The dual eigenvalue problems of the conformable fractional Sturm–Liouville problems

In this paper, we are concerned with the eigenvalue gap and eigenvalue ratio of the Dirichlet conformable fractional Sturm–Liouville problems. We show that this kind of differential equation satisfies the Sturm–Liouville property by the Prüfer substitution. That is, the nth eigenfunction has $n-1$ n − 1 zero in $( 0,\pi ) $ ( 0 , π ) for $n\in \mathbb{N}$ n ∈ N . Then, using the homotopy argument, we find the minimum of the first eigenvalue gap under the class of single-well potential functions and the first eigenvalue ratio under the class of single-barrier density functions. The result of the eigenvalue gap is different from the classical Sturm–Liouville problem.

Liouville property of fractional Lane-Emden equation in general unbounded domain

Our purpose of this paper is to consider Liouville property for the fractional Lane-Emden equation $$\begin{array}{} \displaystyle (-{\it\Delta})^\alpha u = u^p\quad {\rm in}\quad {\it\Omega},\qquad u = 0\quad {\rm in}\quad \mathbb{R}^N\setminus {\it\Omega}, \end{array}$$ where α ∈ (0, 1), N ≥ 1, p > 0 and Ω ⊂ ℝN–1 × [0, +∞) is an unbounded domain satisfying that Ωt := {x′ ∈ ℝN–1 : (x′, t) ∈ Ω} with t ≥ 0 has increasing monotonicity, that is, Ωt ⊂ Ωt′ for t′ ≥ t. The shape of Ω∞ := limt→∞ Ωt in ℝN–1 plays an important role to obtain the nonexistence of positive solutions for the fractional Lane-Emden equation.

Analysis of weighted p-harmonic forms and applications

In this paper, we study weighted [Formula: see text]-harmonic forms on smooth metric measure space [Formula: see text] with a weighted Sobolev or a weighted Poincaré inequality. When [Formula: see text] is constant, we derive a splitting theorem for Kähler manifolds with maximal bottom spectrum for the [Formula: see text]-Laplacian. For general [Formula: see text] we also obtain various splitting and vanishing theorems when the weighted curvature operator of [Formula: see text] is bounded below. As applications, we conclude Liouville property for weighted [Formula: see text]-harmonic functions and [Formula: see text]-harmonic maps.