Some results on principal eigenvalues for periodic parabolic problems with weight

Abstract

Let 0 C K w be a bounded domain. We study existence and uniqueness of principal eigenvalues for the Dirichlet periodic parabolic problem with weight Lu = Amu in fi x K when the independent coefficient of the differential operator L is not necessarily positive.

Positive periodic solutions of the weighted $p$-Laplacian with nonlinear sources

“…If m < λ 1 , there exists λ with m < λ < λ 1 and a domain Ω with Ω ⊂⊂ Ω such that λ is the first eigenvalue of the weighted p-Laplacian eigenvalue problem (13) on Ω, and correspondingly, ψ is the first eigenfunction with ψ L ∞ = 1. Further, there exists a constant δ > 0 such that ψ ≥ δ for x ∈ Ω.…”

“…where L is a linear uniformly elliptic operator, m(x, t) is a given weight function, see [5,9,10,13] and the original work of Beltramo and Hess [2,3]. We are quite interested in the case of weighted flux, in which (1) may have degeneracy or singularity due to the weight |x| α and the p-Laplacian when p = 2.…”