Perturbated p-Laplacian on Riemannian manifolds

This paper deals with the nonlinear eigenvalue problem, for perturbated p-Laplacian operator, on a compact Riemannian manifold and determines a gradient estimate of eigenfunction associated with (first) eigenvalue of perturbated p-Laplacian operator. Using this estimate, we find a lower bound for this eigenvalue. In this paper we investigate the first (principal) nonlinear eigenvalue of the perturbated p-Laplacian on compact Riemannian manifolds and provide a lower bound through use of the diameter and the inscribed radius in terms of geometric quantities of manifold, and properties of disturbed term, when the Ricci curvature is non-negative.
There are many results on the lower bound estimates for principal eigenvalues and eigenfunctions for domains in Euclidean space examined in multiple research papers. For a compact manifold with no boundary, for Laplace operator, i.e. p = 2, a sharp lower bound estimate on a compact Riemannian manifold with nonnegative Ricci curvature is known. Through a process of computation which involves Lagrange multipliers, it can be demonstrated.