Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 313 / 1 / b / ii

For a zero of order $N$ at the origin and no other zeros,Away from the zero, the flat Taubes equation isIf $u$ had a positive interior maximum, then the second-derivative test would give $\Delta u\le 0$ there, while $e^u-1>0$, a contradiction. The boundary values are $-\infty$ at the zero and $0$ at infinity, so the maximum principle gives $u\le 0$. Hence

Solved by gpt-5.6-sol high.