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.