A fractional p(x, .)-Laplacian problem involving a singular term

This paper deals with a class of singular problems involving the fractional p(x, .)-Laplace operator of the form {(-Delta)(p(x,.))(s) u(x) = lambda/u(gamma(x)) + u(q(x)+1) in Omega, u > 0, in Omega u = 0 on R-N\Omega, where Omega is a smooth bounded domain in R-N (N >= 3), 0<s<1, lambda is a positive parameter and gamma : R-N -> (0, 1) is a continuous function, p : R-2N -> (1, infinity) is a bounded, continuous and symmetric function, q : R-N -> (1, infinity) is a continuous function. Using the direct method of minimization combined with the theory of fractional Sobolev spaces with variable exponents, we prove that the problem has one positive solution for lambda > 0 small enough. To our best knowledge, this paper is one of the first attempts in the study of singular problems involving fractional p(x, .)-Laplace operators.