A multiplicity result for a nonlinear fractional Schrödinger equation in $mathbb{R}^{N}$ without the Ambrosetti–Rabinowitz condition

In this paper we study the existence and the multiplicity of positive solutions for the following class of fractional Schrödinger equations egin{equation*} arepsilon^{2s} (-Delta)^{s} u + V(x) u = f(u) mbox{ in } mathbb{R}^{N}, end{equation*} where $arepsilon>0$ is a parameter, $sin (0, 1)$, $N>2s$, $V: mathbb{R}^{N} ightarrow mathbb{R}$ is a continuous positive potential, and $f: mathbb{R} ightarrow mathbb{R}$ is a $C^{1}$ superlinear nonlinearity which does not satisfy the Ambrosetti–Rabinowitz condition. The main result is established by using minimax methods and Ljusternik–Schnirelmann theory of critical points.