Guezainia, HoudaGouasmia, Abdelhamid2024-10-162024-10-162024http://dspace.univ-oeb.dz:4000/handle/123456789/19999The main objective of this thesis is to study the existence , non-existence , and regularity of the weak solution of the fractional non-homogenous problem involving fractional and non-homogenous operateurs by using the mountain pass theorem . We divided this work into two chapters: In the first chapter, we begin by providing an overview of the functional analysis used in this work and define the fractional sobolev espaceWs,p(Ù) ,moreover, definemethod of mountain pass theorem . the second chapter study the existence,non-existence, and regularity of the weak solutions.enExistence; Regularity results; Fractional pOther