The Effect of a Perturbation on Brezis-Nirenberg’S Problem

In this article, we consider some critical Brézis-Nirenberg problems in dimension N ≥ 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: (Equation Presented), where B ⊂ RN is a unit ball centred at the origin, N ≥ 3, r = |x|, α ∈ (0, min{N/2, N − 2}), λ is a fixed real parameter and q ∈ [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 ∈ (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. © The Author(s), 2024.