DISCRETE AND CONTINUOUS doi:10.3934/dcdss.2020425 DYNAMICAL SYSTEMS SERIES S Volume 14, Number 10, October 2021 pp. 3479–3495 ON A NONLOCAL PROBLEM INVOLVING THE FRACTIONAL p(x, .)-LAPLACIAN SATISFYING CERAMI CONDITION Elhoussine Azroul, Abdelmoujib Benkirane, and Mohammed Shimi ∗ Sidi Mohamed Ben Abdellah University, Faculty of Sciences Dhar El Mahraz Laboratory of Mathematical Analysis and Applications, Fez, Morocco B.P. 1796 Fez-Atlas, 30003 MOROCCO. Abstract. The present paper deals with the existence and multiplicity of so- lutions 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 un- bounded, we consider the Cerami sequences. The main results are established by means of mountain pass theorem and Fountain theorem with Cerami con- dition. 1. Introduction. For several years great effort has been devoted to the study of linear and nonlinear equations involving fractional derivatives of functions of one or several variables, and in particular fractional Laplacian (−Δ) s ,s ∈ (0, 1), is a more recent phenomenon. This nonlocal operator can be defined using Fourier analysis, functional calculus, singular integrals, or L´ evy processes, etc. Its inverse is closely related to the famous potentials introduced by Marcel Riesz in the late 1930s. For a deeper comprehension, we refer the reader to [18] in which the authors estab- lished the equivalence between ten different definitions of fractional Laplacian. So this type of operator arises in a quite natural way in many different contexts, such as phase transitions, stratified materials, anomalous diffusion, crystal dislocation, conservation laws, ultrarelativistic limits of quantum mechanics, quasi-geostrophic flows, multiple scattering, minimal surfaces, materials science, water waves, geo- physical fluid dynamics, and mathematical finance (see for instance [10, 21, 22]). In the last few years, much research has tended to focus on the nonlocal nonlinear case. More precisely, the problems involving the fractional p-Laplacian operator (−Δ) s p have been investigated by many papers, see for example [5, 11, 14] and the references therein. More recently, there are many researchers extended the constant case to include the fractional variable exponent case [2, 3, 4, 6, 9, 16] where the au- thors introduced some definitions and basic properties of new fractional Sobolev spaces with variable exponents and obtained some existence results for related non- local fractional problems with variable exponents by means of different variational methods. For our first goal, we aim to keep on the study of such nonlocal problems 2020 Mathematics Subject Classification. Primary: 35R11, 47G30; Secondary: 35S15, 35A15. Key words and phrases. Fractional p(x, .)-Laplacian operator, Nonlocal problem, Cerami con- dition, Mountain pass theorem, Fountain theorem. ∗ Corresponding author: Mohammed Shimi. 3479