Multiplicity and concentration of positive solutions for fractional nonlinear Schrödinger equation

In this paper we consider the multiplicity and concentration behavior of positive solutions for the following fractional nonlinear Schrodinger equation \begin{document}$\left\{ \begin{align} &{{\varepsilon }^{2s}}{{\left( -\Delta \right)}^{s}}u+V\left( x \right)u = f\left( u \right)\ \ \ \ \ \ x\in {{\mathbb{R}}^{N}}, \\ u'>where \begin{document}$\varepsilon$\end{document} is a positive parameter, \begin{document}$(-Δ)^{s}$\end{document} is the fractional Laplacian, \begin{document}$s ∈ (0,1)$\end{document} and \begin{document}$N> 2s$\end{document} . Suppose that the potential \begin{document}$V(x) ∈\mathcal{C}(\mathbb{R}^{N})$\end{document} satisfies \begin{document}$\text{inf}_{\mathbb{R}^{N}} V(x)>0$\end{document} , and there exist \begin{document}$k$\end{document} points \begin{document}$x^{j} ∈ \mathbb{R}^{N}$\end{document} such that for each \begin{document}$j = 1,···,k$\end{document} , \begin{document}$V(x^{j})$\end{document} are strict global minimum. When \begin{document}$f$\end{document} is subcritical, we prove that the problem has at least \begin{document}$k$\end{document} positive solutions for \begin{document}$\varepsilon>0$\end{document} small. Moreover, we establish the concentration property of the solutions as \begin{document}$\varepsilon$\end{document} tends to zero.