The Cheeger constant as limit of Sobolev-type constants

Let \(\Omega \) be a bounded, smooth domain of \({\mathbb {R}}^{N},\) \(N\ge 2.\) For \(1<p<N\) and \(0<q(p)<p^{*}:=\frac{Np}{N-p}\), let

$$\begin{aligned} \lambda _{p,q(p)}:=\inf \left\{ \int _{\Omega }\left| \nabla u\right| ^{p}\textrm{d}x:u\in W_{0}^{1,p}(\Omega ) \ \text {and} \ \int _{\Omega }\left| u\right| ^{q(p)}\textrm{d}x=1\right\} . \end{aligned}$$

We prove that if \(\lim _{p\rightarrow 1^{+}}q(p)=1,\) then \(\lim _{p\rightarrow 1^{+}}\lambda _{p,q(p)}=h(\Omega )\), where \(h(\Omega )\) denotes the Cheeger constant of \(\Omega .\) Moreover, we study the behavior of the positive solutions \(w_{p,q(p)}\) to the Lane–Emden equation \(-{\text {div}} (\left| \nabla w\right| ^{p-2}\nabla w)=\left| w\right| ^{q-2}w,\) as \(p\rightarrow 1^{+}.\)