On an eigenvalue problem for an anisotropic elliptic equation

We study the existence of infinitely many solutions for anisotropic variable exponent problem of the type { −∑i=1N∂xiai(x,∂xiu)+∑i=1Nai(x,u)=λ| u |q(x)−2uinΩ,∂u∂v=0on∂Ω. Where Ω ⊂ RN (N ≥ 3) is a bounded domain with smooth boundary ∂Ω, λ > 0, pi, q are continuous functions on Ω¯ such that pi(x) ≥ 2, ∀x ∈ Ω and i ∈ {1, 2, ….., N}. The main result of this paper establishes the existence of two positive constants λ0 and λ1 with λ0 ≤ λ1 such that every λ ∈ (λ1, ∞) is an eigenvalue, while no λ ∈ (0, λ0) can be an eigenvalue of the above problem.