The current article is devoted to the study of the semi-linear partial differential equations (PDEs) with the homogeneous Neumann boundary condition \(\begin{aligned} {\left\{ \begin{array}{ll} \Delta _{g} u + f u^{\alpha }=0, \ \text {in} \ M,\\ \frac{\partial u}{\partial \nu }=0 ,\ \text {on} \ \partial M. \end{array}\right. } \end{aligned}\) The domain M is a smooth compact Riemannian manifold of dimension \(n\ge 3\) with boundary \(\partial M\) . With the concern on its geometric reasoning behind, the positive solution u is expected. Naturally the variational method will be adopted. Due to its obvious necessary conditions on f, we derive some appropriate conditions on the function f and the manifold M to ensure the solvability.
