  In this article we study the fundamental solutions or "$\alpha$-harmonic
functions" for some nonlinear positive homogeneous nonlocal elliptic problems
in conical domains, such as \begin{eqnarray*}\label{ecbir1a1} {\mathcal F
}(u)=0\ \ \hbox{in} \ \ \mathcal{C}_\omega,\quad u=0\ \ \hbox{in} \ \
\mathbb{R}^n\setminus \mathcal{C}_\omega ,\ \ \end{eqnarray*} where $\omega$ is
a proper $C^2$ domain in $S^{N-1}$ for $ N\geq 2$,
$\mathcal{C}_\omega:=\{x\,:\,x\neq 0, {|x|^{-1}}x\in \omega\}$ is the cone-like
domain related to $\omega$, and ${\mathcal F }$ is an extremal fully nonlinear
integral operator. We prove the existence of two fundamental solutions that are
homogeneous and do not change signs in the cone; one is bounded at the origin
and the other at infinity.
  As an application, we use the fundamental solutions obtained to prove
Liouville type theorems in cones for supersolutions of the Lane-Emden-Fowler
equation in the form \begin{eqnarray*}\label{eq 0.2} {\mathcal F }(u)+u^p = 0\
\ \hbox{in} \ \ \mathcal{C}_\omega, \quad u=0\ \ \hbox{in} \ \
\mathbb{R}^n\setminus \mathcal{C}_\omega.\ \ \end{eqnarray*} We also prove a
generalized Hopf type lemma in domains with corners. Most of our results are
new even when ${\mathcal F }$ is the fractional Laplacian operator.
