标题： 流形上半线性方程的径向解的存在性和唯一性 显示英文标题

标题： Existence and uniqueness of radial solutions of semi-linear equations on manifolds

摘要： 我们研究定义在黎曼流形$M$上的二阶半线性偏微分方程解的存在性和唯一性。 通过结合微分几何和分析技术，我们通过群作用的轨道建立了常数解的存在性和唯一性。 我们的方法将此类问题转化为一个与群作用正交的一维子流形$\Sigma$上的等价问题。 这种约简使我们进入一维设置，在此我们可以应用常微分方程理论中的不同结果。 我们的框架具有通用性，包括极坐标作用或指数坐标设置，具体例子如球面、旋转曲面等。 显示英文摘要

摘要： We investigate the existence and uniqueness of solutions for second-order semi-linear partial differential equations defined on a Riemannian manifold $M$. By combining differential geometry and analysis techniques, we establish the existence and uniqueness of constant solutions through the orbits of a group action. Our approach transforms such problems into equivalent ones over a submanifold $\Sigma$ of dimension one, which is transversal to the group action. This reduction leads us to a one-dimensional setting, where we can apply different results from the theory of ordinary differential equations. Our framework is versatile and includes the setups of polar actions or exponential coordinates, with particular examples such as the sphere, surfaces of revolution, and others.