A Global multiplicity result for a very singular critical nonlocal equation

In this article, we show the global multiplicity result for the following nonlocal singular problem \begin{equation*} (P_\la):;\quad (-\De)^s u = u^{-q} + \la u^{{2^_s}-1}, \quad u>0 ; $\text{in}$; \Om,\quad u = 0 ; \mbox{in}; \mb R^n \setminus\Om, \end{equation} where $\text{Om}$ is a bounded domain in $\text{mb}{R}^n$ with smooth boundary $\mathrm{\partial }\text{Om}$, $n>2s,s\in (0,1),\text{la}>0,q>0$ satisfies $q(2s-1)<(2s+1)$ and $2_s^{\ast }=\frac{2n}{n-2s}$. Employing the variational method, we show the existence of at least two distinct weak positive solutions for $(P_{\text{la}})$ in $X_0$ when $\text{la}\in (0,\text{La})$ and no solution when $\text{la}>\text{La}$, where $\text{La}>0$ is appropriately chosen. We also prove a result of independent interest that any weak solution to $(P_{\lambda })$ is in $C^{\alpha }({\mathbb{R}}^n)$ with $\alpha =\alpha (s,q)\in (0,1)$. The asymptotic behaviour of weak solutions reveals that this result is sharp.