Transition between nonlinear and linear eigenvalue problems

Abstract We study convergence of variational solutions of the nonlinear eigenvalue problem − Δ u = λ | u | p − 2 u , u ∈ H 0 1 ( Ω ) , as p ↓ 2 or as p ↑ 2 , where Ω is a bounded domain in R N with smooth boundary. It turns out that if λ is not an eigenvalue of −Δ then the solutions either blow up or vanish according to p ↓ 2 or p ↑ 2 , while if λ is an eigenvalue of −Δ then the solutions converge to the associated eigenspace.