Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: Two approaches

In this paper, we study the asymptotic behavior as x1→+∞ of solutions of semilinear elliptic equations in quarter- or half-spaces, for which the value at x1=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 RN. The second one is based on the theory of dynamical systems.