Nonlinear eigenvalue problems for a class of quasilinear operator on complete Riemannian manifolds
Bin Shen, Yuhan Zhu
TL;DR
The paper studies nonlinear eigenvalue problems for a broad class of quasilinear operators on complete Riemannian manifolds with a Ricci curvature lower bound. It formulates $Qu = \operatorname{div}(\mathcal{F}(u^2,|\nabla u|^2)\nabla u)$ and analyzes $Qu + \lambda f(u) = 0$, deriving Cheng–Yau type gradient estimates via Nash–Moser iteration under separability and finite degree conditions on $\mathcal{F}$. A Liouville-type theorem shows that on manifolds with nonnegative Ricci curvature, any bounded positive solution with $\lambda\neq 0$ is ruled out, implying unbounded eigenfunctions for nonzero eigenvalues in the noncompact setting. The framework is applied to nonlinear eigenvalue problems such as $\Delta_p\bigl(\sum a_i u^{q_i}\bigr) + \lambda u^r = 0$, establishing explicit gradient bounds and unboundedness for nonzero eigenvalues, and unifying several results for $p$-Laplacian and porous-medium type equations, with extensions to anisotropic or Finsler-like contexts.
Abstract
In this manuscript, we study the nonlinear eigenvalue problem on complete Riemannian manifolds with Ricci curvature bounded from below, to find the unknowns $λ$ and $u$, such that $$ Qu + λf(u) = 0 $$ where $λ$ 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.} $Δ_p u^q = λu^r$), and generally, $Δ_{p}\left(\sum_{i=1}^{m}a_iu^{q_i}\right)+λu^r = 0$.