Regularity of Radial Extremal Solutions for Some Non-Local Semilinear Equations

Abstract

We express our sincere gratitude to the referee for a careful reading of the manuscript.International audienceWe investigate stable solutions of elliptic equations of the type \begin{equation*} \left \{ \begin{aligned} (-\Delta)^s u&=\lambda f(u) \qquad {\mbox{ in $B_1 \subset \R^{n}$}} \\ u&= 0 \qquad{\mbox{ on $\partial B_1$,}}\end{aligned}\right . \end{equation*} where $n\ge2$, $s \in (0,1)$, $\lambda \geq 0$ and $f$ is any smooth positive superlinear function. The operator $(-\Delta)^s$ stands for the fractional Laplacian, a pseudo-differential operator of order $2s$. According to the value of $\lambda$, we study the existence and regularity of weak solutions $u$

Lotka–Volterra models with fractional diffusion

“…As proved in [9] [Section 2.1], for each u ∈ V α 0 (Ω), there exists a unique v ∈ H α 0 (C), called its α−harmonic extension such that…”

“…Let us point some remarks. The sub-supersolution method has been used previously in non-linear fractional diffusion problem, see for instance [3] and [9]. In both papers, the method is consequence of a maximum principle and a classical iterative argument.…”

“…Moreover, by Trace Theorem (see Proposition 2.1 in [9]) and embeddings for fractional Sobolev spaces (see Theorem 6.7 in [12]) it follows that…”

“…Along the paper, the following maximum principle will be very useful, see Lemma 2.5 in [9] for a related result.…”

Periodic solutions for the non-local operator $(-\Delta+m^{2})^{s}-m^{2s}$ with $m\geq 0$

“…We recall that this argument is an adaptation of the idea originally introduced in [11] to study the fractional Laplacian in R N (see also [7,8]) and subsequently generalized for the case of the fractional Laplacian on bounded domain (see [10,13]). …”