Eigenvalue problems for anisotropic equations involving a potential on Orlicz-Sobolev type spaces

Abstract

Abstract. In this paper we consider an eigenvalue problem that involves a nonhomogeneous elliptic operator, variable growth conditions and a potential V on a bounded domain in R N (N ≥ 3) with a smooth boundary. We establish three main results with various assumptions. The first one asserts that any λ > 0 is an eigenvalue of our problem. The second theorem states the existence of a constant λ * > 0 such that any λ ∈ (0, λ * ] is an eigenvalue, while the third theorem claims the existence of a constant λ * > 0 such that every λ ∈ [λ * , ∞) is an eigenvalue of the problem.

Multiplicity of solutions for a nonhomogeneous problem involving a potential in Orlicz-Sobolev spaces

“…Many researchers have studied the existence of solutions for eigenvalue problems involving nonhomogeneous operators in the divergence form in Orlicz-Sobolev spaces by means of variational methods, see for examples [3,19,20,21]. In [8] where Ω is a bounded domain in R N , g ∈ C(Ω × R, R) and the function ϕ(s) = sa(|s|) is an increasing homeomorphism from R onto R. Under appropriate conditions on ϕ, g and the Orlicz-Sobolev conjugate Φ * of Φ(s) = s 0 ϕ(t)dt, they obtained the existence of non-trivial solutions of mountain pass type.…”

Multiple solutions for eigenvalue problems involving an indefinite potential and with (p 1(x), p 2(x)) balanced growth

“…More contributions to the study of the eigenvalue nonlinear elliptic equations in an anisotropic framework were also added by K. Ben Ali, A. Ghanmi, K. Kefi [1], M. Cavalcanti, V. Domingos Cavalcanti, I. Lasiecka, C. Webler [4], M. Cencelj, D. Repovš, Z. Virk [5], Y. Fu, Y. Shan [10], K. Kefi, V. Rȃdulescu [11], D. Repovš [20] and I. Stȃncuţ, I. Stîrcu [21].…”

Improved bounds for solutions of ϕ-Laplacians

“…Even though we do not address these properties here, it is of particular interest the special case of the p-Laplacian operator for which φ(s) = |s| p−2 s and p > 1. The particular case of variable exponents p(x), where p : Ω → (1, +∞) is a bounded function, is treated in [14][15][16]. Special types of nonlinearities in connection with the p-Laplace operator have been considered recently in the article [9].…”