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 ( - Δ ) p ( x , · ) s u ( x ) = λ u γ ( x ) + u q ( x ) - 1 in Ω , u > 0 , in Ω u = 0 on R N \ Ω , where Ω is a smooth bounded domain in R N ( N ≥ 3 ), 0 < s < 1 , λ is a positive parameter and γ : R N ⟶ ( 0 , 1 ) is a continuous function, p : R 2 N ⟶ ( 1 , ∞ ) is a bounded, continuous and symmetric function, q : R N ⟶ ( 1 , ∞ ) 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 λ > 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.