In this paper, we study the asymptotic behavior as $x_1\to+\infty$ ofsolutions of semilinear elliptic equations in quarter- or half-spaces, forwhich the value at $x_1=0$ is given. We prove the uniqueness and characterizethe 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 isbased on the maximum principle, the sliding method and some new Liouville typeresults for elliptic equations in the half-space or in the wholespace~$\mathbb{R}^N$. The second one is based on the theory of dynamicalsystems.
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机译：在本文中，我们研究四分之一或半空间中半线性椭圆型方程解的xx \ to + \ infty $的渐近行为，并给出$ x_1 = 0 $的值。我们证明了唯一性，并刻画了无穷大时解的一维或恒定轮廓。为此，我们使用两种不同的方法。第一个是纯PDE方法，它基于最大原理，滑动方法和半空间或全空间〜$ \ mathbb {R} ^ N $中椭圆方程的一些新的Liouville型结果。第二个是基于动力学系统的理论。
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