Abstract <p>In the paper, we study the sufficient condition for the entire solution of a certain type of partial differential equation in <InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7237_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}^{m}\)</EquationSource>  </InlineEquation>. The main objective of this paper is to investigate the question raised by Berenstein et al [<CitationRef CitationID="CR1">1</CitationRef>]. That is, we tackle the situation when a nonconstant entire function in <InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7237_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}^{m}\)</EquationSource>  </InlineEquation> shares two values ignoring multiplicities with its partial differential polynomial. Aside from this, we also solve a specific type of nonlinear partial differential equation and our obtained results are the extensions of the results of Li and Yang [<CitationRef CitationID="CR2">2</CitationRef>] and Li and Yang [<CitationRef CitationID="CR3">3</CitationRef>] from <InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7237_Article_IEq5.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}\)</EquationSource>  </InlineEquation> to <InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="11957_2025_7237_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="24" /> </InlineMediaObject> <EquationSource Format="TEX">\(\mathbb{C}^{m}\)</EquationSource>  </InlineEquation>.</p>

Sufficient Condition for Entire Solution of a Certain Type of Partial Differential Equation in \(\boldsymbol{\mathbb{C}}^{\boldsymbol{m}}\)

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发表日期：

2025年10月10日

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S. Majumder,
P. Das,
D. Pramanik

摘要

Abstract

In the paper, we study the sufficient condition for the entire solution of a certain type of partial differential equation in \(\mathbb{C}^{m}\) . The main objective of this paper is to investigate the question raised by Berenstein et al [1]. That is, we tackle the situation when a nonconstant entire function in \(\mathbb{C}^{m}\) shares two values ignoring multiplicities with its partial differential polynomial. Aside from this, we also solve a specific type of nonlinear partial differential equation and our obtained results are the extensions of the results of Li and Yang [2] and Li and Yang [3] from \(\mathbb{C}\) to \(\mathbb{C}^{m}\) .