## Abstract

In this paper, by using a new type of Carleman formula with respect to a certain Laplace operator, we estimate the growth property for solutions of certain Laplace equations defined in a smooth cone.

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## 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

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

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])

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

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*

*and*

*then*

*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*

*and*

*then*

*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*

*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*,

*and*

### Lemma 2

(See [9], Lemma 4)

*We have*

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

*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*

*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

It is easy to see that

and

from (1.3).

We remark that

We have

and

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

It follows from (3.6) that

which shows that

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

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

from (3.8), where

and

We have the following estimates:

and

We consider the inequality

where

and

We first have

from (3.7).

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

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

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

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

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

where

Since

we have

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

Since

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

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

So

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

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

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

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

It follows from (3.8) that

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

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

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

which also gives (3.18).

We finally have

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## Acknowledgements

The authors are grateful to the two anonymous referees for their valuable comments, which led to a much improved version of the manuscript.

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### Authors’ contributions

MZ completed the main study. BH carried out the results of this article. JW verified the calculation. All the authors read and approved the final manuscript.

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Huang, B., Wang, J. & Zylbersztejn, M.H. An extension of the estimation for solutions of certain Laplace equations.
*J Inequal Appl* **2016**, 167 (2016). https://doi.org/10.1186/s13660-016-1109-x

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DOI: https://doi.org/10.1186/s13660-016-1109-x