{n}=r\cos\theta_{1}\).
Let \(\mathbf{S}^{n1}\) and \(\mathbf{S}_{+}^{n1}\) denote the unit sphere and the upper half unit sphere, respectively. For \(\Omega\subset\mathbf{S}^{n1}\), a point \((1,\Theta)\) on \(\mathbf{S}^{n1}\), 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 \(\Omega\subset \mathbf{S}^{n1}\). Especially, the set \(\mathbf{R}_{+}\times\Omega\) by \(C_{n}(\Omega)\), where \(\mathbf{R}_{+}\) is the set of positive real number and \(\Omega\subset \mathbf{S}^{n1}\).
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 \(\partial{C_{n}(\Omega)}\{O\}\).
Let \(\Delta^{\ast}\) be the spherical part of the Laplace operator Δ (see [1]),
and Ω be a domain on \(\mathbf{S}^{n1}\) with smooth boundary. We consider the Dirichlet problem (see [2], p.41)
The least positive eigenvalue of the above boundary value problem is denoted by λ and the normalized positive eigenfunction corresponding to λ by \(\varphi(\Theta)\), \(\int_{\Omega}\varphi^{2}(\Theta)\,d\Omega=1\), where dΩ denotes the \((n1)\)dimensional volume element.
We put a rather strong assumption on Ω: if \(n\geq3\), then Ω is a \(C^{2,\alpha}\)domain (\(0<\alpha<1\)) on \(\mathbf{S}^{n1}\) surrounded by a finite number of mutually disjoint closed hypersurfaces (e.g. see [3], pp.8889, 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 \(n=3\).
Let I be the identical operator. If \(a\in\mathscr{A}_{a}\), then the stationary Schrödinger operator
can be extended in the usual way from the space \(C_{0}^{\infty}(C_{n}(\Omega))\) to an essentially selfadjoint operator on \(L^{2}(C_{n}(\Omega))\) (see [4], Chapter 13). We will denote it \(SSE_{a}\) as well. This last one has a GreenSch function \(G_{\Omega}^{a}(P,Q)\) which is positive on \(C_{n}(\Omega)\) and its inner normal derivative \(\partial G_{\Omega}^{a}(P,Q)/{\partial n_{Q}}\geq0\), where \({\partial}/{\partial n_{Q}}\) denotes the differentiation at Q along the inward normal into \(C_{n}(\Omega)\).
In this paper, we are concerned with the weak solutions of the inequality
where \(P=(r,\Theta)\in C_{n}(\Omega)\).
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]).
Solutions of the ordinary differential equation
play an essential role in this paper. It is well known (see, for example, [12]) that if the potential \(a\in\mathscr{A}_{a}\), then equation (1.2) has a fundamental system of positive solutions \(\{V,W\}\) such that V is nondecreasing with
and W is monotonically decreasing with
Denote
then the solutions to equation (1.2) have the following asymptotic (see [3]):
Let \(u(P) \) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) be a function. We introduce the following notations: \(u^{+}=\max\{u,0\}\), \(u^{}=\min\{u,0\}\), \(M_{u}(r)=\sup_{\Theta\in \Omega}u(P)\), \(l=\max_{\Theta\in\Omega}\varphi(\Theta)\),
For any two positive numbers δ an r, we put
and
The integral
is denoted by \(N_{u}(r)\), when it exists. The finite or infinite limits
are denoted by \(\mu_{u}\) and \(\eta_{u}\), respectively, when they exist.
We shall say that \(u(P)\) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) satisfies the PhragménLindelöf boundary condition on \(S_{n}(\Omega)\), if
for every \(Q\in S_{n}(\Omega)\).
Throughout this paper, unless otherwise specified, we will always assume that \(u(P)\in SbH_{a}(\Omega)\) and satisfy the PhragménLindelöf boundary condition on \(S_{n}(\Omega)\). Recently, about the PhragménLindelöf theorems for subfunctions in a cone, Qiao and Deng (see [9], Theorem 3) proved the following result.
Theorem A
If
then
for any \(P=(r,\Theta)\in C_{n}(\Omega)\).
A stronger version of a PhragménLindelöf type theorem is also due to Qiao and Deng (see [9], Theorem 3).
Theorem B
If
and
then
for any \(P=(r,\Theta)\in C_{n}(\Omega)\).
However, they do not tell us in [9] whether or not the limit
exists. In this paper, we first of all answer this question positively and prove the following result.
Theorem 1
If (1.3) is satisfied, then the limit \(B_{u} \) (\(0\leq B_{u}\leq+\infty\)) exists and
where
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
If
then
In [9], the authors gave the properties of the positive part of weak solutions satisfying the PhragménLindelöf boundary condition on \(S_{n}(\Omega)\). Finally, we shall show one of the properties of its negative part.
From the remark, we have
Since
Theorem 2 follows immediately.
Theorem 2
Under the conditions of Theorem B, if \(\eta_{u}\geq0\), then
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 nondecreasing on \((0,+\infty)\).

(3)
If \(\mu_{u}\leq0\), then \(W^{1}(r)N_{u}(r)\) is nonincreasing on \((0,+\infty)\).
Lemma 2
If (1.3) is satisfied and there exists a positive number R such that \(u(P)\leq0\) for any \(P=(r,\Theta)\in C_{n}(\Omega;(0,R))\), then for any positive number δ, we have
for any \(P=(r,\Theta)\in C_{n}(\Omega)\).
Proof
Let δ be any given positive number and \(\{r_{k}\}\) be a sequence such that
Then we have
for any \(0< r_{k}< R\) and hence
Thus we obtain (2.1) from Theorem B. □
Lemma 3
Under the conditions of the corollary, \(L_{u}>\infty\) and \(J_{u}=L_{u}\).
Proof
It is evident that
Hence, we shall prove that \(J_{u}=L_{u}\) under the assumption that \(L_{u}<+\infty\). Since (1.3) and (1.4) are satisfied and (1.8) gives \(\mu_{u}\leq0\), we have
for any \(P=(r,\Theta)\in C_{n}(\Omega)\) from Theorem B, which gives
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 DirichletSch 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
If
are satisfied, then
Proof
Take any \(P=(r,\Theta)\in C_{n}(\Omega)\) and any pair of numbers \(R_{1}\), \(R_{2}\) satisfying \(0<2R_{1}<r<\frac {1}{2}R_{2}<\infty\). If \(\psi(P)\) is a boundary function on \(\partial {C_{n}(\Omega;(R_{1},R_{2}))}\) satisfying
then we have
where \(G_{C_{n}(\Omega;(R_{1},R_{2}))}^{a}(\cdot,\cdot)\) is the GreenSch function on \(C_{n}(\Omega;(R_{1},R_{2}))\) with the pole at P.
Here we use the following inequalities (see [1], p.124):
and
where \(c_{1}\) and \(c_{2}\) are two positive constants.
Then we have
where \(c_{3}\) and \(c_{4}\) are two positive constants.
As \(R_{1}\rightarrow0\) and \(R_{2}\rightarrow\infty\) in (2.5), we obtain
from Lemma 1, which gives the conclusion of Lemma 4 from (2.4). □
3 Proof of Theorem 1
Put
Since
and Lemma 1 gives
we immediately see that \(\tau>\infty\).
Now we distinguish two cases.
Case 1 \(\tau=+\infty\).
In this case \(B_{u}\) exists and is equal to +∞. It is obvious that for any positive number r
which gives \(L_{u}=+\infty\).
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 \(0\leq L_{u}<+\infty\).
There exists a positive number \(R_{\epsilon}\) such that
for any \(\epsilon>0\), where \(P=(r,\Theta)\in C_{n}(\Omega;(0,R_{\epsilon}))\).
This gives
Now, assume that \(\tau< L_{u}l\). There exist a positive number \(\delta_{1}\) and a set \(E_{u}\subset\Omega\) such that
and
for \(\Theta\in E_{u}\).
We define \(v_{1}(P) \) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) by
and apply Lemma 2 to \(v_{1}(P)\). It gives
for any \(P=(r,\Theta)\in C_{n}(\Omega)\).
So we have
If we can show that
then we have a contradiction.
To prove (3.6), take a sequence \(\{r_{k}\}\), with \(\lim_{k\rightarrow\infty}r_{k}=0\), such that
for any \(\Theta\in E_{u}\), which gives
Hence
Thus from (3.3) we can simultaneously prove the existence of \(B_{u}\) and (1.6).
Subcase 2.2 \(\infty\leq L_{u}<0\).
Take any small number \(\epsilon>0\) satisfying \(L_{u}+\epsilon<0\). There exists a positive number \(R_{\epsilon}\) such that
for any \(P=(r,\Theta)\in C_{n}(\Omega;(0,R_{\epsilon}))\).
This gives
Now suppose that \(\tau<0\). There are a sequence \(\{r_{k}\}\) tending to 0 and a positive number \(\delta_{2}\) such that
Define \(v_{2}(P) \) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) by
and apply Lemma 2 to \(v_{2}(P)\). Then we obtain
for any \(P=(r,\Theta)\in C_{n}(\Omega)\), which gives
If we can show that
then we have a contradiction.
To prove (3.8), write
It is evident that
For every \(\Theta\in F_{u}\), we have
which shows that
Hence we have
Thus we can prove that \(\tau\geq0\). With (3.7), this also gives the existence of \(B_{u}\) and
Lastly, we shall show that (1.7) holds.
If \(\eta_{u^{+}}=+\infty\), then it is evident that \(B_{u^{+}}=+\infty\). This together with (3.2) gives \(L_{u^{+}}=+\infty\). Since
we know 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).
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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. 2013ZD011) 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. 2015CXTD09).
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Wan, L. Some remarks on PhragménLindelöf theorems for weak solutions of the stationary Schrödinger operator. Bound Value Probl 2015, 239 (2015). https://doi.org/10.1186/s1366101505080
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DOI: https://doi.org/10.1186/s1366101505080