Existence and multiplicity of solutions for a Schrodinger type equations involving the fractional p(x)-Laplacian

We are concerned with the following Schrodinger type equation with variable exponents (-Delta(p(x)))(s)u + V(x)vertical bar u vertical bar(p(x)-2)u = f(x, u) in R-N, where (-Delta(p(x)))(s) is the fractional p(x)-Laplace operator, s is an element of (0, 1), V : R-N -> (0, +infinity) is a continuous potential function, and f : R-N x R -> R satisfies the Carathe'odory condition. We study the nonlinearity of this equation which is superlinear but does not satisfy the Ambrosetti-Rabinowitz type condition. By using variational techniques and the fountain theorem, we obtain the existence and multiplicity of nontrivial solutions. Furthermore, we show that the problem has a sequence of solutions with high energies.