Nonlinear eigenvalue problems for a class of quasilinear operator on complete Riemannian manifolds

TL;DR

This paper investigates nonlinear eigenvalue problems involving quasilinear operators on complete Riemannian manifolds, extending gradient estimates and analyzing eigenfunction behavior for various nonlinear equations.

Contribution

It generalizes the Cheng--Yau gradient estimate to quasilinear operators and explores eigenvalue and eigenfunction properties for a broad class of nonlinear equations.

Findings

Non-zero eigenvalues can lead to unbounded eigenfunctions.

The results encompass equations like the p-porous medium and generalized p-Laplacian equations.

The study extends gradient estimates to a wider class of quasilinear operators.

Abstract

In this manuscript, we study the nonlinear eigenvalue problem on complete Riemannian manifolds with Ricci curvature bounded from below, to find the unknowns $\lambda$ and $u$, such that $$ Qu + \lambda f(u) = 0 $$ where $\lambda$ is an eigenvalue of $u$, with respect to the quasilinear operator $Qu = \operatorname{div} (\mathcal{F}(u^2, |\nabla u|^2)\nabla u)$ and nonlinar function $f(\cdot)\neq 0$. We generalize the Cheng--Yau gradient estimate in \cite{shen2025feasibilitynashmoseriterationchengyautype} and demonstrate that under certain conditions, a non-zero eigenvalue gives rise to unbounded eigenfunction $u$. Our new result also covers more quasilinear equations like $p$-porous medium equation (\textit{i.e.} $\Delta_p u^q = \lambda u^r$), and generally, $\Delta_{p}\left(\sum_{i=1}^{m}a_iu^{q_i}\right)+\lambda u^r = 0$.