The effect of a perturbation on Brezis-Nirenberg’s problem

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: \begin{equation*} \left\{ \begin{array}{rllll} -\Delta u & = & \lambda u^{q-1} + u^{2^*+ r^\alpha -1} & \mbox{in} & B, \\ u & \gt & 0 & \mbox{in} & B, \\ u&=&0 & \mbox{on} & \partial B,\\ \end{array} \right. \end{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 r α 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.