We study the first eigenvalue of the $p$-Laplacian under the Dirichlet boundary condition. For a convex domain, we give an a priori estimate for the first eigenvalue in terms of the radius $d$ of the maximum ball contained in the domain. As a consequence, we prove that the first eigenvalue diverges to infinity as $p\to\infty$ if the domain is convex and $d\leq 1$. Moreover, we show that in the annulus domain $a < |x| < b$, the first eigenvalue diverges to infinity if $b-a\leq 2$ and converges to zero if $b-a>2$.
"A priori estimate for the first eigenvalue of the $p$-Laplacian." Differential Integral Equations 28 (9/10) 1011 - 1028, September/October 2015. https://doi.org/10.57262/die/1435064548