Stable and periodic solutions to nonlinear equations with fractional diffusion

Abstract

The aim of this thesis is to study stable solutions to nonlinear elliptic equations involving the fractional Lapacian. More precisely, we study the extremal solution for the problem $(\Delta )^s u = \lambda f(u)$ in $\Omega$, $u \equiv 0 $ in $\R^n \setminus \Omega$, where $\lambda > 0$ is a parameter and $s \in (0,1)$. The main result of this work, which is new, is the following: we prove that when $s=1/2$ and $\Omega = B_1$, then the extremal solution is bounded whenever $n \leq 8$.