Harnack type inequalities and quantization for the uniformly elliptic Liouville equation

We extend the Harnack type inequality proved in C. R. Acad. Sci. Paris 315(2) (1992), 159–164, to the solutions of

-div(A∇u)=Ve^u in Ω,

with no boundary conditions. Here A is a symmetric, uniformly elliptic matrix and Ω⊂R² is open and bounded. As an application we are able to generalize the quantization results of Ind. Univ. Math. J. 43(4) (1994), 1255–1270, to the uniformly elliptic case.