The present paper deals with the existence and multiplicity of solutions for a class of fractional $ p(x,.) $-Laplacian problems with the nonlocal Dirichlet boundary data, where the nonlinearity is superlinear but does not satisfy the usual Ambrosetti-Rabinowitz condition. To overcome the difficulty that the Palais-Smale sequences of the Euler-Lagrange functional may be unbounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami condition.
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