ON THE ASYMPTOTICS OF SOLUTIONS OF THE LANE-EMDEN PROBLEM FOR THE p-LAPLACIAN

ABSTRACT. In this paper we consider the Lane–Emden problem adapted for the p-Laplacian{ −∆pu = λ |u|q−2u, in Ω, u = 0, on ∂Ω, where Ω is a bounded domain in Rn, n> 2, λ> 0 and p< q< p ∗ (with p ∗ = npn−p if p< n, and p ∗ = ∞ otherwise). After some recalls about the existence of ground state and least energy nodal solutions, we prove that, when q → p, accumulation points of ground state solutions or of least energy nodal solutions are, up to a "good " scaling, respectively first or second eigenfunctions of −∆p. 1