Fractional Musielak spaces: a class of non-local elliptic system involving generalized nonlinearity

In this research, we analyze the existence and multiplicity of nonnegative solutions for a class of non-local elliptic system with Dirichlet boundary conditions. The problem’s nonlinearity does not typically satisfy the Ambrosetti–Rabinowitz condition and includes various non-linear terms characterized by concave-convex variables and variable exponent functions. We use minimization techniques and the properties of a monotone operator of type \((S)^+\) ${(S)}^{+}$ to establish the existence of positive solutions for the non-local elliptic system within fractional Musielak spaces. Also, we show an embedding result from fractional Musielak spaces to the weighted Musielak spaces. Our main results generalize some recent findings in the literature to non-smooth cases.