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Applying the theory of the vector valued Musielak-Orlicz spaces, we give a sufficient condition in order that a quasilinear elliptic operator with the variational structure is of type (S)+, which is stated by the term of the uniform convexity of the corresponding Musielak-Orlicz function. As an application of this result, we obtain the (S)+ property of the operators of the p(x)-Laplacian type and some results on the existence and multiplicity for the equations of the p(x)-Laplacian type, where the operators of the p(x)-Laplacian type are a generalization of the p(x)-Laplacian operator.
Keywords:Musielak-Orlicz space; uniform convexity; mapping of type (S)+; quasilinear elliptic operator; equation of p(x)-Laplacian type