Elliptic Equation with Singular Terms on Riemannian Manifold

Abstract

In this paper we deal with the following equation: \({\Delta_g u -c u + \sum_{i=2}^l c_i u^i + \sum_{i=1}^{m} c_{-i} u^{-i} =0 }\) on a three-dimensional Riemannian manifold \({(\Sigma, g)}\) . We assume that the volume of Σ, the norm \({||c_i||_{H^2(\Sigma)}}\) , \({||\nabla c||_{L^2(\Sigma)}}\) and \({||c-1||_{L^{\infty}(\Sigma)}}\) are small enough. Using a rather simple argument we show the existence of solution to the problem.