On the convergence of the sequence of solutions for a family of eigenvalue problems

The asymptotic behavior of the sequence { u n } of positive first eigenfunctions for a class of eigenvalue problems is studied in a bounded domain Ω ⊂ R N with smooth boundary ∂ Ω. We prove u n → ‖ δ ‖L 2 ( Ω ) − 1 δ , where δ is the distance function to ∂ Ω. Our study complements some earlier results by Payne and Philippin, Bhattacharya, DiBenedetto, and Manfredi, and Kawohl obtained in relation with the “ torsional creep problem .”