A critical fractional equation with concave-convex power nonlinearities

In this work we study the following fractional critical problem $$(P_{\lambda })=\{\begin{array}{ll}(-\mathrm{\Delta }{)}^{s}u=\lambda u^q+u^{2_s^{\ast }-1},u>0& in\mathrm{\Omega }\\ u=0& in{\text{RR}}^n\setminus \mathrm{\Omega },\end{array}$$ where $\mathrm{\Omega }\subset {\mathbb{R}}^n$ is a regular bounded domain, $\lambda >0$, $0<s<1$ and $n>2s$. Here $(-\mathrm{\Delta }{)}^s$ denotes the fractional Laplace operator defined, up to a normalization factor, by $$-(-\mathrm{\Delta }{)}^{s}u(x)=\mathrm{P}\mathrm{.}\mathrm{V}\mathrm{.}{\int }_{{\text{RR}}^n}\frac{u(x+y)+u(x-y)-2u(x)}{\mathrm{\mid }y{\mathrm{\mid }}^{n+2s}}dy,x\in {\text{RR}}^n\mathrm{.}$$ 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 ($1<q<2_s^{\ast }-1$). These two cases will be treated separately.