On the blow-up of solutions to Liouville-type equations (Q2789498)

This paper deals with the following perturbed Liouville-type problem: NEWLINE\[NEWLINE -\Delta_g u =\rho f(u) - c_{\rho}\qquad \text{in }M, NEWLINE\]NEWLINE NEWLINE\[NEWLINE \int_{M} u dx = 0, NEWLINE\]NEWLINE \noindent where \((M, g)\) is a compact Riemannian 2-manifold without boundary, \(c_{\rho} = \frac{\rho}{|M|} \int_{\Omega} f(u) dx\), \(dx\) being the volume element on \(M\), and \(\Delta_g\) denotes the Laplace-Beltrami operator.NEWLINENEWLINEConditions are derived on the regular nonlinearities \(f\) under which the blow-up points are located. Furthermore, a quick proof of the mass quantization is obtained.