A critical fractional equation with concave-convex power nonlinearities

In this work we study the following fractional critical problem $$(P_{\lambda})=\left\{\begin{array}{ll} (-\Delta)^s u=\lambda u^{q} + u^{2^*_{s}-1}, \quad u{>}0 & \mbox{in} \Omega\\ u=0 & \mbox{in} \RR^n\setminus \Omega\,, \end{array}\right.$$ where $\Omega\subset \mathbb{R}^n$ is a regular bounded domain, $\lambda>0$, $0<s<1$ and $n>2s$. Here $(-\Delta)^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$-(-\Delta)^s u(x)={\rm P. V.} \int_{\RR^n}\frac{u(x+y)+u(x-y)-2u(x)}{|y|^{n+2s}}\,dy, \quad x\in \RR^n.$$ Our main results show the existence and multiplicity of solutions to problem $(P_\lambda)$ for different values of $\lambda$. The dependency on this parameter changes according to whether we consider the concave power case ($0<q<1$) or the convex power case ($1<q<2^*_s-1$). These two cases will be treated separately.