Let \((M,g)\) be a compact Riemann surface without boundary and let \(\Delta_{g}\) be the associated Laplace-Beltrami operator. The main purpose of the paper is to provide a careful analysis of blowup solutions \((\xi_{n})\) in \(C^{2}(M)\) to the problems \[ - \Delta_{g} \xi_{n} = \lambda_{n} (V_{n} e^{\xi_{n}} - W_{n}) \quad\text{on \(M\)} , \qquad \int_{M} V_{n} e^{\xi_{n}} dv_{g} = 1 , \] where \((\lambda_{n})\) is convergent in \({\mathbb R}\) and \(V_{n},W_{n}\) are subjected to suitable uniform estimates. Such a result turns out to be a crucial step in the study of some semilinear elliptic problems, which originated from some Chern-Simons Higgs model. The main result is derived by a Harnack type inequality for a semilinear elliptic equation in dimension two, which is proved by the technique of moving planes.