An example of a blow-up sequence for \(-\Delta{}u = V(x)e^ u\) (Q1198725)

\textit{H. Brezis} and \textit{F. Merle} [Commun. Partial Differ. Equations 16, No. 8/9, 1223-1253 (1991; Zbl 0746.35006)] have shown that if \((u_ n)\) is a sequence of solutions of the system \[ -\Delta u=V(x)e^ u\text{ in }\Omega\subseteq\mathbb{R}^ 2; \qquad u=0\text{ on } \partial\Omega \qquad (V\in L^ \infty(\Omega),\;\Omega\text{ bounded}) \] for which \(\| V_ n(x)\|_{L^ \infty}<C\), (*) \(V_ n(x)\geq 0\) on \(\Omega\) and \(\| e^{u_ n}\|_{L^ 1}\leq C\), then \((u_ n)\) is bounded in \(L_{\text{loc}}^ \infty(\Omega)\). In this paper it is shown that condition (*) is essential. A counterexample is given when it does not hold.