Multiplicity of solutions for fractional q(center dot)-laplacian equations

In this paper, we deal with the following elliptic-type problem {(-Delta)(q(center dot))(s(center dot)) u + lambda Vu = alpha vertical bar u vertical bar(p(center dot)-2) u + beta vertical bar u vertical bar(k(center dot)-2) u in Omega, u = 0 in R-n\ Omega, where q(center dot) : (Omega) over bar x (Omega) over bar -> R is a measurable function and s(center dot) : R-n x R-n -> (0, 1) is a continuous function, n > q(x, y)s(x, y) for all ( x, y) is an element of Omega x Omega, (-Delta)(q(center dot))(s(center dot)) is the variable-order fractional Laplace operator, and V is a positive continuous potential. Using the mountain pass category theorem and Ekeland's variational principle, we obtain the existence of at least two different solutions for all lambda > 0. Besides, we prove that these solutions converge to two of the infinitely many solutions of a limit problem as lambda -> +infinity.