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 $p\rightarrow1^{ +}.$
Categories: math.AP
