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