In this article, we consider some critical Brézis-Nirenberg problems in dimension $N \geq 3$ that do not have a solution. We prove that a supercritical perturbation can lead to the existence of a positive solution. More precisely, we consider the equation:

where $B \subset \mathbb{R}^N$ is a unit ball centred at the origin, $N\geq 3$, $r=\vert x \vert$, $\alpha \in (0,\min\{N/2,N-2\})$, λ is a fixed real parameter and $q\in [2,2^*]$. This class of problems can be interpreted as a perturbation of the classical Brézis–Nirenberg problem by the term rα at the exponent, making the problem supercritical when $r \in (0,1)$. More specifically, we study the effect of this supercritical perturbation on the existence of solutions. In particular, when N = 3, an interesting and unexpected phenomenon occurs. We obtain the existence of solutions for λ in a range where the Brézis–Nirenberg problem has no solution.