On an eigenvalue problem involving variable exponent growth conditions Original Research Article

We study the problem View the MathML source−Δu−εdiv((1+|∇u|2)p(x)−22∇u)=λ(u+ε) in ΩΩ, u=0u=0 on ∂Ω∂Ω, where ΩΩ is a bounded domain in RNRN, View the MathML sourcep:Ω¯→(1,2) is a continuous function and λλ and εε are two positive constants. We prove that for any ε>0ε>0 each λ∈(0,λ1)λ∈(0,λ1) is an eigenvalue of the above problem, where λ1λ1 is the principal eigenvalue of the Laplace operator on ΩΩ. Moreover, for each eigenvalue λ∈(0,λ1)λ∈(0,λ1) it corresponds a unique eigenfunction. The proofs will be based on the Banach fixed point theorem combined with adequate variational techniques.