Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

We study the fractional laplacian problem (-\Delta)^s u &=& u^p -\epsilon u^q \quad$\text{in }$\quad \Omega, u &\in& H^s(\Omega)\cap L^{q+1}(\Omega),u &>&0 \quad$\text{in }$\quad \Omega, u&=&0 \quad$\text{in}$\quad $\mathbb{R}$^N\setminus\Omega, where $s\in (0,1)$, $q>p\ge \frac{N+2s}{N-2s}$ and $ϵ>0$ is a parameter. Here $\mathrm{\Omega }\subseteq {\mathbb{R}}^N$ is a bounded star-shaped domain with smooth boundary and $N>2s$. We establish existence of a variational positive solution $u_{ϵ}$ and characterize the asymptotic behaviour of $u_{ϵ}$ as $ϵ\to 0$. When $p=\frac{N+2s}{N-2s}$, we describe how the solution $u_{ϵ}$ concentrates and blows up at a interior point of the domain. Furthermore, we prove the local uniqueness of solution of the above problem when $\mathrm{\Omega }$ is a convex symmetric domain of ${\mathbb{R}}^N$ with $N>4s$ and $p=\frac{N+2s}{N-2s}$.