Comparison estimates on the first eigenvalue of a quasilinear elliptic system

Abstract

Abstract We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and the inequality of Faber–Krahn for the first eigenvalue of a ( p , q ) {(p,q)} -Laplacian are recovered. Lastly, we reprove a Cheeger-type estimate for the p-Laplacian, 1 < p < ∞ {1<p<\infty} , from where a lower bound estimate in terms of Cheeger’s constant for the first eigenvalue of a ( p , q ) {(p,q)} -Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger’s constant as p , q → 1 , 1 {p,q\to 1,1} .