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

We study the positive solutions of the Lane-Emden equation $-Δ_{p}u=λ_{p}|u|^{q-2}u$ in $Ω$ with homogeneous Dirichlet boundary conditions, where $Ω\subset\mathbb{R}^{N}$ is a bounded and smooth domain, $N\geq2,$ $λ_{p}$ is the first eigenvalue of the $p$-Laplacian operator $Δ_{p}$ and $q$ is close to $p>1.$ We prove that any family of positive solutions of this problem converges in $C^{1}(\barΩ)$ to the function $θ_{p}e_{p}$ when $q\rightarrow p,$ where $e_{p}$ is the positive and $L^{\infty}$-normalized first eigenfunction of the $p$-Laplacian and $θ_{p}:=\exp(|e_{p}|_{L^{p}(Ω)}^{-p}\int_Ωe_{p}$ A consequence of this result is that the best constant of the immersion $W_{0}^{1,p}(Ω)\hookrightarrow L^{q}(Ω)$ is differentiable at $q=p.$ Previous results on the asymptotic behavior (as $q\rightarrow p$) of the positive solutions of the non-resonant Lane-Emden problem (i.e. with $λ_{p}$ replaced by a positive $λ\neqλ_{p}$) are also generalized to the space $C^{1}$ and to arbitrary families of these solutions. Moreover, if $u_{λ,q}$ denotes a solution of the non-resonant problem for an arbitrarily fixed $λ>0,$ we show how to obtain the first eigenpair of the $p$-Laplacian as the limit in $C^{1}(\barΩ),$ when $q\rightarrow p$, of a suitable scaling of the pair $(λ,u_{λ,q}).$ For computational purposes the advantage of this approach is that $λ$ does not need to be close to $λ_{p}.$ Finally, an explicit estimate involving $L^{\infty}$ and $L^{1}$ norms of $u_{λ,q}$ is also deduced using set level techniques.

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