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