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 $$\left\{\begin{array}{l} -\Delta_{p}u = \lambda |u|^{q-2}u,\quad {\rm in}\,\Omega,\\ \quad\quad u=0, \quad\quad\quad\quad\,{\rm on}\, \partial \Omega,\end{array}\right.$$ where Ω is a bounded domain in \(\mathbb{R}^n\), n ≥ 2, λ > 0 and p < q < p* (with \(p^* = \frac{np}{n-p}\) 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 −Δ p .