A continuous spectrum for nonhomogeneous differential operators in Orlicz-Sobolev spaces

Abstract

Abstract. We study the nonlinear eigenvalue problem −div(a(|∇u|)∇u) = λ|u| q(x)−2 u in Ω, u = 0 on ∂Ω, where Ω is a bounded open set in R N with smooth boundary, q is a continuous function, and a is a nonhomogeneous potential. We establish sufficient conditions on a and q such that the above nonhomogeneous quasilinear problem has continuous families of eigenvalues. The proofs rely on elementary variational arguments. The abstract results of this paper are illustrated by the cases a(t) = t p−2 log(1 + t r ) and a(t) = t p−2 [log(1 + t)] −1 .

Eigenvalue problems for a class of bi-nonlocal equations in Orlicz-Sobolev spaces

“…satisfy the above conditions, where the positive constants t 1 ,t 2 can be chosen in a suitable manner such as (κ + 1)t 1 < αφ 0 ≤ αφ 0 < m 1 (κ+1) m 2 t 2 for the first case in (Q 2 ) and (κ + 1)t 2 < αφ 0 ≤ αφ 0 < m 1 (κ+1) m 2 t 1 for the second one. From conditions (Q 1 )-(Q 2 ), our main results in this paper complement and improve the previous ones [18,19,20], in which the authors considered the local problem with sublinear or superlinear terms. Definition 3.1.…”

“…The purpose of this work is to study bi-nonlocal problem (1.1) with the weighted function V and the general subcritical growth (see conditions (V 0 ) and (Q 1 )-(Q 2 )). For this reasons, the nonlinear term appeared in our problem covers the previous cases [18,19,20] in the sense that it may be sublinear or superlinear at infinity. We will establish the existence of a continuous family of eigenvalues for problem (1.1) in a neighborhood of the origin.…”

“…Motivated by the contribution mentioned above and recent results in [13,14,18,19,20], in this paper, we are interested in the eigenvalue problem for bi-nonlocal equation (1.1) in the Orlicz-Sobolev space. In [18,19,20], the authors considered problem (1.1) in the local case. That is, M ≡ 1 and κ = 0.…”

“…More precisely, using variational methods, Ge [18] and Mihȃilescu and Rȃdulescu [20] considered eigenvalue problem (1.3) in the sublinear case when q + < φ 0 and Ge and Geng [19] considered the problem in the superlinear case when φ 0 < q − (see the definitions of q + , q − , φ 0 and φ 0 in Section 2). The purpose of this work is to study bi-nonlocal problem (1.1) with the weighted function V and the general subcritical growth (see conditions (V 0 ) and (Q 1 )-(Q 2 )).…”

Concentration phenomena in nonlinear eigenvalue problems with variable exponents and sign-changing potential

“…The interest in analyzing this kind of problem is motivated by some recent advances in the study of eigenvalue problems involving variable exponent growth conditions. We refer especially to the results in [13,14,18,19,20,22,24,25,26]. Problem (1) can be placed in the context of the above results, since in the particular case when q 1 (x) = q 2 (x) = q(x) for any x ∈ and V ≡ 0 in , it has been studied in [23].…”