The Cheeger constant as limit of Sobolev-type constants

Let $\Omega$ be a bounded, smooth domain of $\mathbb{R}^{N},$ $N\geq2.$ For $1<p<N$ and $0<q(p)<p^{\ast}:=\frac{Np}{N-p}$ let $$\lambda_{p,q(p)}:=\inf\left\{ \int_{\Omega}\left\vert \nabla u\right\vert ^{p}\mathrm{d}x:u\in W_{0}^{1,p}(\Omega)\text{ \ and \ }\int_{\Omega }\left\vert u\right\vert ^{q(p)}\mathrm{d}x=1\right\} .$$ We prove that if $\lim_{p\rightarrow1^{+}}q(p)=1,$ 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 $-\operatorname{div}(\left\vert \nabla w\right\vert ^{p-2}\nabla w)=\left\vert w\right\vert ^{q-2}w,$ as $p\rightarrow1^{+}.$