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WE discuss the existence and uniqueness of solutions of the quasilinear elliptic equations in Sobolev spaces with variable exponent. These solutions are obtained by the p(.)-obstacle problem. WE study regularity properties of weak solutions and we prove the Harnack's inequality and continuity of solution. We show by proving a comparison principle that Keller-Osserman property is valid and we discuss the existence of Evans functions for solutions to the quasilinear elliptic equations in Sobolev spaces with variable exponent.