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In this paper, we prove the existence and multiplicity of solutions for alarge class of quasilinear problems on a nonreflexive Orlicz-Sobolev space.Here, we use the variational methods developed by Szulkin combined with someproperties of the weak$^*$ topology.

Multiple solutions for two classes of quasilinear problems defined on a nonreflexive Orlicz-Sobolev space