一类半线性椭圆型方程边值问题的可解性研究

Solvability for Certain Semilinear Elliptic Equations with Boundary Value

非线性偏微分方程(组)是现代微分方程研究中的重中之重,在解决物理学、生态学、气动力学等领域问题中起到重要作用。但非线性偏微分方程求解难度很大,本文利用Leray-Schauder不动点定理证明了一类半线性椭圆型方程边值问题解的存在性,并对非线性项在满足两种不同情形时,证明了其解的唯一性;并且讨论了若干个条件在不同定理中使用的情况,利用确界原理和格林第一公式得出了4个重要定理。

Nonlinear partial differential equations (pdes) are the most important in the study of modern differential equations, and play an important role in solving problems in physics, ecology, aerodynamics and other fields. However, it is very difficult to solve nonlinear partial differential equations.In this paper, we prove the existence of solutions for a class of semilinear elliptic boundary value problems by using Leray-Schauder fixed point theorem, and prove the uniqueness of solutions for nonlinear terms when the nonlinear term satisfies two different conditions. In addition, the application of several conditions in different theorems is discussed, and four important theorems are obtained by using the definite bound principle and green's first formula.