1 Introduction and results

Let \(\mathbf{R}^{n} \) (\(n\geq2\)) be the n-dimensional Euclidean space. A point in \(\mathbf{R}^{n}\) is denoted by \(V=(X,y)\), where \(X=(x_{1},x_{2},\ldots,x_{n-1})\). The boundary and the closure of a set E in \(\mathbf{R}^{n}\) are denoted by ∂E and E̅, respectively.

We introduce a system of spherical coordinates \((l,\Lambda)\), \(\Lambda=(\theta_{1},\theta_{2},\ldots,\theta_{n-1})\), in \(\mathbf{R}^{n}\) which are related to cartesian coordinates \((x_{1},x_{2},\ldots,x_{n-1},y)\) by \(y=l\cos\theta_{1}\).

The unit sphere in \(\mathbf{R}^{n}\) is denoted by \(\mathbf{S}^{n-1}\). For simplicity, a point \((1,\Lambda)\) on \(\mathbf{S}^{n-1}\) and the set \(\{\Lambda; (1,\Lambda)\in\Gamma\}\) for a set Γ, \(\Gamma\subset\mathbf{S}^{n-1}\) are often identified with Λ and Γ, respectively. For two sets \(\Xi\subset\mathbf{R}_{+}\) and \(\Gamma\subset \mathbf{S}^{n-1}\), the set

$$\bigl\{ (l,\Lambda)\in\mathbf{R}^{n}; l\in\Xi,(1,\Lambda) \in\Gamma\bigr\} $$

in \(\mathbf{R}^{n}\) is simply denoted by \(\Xi\times\Gamma\).

We denote the set \(\mathbf{R}_{+}\times\Gamma\) in \(\mathbf{R}^{n}\) with the domain Γ on \(\mathbf{S}^{n-1}\) by \(T_{n}(\Gamma)\). We call it a cone. The sets \(I\times\Gamma\) and \(I\times\partial{\Gamma}\) with an interval on R are denoted by \(T_{n}(\Gamma;I)\) and \(\mathcal{S}_{n}(\Gamma;I)\), respectively. By \(\mathcal{S}_{n}(\Gamma; l)\) we denote \(T_{n}(\Gamma)\cap S_{l}\). We denote \(\mathcal{S}_{n}(\Gamma; (0,+\infty))\) by \(\mathcal{S}_{n}(\Gamma)\).

Let \(\mathbb{G}_{\Gamma}(V,W)\) (\(P, Q\in T_{n}(\Gamma)\)) be the Green function in \(T_{n}(\Gamma)\). Then the ordinary Poisson formula in \(T_{n}(\Gamma)\) is defined by

$$c_{n}\mathbb{PI}_{\Gamma}(V,W)=\frac{\partial\mathbb{G}_{\Gamma }(V,W)}{\partial n_{W}}, $$

where \({\partial}/{\partial n_{W}}\) denotes the differentiation at Q along the inward normal into \(T_{n}(\Gamma)\). Here, \(c_{2}=2\) and \(c_{n}=(n-2)w_{n}\) when \(n\geq3\), where \(w_{n}\) is the surface area of \(\mathbf{S}^{n-1}\).

Let \(\Delta_{n}^{*}\) be the spherical version of the Laplace operator and Γ be a domain on \(\mathbf{S}^{n-1}\) with smooth boundary ∂Γ. Consider the Dirichlet problem (see [1])

$$\begin{aligned}& \bigl(\Delta_{n}^{*}+\tau\bigr)\psi=0 \quad\mbox{on } \Gamma, \end{aligned}$$
(1.1)
$$\begin{aligned}& \psi=0 \quad\mbox{on } \partial{\Gamma}. \end{aligned}$$
(1.2)

We denote the least positive eigenvalue of (1.1) and (1.2) by τ and the normalized positive eigenfunction corresponding to τ by \(\psi(\Lambda)\). In the sequel, for the sake of brevity, we shall write χ instead of \(\aleph^{+}-\aleph^{-}\), where

$$2\aleph^{\pm}=-n+2\pm\sqrt{(n-2)^{2}+4\tau}. $$

We use the standard notations \(h^{+}=\max\{h,0\}\) and \(h^{-}=-\min\{h,0\}\). All constants appearing in the expressions in the following sections will be always written M, because we do not need to specify them. Throughout this paper, we will always assume that \(\eta (t)\) and \(\rho(t)\) are nondecreasing real-valued functions on an interval \([1,+\infty)\) and \(\rho(t)> \aleph^{+}\) for any \(t\in[1,+\infty)\).

Recently, Li and Vetro (see [2], Theorem 1) obtained the lower bounds for functions harmonic in a smooth cone. Similar results for solutions of p-Laplace equations under Neumann boundary condition, we refer the reader to the papers by Guo and Gao (see [3]) and Rao and Pu (see [4]).

Theorem A

Let K be a constant, \(h(V)\) (\(V=(R,\Lambda)\)) be harmonic on \(T_{n}(\Gamma)\) and continuous on \(\overline{T_{n}(\Gamma)}\). If

$$h(V)\leq KR^{\rho(R)},\qquad V=(R,\Lambda)\in T_{n}\bigl(\Gamma;(1, \infty)\bigr) $$

and

$$h(V)\geq-K,\quad R\leq1,\qquad V=(R,\Lambda) \in \overline{T_{n}(\Gamma)}, $$

then

$$h(V)\geq-KM\bigl(1+\rho(R)R^{\rho(R)}\bigr)\psi^{1-n}(\Lambda), $$

where \(V\in T_{n}(\Gamma)\) and M is a constant independent of K, R, \(\psi(\Lambda)\), and the function \(h(V)\).

In this paper, we shall extend Theorem A to solutions of a certain Laplace equation (see [5] for the definition of this Laplace equation).

Theorem 1

Let \(h(V)\) (\(V=(R,\Lambda)\)) be solutions of certain Laplace equation defined on \(T_{n}(\Gamma)\) and continuous on \(\overline{T_{n}(\Gamma)}\). If

$$ h(V)\leq\eta(R)R^{\rho(R)},\qquad V=(R,\Lambda)\in T_{n}\bigl( \Gamma;(1,\infty )\bigr), $$
(1.3)

and

$$ h(V)\geq-\eta(R),\quad R\leq1,\qquad V=(R,\Lambda) \in \overline{T_{n}(\Gamma)}, $$
(1.4)

then

$$h(V)\geq-M\eta(R) \bigl(1+\rho(cR)R^{\rho(cR)}\bigr)\psi^{1-n}( \Lambda), $$

where \(V\in T_{n}(\Gamma)\), c is a real number satisfying \(c\geq1\) and M is a constant independent of R, \(\psi(\Lambda)\), the functions \(\eta(R)\) and \(h(V)\).

Remark

In the case \(c\equiv1\) and \(\eta(R)\equiv K\), where K is a constant, Theorem 1 reduces to Theorem A.

2 Lemmas

In order to prove our result, we first introduce a new type of Carleman formula for functions harmonic in a cone (see [6]). For the Carleman formula for harmonic functions and its application, we refer the reader to the paper by Yang and Ren (see [7], Lemma 1). Recently, it has been extended to Schrödinger subharmonic functions in a cone (see [8], Lemma 1). For applications, we also refer the reader to the paper by Wang et al. (see [8], Theorem 2).

Lemma 1

Let h be harmonic on a domain containing \(T_{n}(\Gamma;(1,R))\), where \(R>1\). Then

$$\chi \int_{S_{n}(\Gamma;R)}h\psi R^{\aleph^{-}-1}\,dS_{R} + \int_{S_{n}(\Gamma;(1,R))}h \bigl(t^{\aleph^{-}}-t^{\aleph^{+}}R^{-\chi} \bigr){\partial\psi}/{\partial n}\,d\sigma_{W}+d_{1}+d_{2}R^{-\chi}=0, $$

where \(dS_{R}\) denotes the \((n-1)\)-dimensional volume elements induced by the Euclidean metric on \(S_{R}\), \({\partial}/{\partial n}\) denotes differentiation along the interior normal,

$$d_{1}= \int_{S_{n}(\Gamma;1)}\aleph^{-}h\psi-\psi({\partial h}/{\partial n})\,dS_{1} $$

and

$$d_{2}= \int_{S_{n}(\Gamma;1)}\psi({\partial h}/{\partial n})-\aleph ^{+}h \psi \,dS_{1}. $$

Lemma 2

(See [9], Lemma 4)

We have

$$\begin{aligned} \mathcal{PI}_{\Gamma}(V,W)\leq M r^{\aleph^{-}}t^{\aleph^{+}-1}\psi( \Lambda)\frac{\partial\psi( \Phi)}{\partial n_{\Phi}} \end{aligned}$$

for any \(V=(r,\Lambda)\in T_{n}(\Gamma)\) and any \(W=(t,\Phi)\in S_{n}(\Gamma)\) satisfying \(0<\frac{t}{r}\leq\frac{4}{5}\).

$$\begin{aligned} \mathcal{PI}_{\Gamma}(V,W)\leq M\frac{\psi(\Lambda)}{t^{n-1}}\frac{\partial\psi( \Phi)}{\partial n_{\Phi}}+M \frac{r\psi(\Lambda)}{|P-Q|^{n}}\frac{\partial\psi( \Phi)}{\partial n_{\Phi}} \end{aligned}$$

for any \(V=(r,\Lambda)\in T_{n}(\Gamma)\) and any \(W=(t,\Phi)\in S_{n}(\Gamma; (\frac{4}{5}r,\frac{5}{4}r))\).

Let \(G_{\Gamma,R}(V,W)\) be the Green function of \(T_{n}(\Gamma,(0,R))\). Then

$$\begin{aligned} \frac{\partial G_{\Gamma,R}(V,W)}{\partial R} \leq M r^{\aleph^{+}}R^{\aleph^{-}-1}\psi(\Lambda)\psi(\Phi), \end{aligned}$$

where \(V=(r,\Lambda)\in T_{n}(\Gamma)\) and \(Q=(R,\Phi)\in S_{n}(\Gamma;R)\).

3 Proof of Theorem 1

We first apply Lemma 1 to \(h=h^{+}-h^{-}\) and obtain

$$\begin{aligned} &\chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{+}R^{\aleph^{-}-1}\psi d S_{R}+ \int _{\mathcal{S}_{n}(\Gamma;(1,R))}h^{+} \bigl(t^{\aleph^{-}}-t^{\aleph ^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W}+d_{1}+d_{2}R^{-\chi} \\ &\quad=\chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{-}R^{\aleph^{-}-1}\psi d S_{R}+ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-} \bigl(t^{\aleph ^{-}}-t^{\aleph^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W}, \end{aligned}$$
(3.1)

It is easy to see that

$$ \chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{+}R^{\aleph^{-}-1}\psi d S_{R} \leq M\eta(R)R^{\rho(cR)-\aleph^{+}} $$
(3.2)

and

$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{+} \bigl(t^{\aleph^{-}}-t^{\aleph ^{+}}R^{-\chi} \bigr){\partial\psi}/{\partial n}\,d\sigma_{W}\leq M \eta(R)R^{\rho(cR)-\aleph ^{+}} $$
(3.3)

from (1.3).

We remark that

$$ d_{1}+d_{2}R^{-\chi} \leq M\eta(R)R^{\rho(cR)-\aleph^{+}}. $$
(3.4)

We have

$$ \chi \int_{\mathcal{S}_{n}(\Gamma;R)}h^{-}R^{\aleph^{-}-1}\psi d S_{R} \leq M\eta(R)R^{\rho(cR)-\aleph^{+}} $$
(3.5)

and

$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-} \bigl(t^{\aleph^{-}}-t^{\aleph ^{+}}R^{-\chi} \bigr) {\partial\psi}/{\partial n}\,d\sigma_{W} \leq M \eta(R)R^{\rho(cR)-\aleph^{+}} $$
(3.6)

from (3.1), (3.2), (3.3), and (3.4).

It follows from (3.6) that

$$\int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-}t^{\aleph^{-}}\frac{\partial \psi}{\partial n}\,d\sigma_{W}\leq M\eta(R) \frac{(\rho(cR)+1)^{\chi}}{(\rho(cR)+1)^{\chi}-(\rho(cR))^{\chi}} \biggl(\frac{\rho(cR)+1}{\rho(cR)}R \biggr)^{\rho(\frac{\rho(cR)+1}{\rho (cR)}R)-\aleph^{+}}, $$

which shows that

$$ \int_{\mathcal{S}_{n}(\Gamma;(1,R))}h^{-}t^{\aleph^{-}}{\partial\psi }/{ \partial n}\,d\sigma_{W} \leq M\eta(R)\rho(cR)R^{\rho(cR)-\aleph^{+}}. $$
(3.7)

By the Riesz decomposition theorem (see [10]), we have

$$\begin{aligned} -h(V)={}& \int_{\mathcal{S}_{n}(\Gamma;(0,R))}\mathcal{PI}_{\Gamma }(V,W)-h(W)\,d\sigma_{W} \\ &{}+ \int_{\mathcal{S}_{n}(\Gamma;R)}\frac{\partial \mathbb{G}_{\Gamma,R}(V,W)}{\partial R}-h(W)\,dS_{R}, \end{aligned}$$
(3.8)

where \(V=(l,\Lambda)\in T_{n}(\Gamma;(0,R))\).

We next distinguish three cases.

Case 1. \(V=(l,\Lambda)\in T_{n}(\Gamma;({5}/{4},\infty ))\) and \(R={5l}/{4}\).

Since \(-h(V)\leq h^{-}(V)\), we have

$$ -h(V)=\sum_{i=1}^{4} U_{i}(V) $$
(3.9)

from (3.8), where

$$\begin{aligned}& U_{1}(V)= \int_{\mathcal{S}_{n}(\Gamma;(0,1])}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \\& U_{2}(V)= \int_{\mathcal{S}_{n}(\Gamma;(1,{4l}/{5}])}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \\& U_{3}(V)= \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}, \end{aligned}$$

and

$$U_{4}(V)= \int_{\mathcal{S}_{n}(\Gamma;R)}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}. $$

We have the following estimates:

$$ U_{1}(V)\leq M\eta(R)\psi(\Lambda) $$
(3.10)

and

$$ U_{2}(V) \leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi( \Lambda) $$
(3.11)

from Lemma 2 and (3.7).

We consider the inequality

$$ U_{3}(V)\leq U_{31}(V)+U_{32}(V), $$
(3.12)

where

$$U_{31}(V)=M \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\frac{-h(W) \psi(\Lambda)}{t^{n-1}}\frac{\partial\phi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W} $$

and

$$U_{32}(V)=Mr\psi(\Lambda) \int_{\mathcal{S}_{n}(\Gamma;({4l}/{5},R))}\frac{-h(W) l\psi(\Lambda)}{|V-W|^{n}} \frac{\partial\phi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W}. $$

We first have

$$ U_{31}(V) \leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi(\Lambda) $$
(3.13)

from (3.7).

We shall estimate \(U_{32}(V)\). Take a sufficiently small positive number d such that

$$\mathcal{S}_{n}\bigl(\Gamma;({4l}/{5},R)\bigr)\subset B(P,{l}/{2}) $$

for any \(V=(l,\Lambda)\in\Pi(d)\), where

$$\Pi(d)=\Bigl\{ V=(l,\Lambda)\in T_{n}(\Gamma); \inf _{(1,z)\in\partial\Gamma}\bigl|(1,\Lambda)-(1,z)\bigr|< d, 0< r< \infty\Bigr\} , $$

and divide \(T_{n}(\Gamma)\) into two sets \(\Pi(d)\) and \(T_{n}(\Gamma)-\Pi(d)\).

If \(V=(l,\Lambda)\in T_{n}(\Gamma)-\Pi(d)\), then there exists a positive \(d'\) such that \(|V-W|\geq{d}'l\) for any \(Q\in \mathcal{S}_{n}(\Gamma)\), and hence

$$ U_{32}(V) \leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi( \Lambda), $$
(3.14)

which is similar to the estimate of \(U_{31}(V)\).

We shall consider the case \(V=(l,\Lambda)\in\Pi(d)\). Now put

$$H_{i}(V)=\bigl\{ Q\in\mathcal{S}_{n}\bigl( \Gamma;({4l}/{5},R)\bigr); 2^{i-1}\delta(V) \leq|V-W|< 2^{i}\delta(V)\bigr\} , $$

where

$$\delta(V)=\inf_{Q\in \partial{T_{n}(\Gamma)}}|V-W|. $$

Since

$$\mathcal{S}_{n}(\Gamma)\cap\bigl\{ Q\in\mathbf{R}^{n}: |V-W|< \delta(V)\bigr\} =\varnothing, $$

we have

$$U_{32}(V)=M\sum_{i=1}^{i(V)} \int_{H_{i}(V)}\frac{-h(W)r\psi(\Lambda)}{|V-W|^{n}}\frac {\partial \psi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W}, $$

where \(i(V)\) is a positive integer satisfying \(2^{i(V)-1}\delta(V)\leq\frac{r}{2}<2^{i(V)}\delta(V)\).

Since

$$r\psi(\Lambda)\leq M\delta(V), $$

where \(V=(l,\Lambda)\in T_{n}(\Gamma)\), similar to the estimate of \(U_{31}(V)\) we obtain

$$\int_{H_{i}(V)}\frac{-h(W)r\psi(\Lambda)}{|V-W|^{n}}\frac{\partial \psi( \Phi)}{\partial n_{\Phi}}\,d\sigma_{W} \leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi^{1-n}( \Lambda) $$

for \(i=0,1,2,\ldots,i(V)\).

So

$$ U_{32}(V)\leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi^{1-n}( \Lambda). $$
(3.15)

From (3.12), (3.13), (3.14), and (3.15) we see that

$$ U_{3}(V)\leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi^{1-n}( \Lambda). $$
(3.16)

On the other hand, we have from Lemma 2 and (3.5)

$$ U_{4}(V) \leq M\eta(R)R^{\rho(cR)}\psi(\Lambda). $$
(3.17)

We thus obtain from (3.10), (3.11), (3.16), and (3.17)

$$ -h(V)\leq M\eta(R) \bigl(1+\rho(cR)R^{\rho(cR)}\bigr)\psi^{1-n}( \Lambda). $$
(3.18)

Case 2. \(V=(l,\Lambda)\in T_{n}(\Gamma;({4}/{5},{5}/{4}])\) and \(R={5l}/{4}\).

It follows from (3.8) that

$$-h(V)= U_{1}(V)+U_{5}(V)+U_{4}(V), $$

where \(U_{1}(V)\) and \(U_{4}(V)\) are defined in Case 1 and

$$U_{5}(V)= \int_{\mathcal{S}_{n}(\Gamma;(1,R))}\mathcal {PI}_{\Gamma}(V,W)-h(W)\,d\sigma_{W}. $$

Similar to the estimate of \(U_{3}(V)\) in Case 1 we have

$$U_{5}(V)\leq M\eta(R)\rho(cR)R^{\rho(cR)}\psi^{1-n}( \Lambda), $$

which together with (3.10) and (3.17) gives (3.18).

Case 3. \(V=(l,\Lambda)\in T_{n}(\Gamma;(0,{4}/{5}])\).

It is evident from (1.4) that we have

$$-h\leq\eta(R), $$

which also gives (3.18).

We finally have

$$h(V)\geq -\eta(R)M\bigl(1+\rho(cR)R^{\rho(cR)}\bigr)\psi^{1-n}( \Lambda) $$

from (3.18), which is the conclusion of Theorem 1.