In this note, we recall an example adapted from an elegant paper due to Y.Y. Li and I. Shafrir published in the Indiana Univ. Math. J. in 1994 [here].
Let us consider the asymptitic behavior of sequences of solutions of
on a bounded domain with a non-negative continuous function. For each solution , we call
the mass of (over a ball ). The terminology limiting mass will be referred to the limit .
For simplicity, we assume . Given , we are going to construct a sequence of solutions which blows up exactly at points, say at where the unit disc of . Our equation reads as
in . Using the Liouville formula for solutions of the above equation, we get
with an holomorphic function such that .
For some fixed distinct points , we consider the functions as follows
and the corresponding given by the Liouville formula.
It is obvious to see that does blow up at but it is not defined at the points the zeros of the polynomial . So we choose a smooth bounded and simply connected domain containing but avoiding .
Let be the Riemann mapping of onto . We define on the sequence
It is easy to see that sequence is a sequence of solutions on which blows up at