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 *d*Ω 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*

$$ \lim_{r\rightarrow0}\frac{N_{u^{-}}(r)}{W(r)}=0. $$