Existence and Multiplicity of positive solutions of certain nonlocal scalar field equations

We study existence and multiplicity of positive solutions of the following class of nonlocal scalar field equations: \begin{equation} \tag{$\mathcal{P}$} \left{\begin{aligned} (-\Delta)^s u + u &= a(x) |u|^{p-1}u+f(x);;$\text{in}$;$\mathbb{R}$^{N}, u &\in H^{s}{($\mathbb{R}$^{N})} \end{aligned} \right. \end{equation} where $s\in (0,1)$, $N>2s$, $1<p<2_s^{\ast }-1:=\frac{N+2s}{N-2s}$, $0<a\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $f\in H^{-s}($\mathbb{R}$^N)$isanonnegativefunctionali\mathrm{.}e\mathrm{.},${\langle}f,u{\rangle} \geq 0$whenever$u$isanonnegativefunctionin$H^s($\mathbb{R}$^N)$. We prove existence of a positive solution when $f\equiv 0$ under certain asymptotic behavior on the function $a\mathrm{.}$ Moreover, when $a(x)\ge 1$, $a(x)\to 1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{-s}({\mathbb{R}}^N)}$ is small enough (but $f\equiv ̸0$), then we show that the above equation admits at least two positive solutions. Finally, we establish existence of three positive solutions to the above equation, under the condition that $a(x)\le 1$ with $a(x)\to 1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_{H^{-s}({\mathbb{R}}^N)}$ is small enough (but $f\equiv ̸0$).