On multiplicity of positive solutions for nonlocal equations with critical nonlinearity

This paper deals with existence and multiplicity of positive solutions to the following class of nonlocal equations with critical nonlinearity: \begin{equation} \tag{$\mathcal{E}$} (-\Delta)^s u = a(x) |u|^{2^*_s-2}u+f(x);;$\text{in}$;$\mathbb{R}$^{N}, \quad u \in \dot{H}^s($\mathbb{R}$^{N}), \end{equation} where $s\in (0,1)$, $N>2s$, $2_s^{\ast }:=\frac{2N}{N-2s}$, $0<a\in L^{\mathrm{\infty }}({\mathbb{R}}^N)$ and $f$ is a nonnegative nontrivial functional in the dual space of ${\dot{H}}^s$. We prove existence of a positive solution whose energy is negative. Further, under the additional assumption that $a$ is a continuous function, $a(x)\ge 1$ in ${\mathbb{R}}^N$, $a(x)\to 1$ as $\mathrm{\mid }x\mathrm{\mid }\to \mathrm{\infty }$ and $\mathrm{\parallel }f{\mathrm{\parallel }}_{{\dot{H}}^s({\mathbb{R}}^N{)}^{\mathrm{\prime }}}$ is small enough (but $f\equiv ̸0$), we establish existence of at least two positive solutions to ($\mathcal{E}$).