Profile of solutions for nonlocal equations with critical and supercritical nonlinearities

Abstract

We study the fractional laplacian problem (-Δ)^s u &=& u^p -εu^q \quad\text{in }\quad Ω, u &\in& H^s(Ω)\cap L^{q+1}(Ω),u &>&0 \quad\text{in }\quad Ω, u&=&0 \quad\text{in}\quad \mathbb{R}^N\setminusΩ, where $s\in(0,1)$, $q>p\geq \frac{N+2s}{N-2s}$ and $ε>0$ is a parameter. Here $Ω\subseteq\mathbb{R}^N$ is a bounded star-shaped domain with smooth boundary and $N> 2 s$. We establish existence of a variational positive solution $u_ε$ 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 $Ω$ is a convex symmetric domain of $\mathbb{R}^N$ with $N>4s$ and $p=\frac{N+2s}{N-2s}$.