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In this paper, we prove, in the spirit of (3, 12, 20, 22, 23), the existence of infinitely many small solutions to the following quasilinear elliptic equation −∆p(x)u + |u| p(x)−2 u = |u| q(x)−2 u +f (x;u) in a smooth bounded domain Ω of RN . We also assume that {q(x) = p∗(x)} ∅, where p ∗ (x) = Np(x)=(N − p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountain- pass lemma due to Kajikiya (22), and property of these solutions are also obtained. where Ω ⊂ R N (N ≥ 3) is a bounded domain with smooth boundary and p(x), q(x) are two continuous functions on Ω, 1 0. λ is a posi- tive parameter, ∆p(x)u := div(|∇u| p(x)−2 ∇u) is the p(x)-Laplacia operator. On the exponent q(x) we assume that is the critical exponent in the sense that

INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH