Remarks on eigenvalue problems involving the -Laplacian

This paper deals with the eigenvalue problem involving the p ( x ) -Laplacian of the form { − div ( | ∇ u | p ( x ) − 2 ∇ u ) = λ | u | q ( x ) − 2 u in Ω , u = 0 on ∂ Ω , where Ω is a bounded domain in R N , p ∈ C 0 ( Ω ¯ ) , inf x ∈ Ω p ( x ) > 1 , q ∈ L ∞ ( Ω ) , 1 ⩽ q ( x ) ⩽ q ( x ) + ε < p * ( x ) for x ∈ Ω , ε is a positive constant, p * ( x ) is the Sobolev critical exponent. It is shown that for every t > 0 , the problem has at least one sequence of solutions { ( u n , t , λ n , t ) } such that ∫ Ω 1 p ( x ) | ∇ u n , t | p ( x ) = t and λ n , t → ∞ as n → ∞ . The principal eigenvalues for the problem in several important cases are discussed especially. The similarities and the differences in the eigenvalue problem between the variable exponent case and the constant exponent case are exposed. Some known results on the eigenvalue problem are extended.