On the role of the range of dispersal in a nonlocal Fisher-KPP equation: an asymptotic analysis

In this paper, we study the asymptotic behavior as $varepsilonto0^+$ of solutions $u_varepsilon$ to the nonlocal stationary Fisher-KPP type equation$$ frac{1}{varepsilon^m}int_{mathbb{R}^N}J_varepsilon(x-y)(u_varepsilon(y)-u_varepsilon(x))mathrm{d}y+u_varepsilon(x)(a(x)-u_varepsilon(x))=0text{ in }mathbb{R}^N, $$where $varepsilon>0$ and $0leq m<2$. Under rather mild assumptions and using very little technology, we prove that there exists one and only one positive solution $u_varepsilon$ and that $u_varepsilonto a^+$ as $varepsilonto0^+$ where $a^+=max{0,a}$. This generalizes the previously known results and answers an open question raised by Berestycki, Coville and Vo. Our method of proof is also of independent interest as it shows how to reduce this nonlocal problem to a local one. The sharpness of our assumptions is also briefly discussed.