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

Messoud Efendiev; Francois Hamel

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arXiv · 1007.3921

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

Messoud Efendiev·Francois Hamel

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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.

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Messoud Efendiev, Francois Hamel. 2010-07-22. Asymptotic behavior of solutions of semilinear elliptic equations in unbounded domains: two approaches. https://arxiv.org/abs/1007.3921

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