Multiplicity of solutions for fractional equation involving the Bessel operator in RN

The aim of this paper is to study the existence of solution to an equation involving the Bessel operator in R-N (I - Delta)(alpha)u + lambda V(x)u = gamma f (x, u), where lambda, gamma are real positive parameters, V : R-N -> R+ is a continuous function, 0 < alpha < 1 with 2 alpha < N, f is a continuous function on R-N x R which does not satisfy the Ambrosetti-Rabinowitz condition. By using the Fountain theorem and Morse theory, we obtain the existence of solutions of the above equation. In our best knowledge, it is the first time that this problem is considered.