On some singular problems involving the fractional p(x,.) -Laplace operator

The purpose of the present paper is to study the existence of solutions for the following nonhomogeneous singular problem involving the fractional p(x,.)-Laplace operator {(-Delta)(p(x,.))(s) u + vertical bar u vertical bar(q(x)-2)u = g(x)u(epsilon-1-.gamma(x)) -/+ lambda f (x, u) in Omega, u = 0 on partial derivative Omega, where Omega is a smooth bounded domain in R-N (N >= 3), 0 < s, epsilon < 1, lambda is a positive parameter and gamma : (Omega) over bar -> (0, epsilon) is a continuous function, p (Omega) over bar x (Omega) over bar -> (1, infinity) is a bounded, continuous and symmetric function, q : (Omega) over bar (1, infinity) is a continuous function, g is an element of L p(s)*(x)-epsilon/p(s)* (x)+ gamma(x)-2 epsilon (Omega) and g(x) > 0 with p(s) * (x) = Np(x,x)/N-sp(x,x). Here, the nonlinearity f is in C1((Omega) over bar x R) and assumed to satisfy suitable assumptions. Using variational methods combined with monotonicity arguments, we obtain the existence of solutions to the problem in a fractional Sobolev space with variable exponent. To our best knowledge, this paper is the first attempt in the study of singular problems involving fractional p(x,.)-Laplace operators.