Complex analytic solutions of certain nonlinear partial differential equations in $\mathbb{C}^n$
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초록
The paper characterizes entire and meromorphic solutions of the nonlinear partial differential equations (PDEs) of the form $u^p+(\mathcal{L}u)^q=e^{g}$ in $\mathbb{C}^n$, where $p\geq 2, q\geq2$ are integers, $g$ is a polynomial in $\mathbb{C}^n$, and $\mathcal{L}u=\sum_{j=1}^{m}\sum_{|I|=j} P_{I}\cdot \frac{\partial^{|I|}u}{\partial z_{n}^{i_{n}} \cdots \partial z_{1}^{i_{1}}}$ is a partial differential polynomial in $u$ with polynomial coefficients. As an application, the method developed here can be used to describe the complex analytic solutions of some nonlinear differential equations.
Abstract
The paper characterizes entire and meromorphic solutions of the nonlinear partial differential equations (PDEs) of the form $u^p+(\mathcal{L}u)^q=e^{g}$ in $\mathbb{C}^n$, where $p\geq 2, q\geq2$ are integers, $g$ is a polynomial in $\mathbb{C}^n$, and $\mathcal{L}u=\sum_{j=1}^{m}\sum_{|I|=j} P_{I}\cdot \frac{\partial^{|I|}u}{\partial z_{n}^{i_{n}} \cdots \partial z_{1}^{i_{1}}}$ is a partial differential polynomial in $u$ with polynomial coefficients. As an application, the method developed here can be used to describe the complex analytic solutions of some nonlinear differential equations.