Fractional Musielak spaces: a class of non-local problem involving concave-convex nonlinearity

In this research, we analyze the existence and multiplicity of nonnegative solutions for a class of non-local elliptic problems with Dirichlet boundary conditions. The nonlinearity of the problem, in general, does not satisfy the Ambrosetti-Rabinowitz condition and is characterized by a concave-convex variable exponent function, exhibiting critical behavior at infinity. Using minimization arguments and Lebourg's mean value theorem, and applying Ekeland's variational principle together with the inverse function theorem, we obtain a ground state solution to the non-local elliptic problem in appropriate fractional Musielak spaces. Our main results generalize some recent findings in the literature to non-smooth cases.