Nonlinear Eigenvalue Problem Without Ambrosetti and Rabinowitz Condition: An Orlicz Space Setting

Abstract

We study the Dirichlet boundary value problem for the p(x)-Laplacian of the formWe introduce a new variational technic that allows us to investigate problem (P) without need of the Ambrosetti and Rabinowitz condition on the nonlinearity f .

Mckean-type Estimates for the First Eigenvalue of the p-Laplacian and (p,q)-Laplacian ‎Operators‎ on Finsler Manifolds

“…Let (M, F, dµ) be a Finsler n-manifold with volume form dµ, Ω ⊂ M a domain with compact closure and nonempty boundary ∂Ω. The first Dirichlet eigenvalue λ 1,p (Ω) of Ω with respect to dµ is defined by: λ 1,p (Ω) = inf Ω (F * (df )) 2 dµ Ω f p dµ…”

Non‐autonomous double phase eigenvalue problems with indefinite weight and lack of compactness

“…When 𝑎 ≡ 1, then (1.1) becomes the so-called 𝑝 & 𝑞 Laplacian problem, which was investigated by Benouhiba and Belyacine [4,5]. A feature of this paper is that we do not assume that the function 𝑎(⋅) is bounded away from zero, that is, we do not require that essinf 𝑥∈ℝ 𝑁 𝑎(𝑥) > 0.…”