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

TL;DR

This paper investigates the long-term behavior of solutions to semilinear elliptic equations in unbounded domains, establishing uniqueness and profile characterization using PDE and dynamical systems methods.

Contribution

It introduces two distinct approaches—PDE techniques and dynamical systems theory—to analyze asymptotic solutions in unbounded domains, providing new Liouville results.

Findings

Proved uniqueness of solutions at infinity.

Characterized solutions as one-dimensional or constant profiles.

Developed two complementary analytical methods.

Abstract

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.