On the resonant Lane-Emden problem for the p-Laplacian

Abstract

We study the positive solutions of the Lane-Emden equation $-{\Delta }_{p}u={\lambda }_p|u{|}^{q-2}u$ in $\Omega$ with homogeneous Dirichlet boundary conditions, where $\Omega \subset {\mathbb{R}}^N$ is a bounded and smooth domain, $N\ge 2,$ ${\lambda }_p$ is the first eigenvalue of the $p$-Laplacian operator ${\Delta }_p$ and $q$ is close to $p>1.$ We prove that any family of positive solutions of this problem converges in $C^1(\bar{\Omega })$ to the function ${\theta }_p{e}_p$ when $q\to p,$ where $e_p$ is the positive and $L^{\infty }$-normalized first eigenfunction of the $p$-Laplacian and ${\theta }_p:=\exp (|e_p{|}_{L^p(\Omega )}^{-p}{\int }_{\Omega }{e}_p$ A consequence of this result is that the best constant of the immersion $W_0^{1,p}(\Omega )\hookrightarrow L^q(\Omega )$ is differentiable at $q=p.$ Previous results on the asymptotic behavior (as $q\to p$) of the positive solutions of the non-resonant Lane-Emden problem (i.e. with ${\lambda }_p$ replaced by a positive $\lambda \ne {\lambda }_p$) are also generalized to the space $C^1$ and to arbitrary families of these solutions. Moreover, if $u_{\lambda ,q}$ denotes a solution of the non-resonant problem for an arbitrarily fixed $\lambda >0,$ we show how to obtain the first eigenpair of the $p$-Laplacian as the limit in $C^1(\bar{\Omega }),$ when $q\to p$, of a suitable scaling of the pair $(\lambda ,u_{\lambda ,q}).$ For computational purposes the advantage of this approach is that $\lambda$ does not need to be close to ${\lambda }_p.$ Finally, an explicit estimate involving $L^{\infty }$ and $L^1$ norms of $u_{\lambda ,q}$ is also deduced using set level techniques.

The paper was improved and completely reformulated. The main result now is valid for any family of positive solutions in the super-linear case (not necessarily ground state)

References in corpus (2)

• Solutions of the Cheeger problem via torsion functions
• Computing the first eigenpair of the p-Laplacian via inverse iteration of sublinear supersolutions