New features of the first eigenvalue on negatively curved spaces

The paper is devoted to the study of fine properties of the first eigenvalue on negatively curved spaces. First, depending on the parity of the space dimension, we provide asymptotically sharp harmonic-type expansions of the first eigenvalue for large geodesic balls in the model n-dimensional hyperbolic space, complementing the results of Borisov and Freitas (2017), Hurtado, Markvorsen and Palmer (2016) and Savo (2008); in odd dimensions, such eigenvalues appear as roots of an inductively constructed transcendental equation. We then give a synthetic proof of Cheng’s sharp eigenvalue comparison theorem in metric measure spaces satisfying a Bishop–Gromov-type volume monotonicity hypothesis. As a byproduct, we provide an example of simply connected, non-compact Finsler manifold with constant negative flag curvature whose first eigenvalue is zero; this result is in a sharp contrast with its celebrated Riemannian counterpart due to McKean (1970). Our proofs are based on specific properties of the Gaussian hypergeometric function combined with intrinsic aspects of the negatively curved smooth/non-smooth spaces.

Reversed S-Shaped Bifurcation Curve for a Neumann Problem

We study the bifurcation and the exact multiplicity of solutions for a class of Neumann boundary value problem with indefinite weight. We prove that all the solutions obtained form a smooth reversed S-shaped curve by topological degree theory, Crandall-Rabinowitz bifurcation theorem, and the uniform antimaximum principle in terms of eigenvalues. Moreover, we obtain that the equation has exactly either one, two, or three solutions depending on the real parameter. The stability is obtained by the eigenvalue comparison principle.

Some eigenvalue comparison theorems of Finsler p-Laplacian

We obtain Cheng type inequality, Cheeger type inequality, Faber–Krahn type inequality and McKean type inequality of [Formula: see text]-Laplacian on a Finsler manifold. These generalize the corresponding theorems in Riemannian geometry and sharpen some results in recent literatures. Moreover, for a complete noncompact Finsler manifold with negative constant flag curvature and vanishing [Formula: see text] curvature, the first eigenvalue is calculated.

Eigenvalue comparison for fractional boundary value problems with the Caputo derivative

We apply the theory for u 0-positive operators to obtain eigenvalue comparison results for a fractional boundary value problem with the Caputo derivative.