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 \(\partial{C_{n}(\Omega)}-\{O\}\).
Let \(\Delta^{\ast}\) be the spherical part of the Laplace operator Δ (see [1]),
$$\Delta=\frac{n-1}{r}\frac{\partial}{\partial r}+\frac{\partial^{2}}{\partial r^{2}}+\frac{\Delta^{\ast}}{r^{2}}, $$
and Ω be a domain on \(\mathbf{S}^{n-1}\) with smooth boundary. We consider the Dirichlet problem (see [2], p.41)
$$\begin{aligned}& \bigl(\Delta^{\ast}+\lambda\bigr)\varphi(\Theta)=0 \quad\mbox{on } \Omega, \\& \varphi(\Theta)=0 \quad\mbox{on } \partial{\Omega}. \end{aligned}$$
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 \((n-1)\)-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 \(\mathbf{S}^{n-1}\) 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 \(n=3\).
Let I be the identical operator. If \(a\in\mathscr{A}_{a}\), then the stationary Schrödinger operator
$$SSE_{a}=-\Delta+a(P)I $$
can be extended in the usual way from the space \(C_{0}^{\infty}(C_{n}(\Omega))\) to an essentially self-adjoint operator on \(L^{2}(C_{n}(\Omega))\) (see [4], Chapter 13). We will denote it \(SSE_{a}\) as well. This last one has a Green-Sch 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
$$ SSE_{a}u(P)\leq0, $$
(1.1)
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
$$ -\Pi''(r)-\frac{n-1}{r} \Pi'(r)+ \biggl( \frac{\lambda}{r^{2}}+a(r) \biggr)\Pi(r)=0,\quad 0< r< \infty, $$
(1.2)
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 non-decreasing with
$$0\leq V(0+)\leq V(r)\nearrow\infty \quad\mbox{as } r\rightarrow+\infty, $$
and W is monotonically decreasing with
$$+\infty=W(0+)>W(r)\searrow0 \quad\mbox{as } r\rightarrow+\infty. $$
Denote
$$\iota_{k}^{\pm}=\frac{2-n\pm\sqrt{(n-2)^{2}+4(k+\lambda)}}{2}, $$
then the solutions to equation (1.2) have the following asymptotic (see [3]):
$$ V(r)\approx r^{\iota_{k}^{+}},\qquad W(r)\approx r^{\iota _{k}^{-}}, \quad\mbox{as } r \rightarrow\infty. $$
Let \(u(P) \) (\(P=(r,\Theta)\in C_{n}(\Omega)\)) be a function. We introduce the following notations: \(u^{+}=\max\{u,0\}\), \(u^{-}=-\min\{u,0\}\), \(M_{u}(r)=\sup_{\Theta\in \Omega}u(P)\), \(l=\max_{\Theta\in\Omega}\varphi(\Theta)\),
$$S_{u}(r)=\sup_{\Theta\in\Omega}\frac{u(P)}{\varphi(\Theta )}, \qquad L_{u}=\limsup_{r\rightarrow0}\frac{S_{u}(r)}{W(r)}, \qquad J_{u}=\sup_{P\in C_{n}(\Omega)}\frac{u(P)}{W(r)\varphi(\Theta)}. $$
For any two positive numbers δ an r, we put
$$E_{0}^{u}(r;\delta)=\bigl\{ \Theta\in\Omega:u(P)\leq-\delta W(r)\bigr\} $$
and
$$\xi_{u}(\delta)=\limsup_{r\rightarrow0} \int_{E_{0}^{u}(r;\delta)}\varphi (\Theta)\,d\Omega. $$
The integral
$$\int_{\Omega}u(r,\Theta)\varphi(\Theta)\,d\Omega, $$
is denoted by \(N_{u}(r)\), when it exists. The finite or infinite limits
$$\lim_{r\rightarrow\infty}\frac{N_{u}(r)}{V(r)} \quad \mbox{and} \quad \lim _{r\rightarrow0}\frac{N_{u}(r)}{W(r)} $$
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 Phragmén-Lindelöf boundary condition on \(S_{n}(\Omega)\), if
$$\limsup_{P\rightarrow Q, Q \in S_{n}(\Omega)} u(P)\leq0 $$
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 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
If
$$\mu_{u^{+}}=\eta_{u^{+}}=0, $$
then
for any
\(P=(r,\Theta)\in C_{n}(\Omega)\).
A stronger version of a Phragmén-Lindelöf type theorem is also due to Qiao and Deng (see [9], Theorem 3).
Theorem B
If
$$ \liminf_{r\rightarrow\infty}\frac{M_{u}(r)}{V(r)}< +\infty $$
(1.3)
and
$$ \liminf_{r\rightarrow 0}\frac{M_{u}(r)}{W(r)}< +\infty, $$
(1.4)
then
$$ u(P)\leq \bigl(\mu_{u}V(r)+\eta_{u}W(r)\bigr) \varphi(\Theta) $$
(1.5)
for any
\(P=(r,\Theta)\in C_{n}(\Omega)\).
However, they do not tell us in [9] whether or not the limit
$$B_{u}=\lim_{r\rightarrow0}\frac{M_{u}(r)}{W(r)} $$
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
$$ B_{u}=(L_{u})^{+}l, $$
(1.6)
where
$$ (L_{u})^{+}=\eta_{u^{+}}. $$
(1.7)
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
$$ \liminf_{r\rightarrow \infty}\frac{M_{u}(r)}{V(r)}\leq0, $$
(1.8)
then
$$ B_{u}=(J_{u})^{+}l. $$
(1.9)
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.
From the remark, we have
$$ \eta_{u^{+}}=L_{u^{+}}=(L_{u})^{+}=(\eta_{u})^{+}. $$
Since
$$ N_{u}(r)=N_{u^{+}}(r)-N_{u^{-}}(r), $$
Theorem 2 follows immediately.
Theorem 2
Under the conditions of Theorem
B, if
\(\eta_{u}\geq0\), then
$$ \lim_{r\rightarrow0}\frac{N_{u^{-}}(r)}{W(r)}=0. $$