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 $s\in (0,1)$, $q>p\ge \frac{N+2s}{N-2s}$ and $\epsilon >0$ is a parameter. Here $\Omega \subseteq {\mathbb{R}}^N$ is a bounded star-shaped domain with smooth boundary and $N>2s$. 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 $p=\frac{N+2s}{N-2s}$.

30 pages

References in corpus (2)

• From the long jump random walk to the fractional Laplacian
• Bubbling solutions for nonlocal elliptic problems

Cited by in corpus (1)

• Nonlocal scalar field equations: qualitative properties, asymptotic profiles and local uniqueness of solutions