Some geometric estimates of the first eigenvalue of quasilinear and (p,q)-Laplace operators

In this paper, we use a particular smooth function $f:\Omega \rightarrow \mathbb{R}$ on a bounded domain $\Omega$ of a Riemannian manifold $M$ to estimate the lower bound of the first eigenvalue for quasilinear operator $Lf=-\Delta_{p}f+V\vert f\vert^{p-2}f$. In this way, we also present a lower bound for the first eigenvalue of the $(p,q)$-Laplacian on compact manifolds.