Let $D \subset \mathbb R^d$ ($d \ge 2$) be a bounded Lipschitz domain, $b \in Bb(D)$
be non-negative and
non-trivial, $a > 0$ and $p > 1$. We study the non-linear equation
$$
−\Delta_\alpha u = au − bu_p \hbox{ in }D, \quad
u = 0 \hbox{ in }D^c, \quad u > 0\hbox{ on }D \tag1
$$
where $\alpha\in (0, 1)$ and $\Delta_\alpha$ is the fractional Laplacian. The operator $\Delta_\alpha$
represents non-local character of dispersal strategy of animals. This type of
dispersal strategy based on Lévy flights has been observed in nature (see, e.g., [3]).
We address the following problems:
(i) The existence and uniqueness problem for (1).
(ii) Let $u_{p,a}$ be a solution to (1). What is the limit of $(u_{p,a})$ when $p \to\infty$?
(iii) What kind of equation solves $u_\infty,a$ (whenever it exists)?
(iv) Do we have uniqueness property for the limit equation?
(v) What do we know about $\sup x \in D u_{\infty,a(x)}$?
These problems have been investigated for the first time by Boccardo
and Murat [1]
(problems (i)–(iii)), with Leray-Lions type operator and with $a = 0$, $b = 1$. Then
the problems have been studied, with general $a, b$ and with the Laplace
operator, by
Dancer, Y. Du, L. Ma [4] (problems (i)–(iii)), by Dancer and Y. Du [2]
(problem (iv)) and Rodrigues and Tavares [6] (problems (i)–(iii)).
We provide a complete answer to problems (i)–(iv) and some estimates
on the supremum appearing in (v). We also discuss behavior of $u_{\infty,a}$ with respect to
parameter $a$ and asymptotic stability of solutions.
Problems (i)-(iv) have never been studied in the literature for
non-local Lévy type operators. To tackle the problems, we propose a new method based on
probabilistic potential theory.
The talk is based on the paper [5].