Existence of positive solutions for fractional Schrödinger equation with general nonlinearities in exterior domains

In this paper, we investigate the existence of positive solutions of the following fractional Schrödinger equation with general nonlinearities: $$\begin{aligned} \left \{ \textstyle\begin{array}{l@{\quad }l@{\quad }l} (-\Delta )^{s} u+\lambda u=f(u), \quad &\text{in}\; \Omega , \\ u=0, \quad &\text{on}\; \mathbb{R}^{N} \backslash \Omega , \end{array}\displaystyle \right . \end{aligned}$$ where $N\ge 2$ , $s\in (0,1)$ , $\Omega \subset \mathbb{R}^{N}$ is an exterior domain, i.e., Ω is an unbounded domain in $\mathbb{R}^{N}$ with $\mathbb{R}^{N} \backslash \Omega $ nonempty and bounded, $\lambda >0$ is a parameter, and $f \in C^{1}(\mathbb{R},\mathbb{R})$ satisfies some technical conditions. We use variational and topological methods to prove that there is a positive solution $u\in H^{s}_{0}(\Omega )$ if $\mathbb{R}^{N} \backslash \Omega $ is small enough. Furthermore, the result can be extended to the fractional Kirchhoff equation with general nonlinearities.