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We are concerned with the study of the existence and multiplicity ofsolutions for Dirichlet boundary value problems involving the (p(x),q(x))-equation and the nonlinearity is superlinear but does not satisfy theusual Ambrossetti-Rabinowitz condition in the framework of Sobolev spaces withvariable exponents in Complete manifolds. The main results are established bymeans of the mountain pass theorem and Fountain theorem with Cerami condition.Moreover, we are giving an example of a (p(x), q(x)) equation that verifies allour demonstrated results.

Nonlinear Elliptic Equations With Variable Exponents Satisfying Cerami Condition