  In this paper, we study the asymptotic behavior as $x_1\to+\infty$ of
solutions of semilinear elliptic equations in quarter- or half-spaces, for
which the value at $x_1=0$ is given. We prove the uniqueness and characterize
the one-dimensional or constant profile of the solutions at infinity. To do so,
we use two different approaches. The first one is a pure PDE approach and it is
based on the maximum principle, the sliding method and some new Liouville type
results for elliptic equations in the half-space or in the whole
space~$\mathbb{R}^N$. The second one is based on the theory of dynamical
systems.
