We prove that the Dirichlet problem for the Lane–Emden equation in a half-space has no positive solution that is monotone in the normal direction. As a consequence, this problem does not admit any positive classical solution that is bounded on finite strips. This question has a long history and our result solves a long-standing open problem. Such a nonexistence result was previously available only for bounded solutions or under a restriction on the power in the nonlinearity. The result extends to general convex nonlinearities.

中文翻译：

---

半空间中Lane–Emden方程的一个Liouville型定理

我们证明了在半空间内的Lane-Emden方程的Dirichlet问题没有在法线方向上为单调的正解。结果，这个问题不容许任何有限带上的正经典解。这个问题源远流长，我们的结果解决了一个长期存在的开放性问题。这种不存在的结果以前仅可用于有界解或在非线性的幂限制下可用。结果扩展到一般的凸非线性。