On a resonant Lane–Emden Problem

Abstract

We study the asymptotic behavior, as q → p, of the positive solutions of the Lane–Emden problem −Δpu = λp |u|q−2 u in Ω, u = 0 on ∂ Ω, where Ω ⊂ RN is a bounded and smooth domain N≥ 2 and λp is the first eigenvalue of the p-Laplacian operator Δp, p> 1 We prove that any family of positive solutions of this problem converges in C1(Ω) to the function θpep when q → p, where ep is the positive and L∞-normalized first eigenfunction of the p-Laplacian and (Formula presented.).