Three remarks on mean field equations

Curvature flow to the Nirenberg problem

“…We only need to exclude the case when K ∞ (q) ≤ 0. It is clear that K ∞ (q) < 0 can not occur since there is no such a solution on the whole plane R 2 [8]. If K ∞ (q) = 0, thenû :=û ∞ is a harmonic function in R 2 .…”

Q-curvature flow with indefinite nonlinearity

“…It can not occur since there is no such a solution on the whole space R 4 (see also the argument in [7]). If Q ∞ (q) = 0, then ∆ R 4û := ∆ R 4û ∞ is a harmonic function in R 4 .…”

“…Since ∆ R 4ū is a continuous function on [0, ∞), we have ∆ R 4ū = A, which gives us thatū = A + Br 2 + Cr −2 for some constants A, B, and C. Again, usingū is regular, we have C = 0 andū = A + Br 2 with B < 0. However, it seems hard to exclude this case without the use of the fact (7).…”