The authors deal with the Brezis-Nirenberg problem: \[ \Delta u+|u|^{p-1}u+\lambda u=0,\text{ in }\mathbb B\quad u=0,\text{ on } \partial\mathbb B,\tag{1} \] where \(\mathbb B\) is the unit ball in \(\mathbb R^N\), \(N\geq 3\), \(p^*=\frac{N+2}{N-2}\). The authors are interested in describing the solutions in the slightly supercritical regime: \(p=p^*+\varepsilon\). To this end, the authors perform a phase plane analysis, which emphasize the common heuristic properties of the Brezis-Nirenberg problem in dimension \(N\geq 3\) and an equation involving the exponential nonlinearity in dimension \(N\geq 2\).