Transition between nonlinear and linear eigenvalue problems

We study convergence of variational solutions of the nonlinear eigenvalue problem−Δu=λ|u|p−2u,u∈H01(Ω), as p↓2 or as p↑2, where Ω is a bounded domain in RN 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.