Resonant Brézis–Nirenberg case

Establish whether the critical ball-mass equation -Δu + Φu(|x|)|u|^{p-2}u = λ|u|^{β-2}u + |u|^{2*−2}u in ℝ^N has a nontrivial weak radial solution when λ equals a scaled eigenvalue λ_k.

Background

The Brézis–Nirenberg-type theorem establishes a nontrivial weak radial solution for positive parameters λ lying strictly between consecutive scaled eigenvalues, and also for 0 < λ < λ_1. It does not cover the resonant values λ = λ_k. The authors explain that the linking geometry and index computation persist at resonance, but they cannot ensure that the associated minimax level remains below the compactness threshold S^{N/2}/N.