Abstract We consider three types of semilinear equations (elliptic, parabolic and hyperbolic) posed in the N-dimensional exterior domain R N \ D , where N ≥ 2 and D is the closed unit ball in R N . A nontrivial Robin boundary condition is imposed on the boundary of D. Using a test function approach with judicious choices of the test functions, we show that the considered problems share a common critical behavior. We discuss separately the cases N = 2 and N ≥ 3 . Moreover, in the case N ≥ 3 , the dependence of the critical exponent on initial data is discussed. To the best our knowledge, the study of the critical behavior in an exterior domain with a nontrivial Robin boundary condition has never been studied in the literature.

中文翻译：

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非平凡罗宾边界条件下某些半线性外部问题全局解的存在与不存在

摘要 我们考虑在 N 维外域 RN \ D 中提出的三种类型的半线性方程（椭圆、抛物线和双曲线），其中 N ≥ 2 且 D 是 RN 中的封闭单位球。在 D 的边界上强加了一个非平凡的 Robin 边界条件。使用测试函数方法和测试函数的明智选择，我们表明所考虑的问题具有共同的关键行为。我们分别讨论 N = 2 和 N ≥ 3 的情况。此外，在 N ≥ 3 的情况下，讨论了临界指数对初始数据的依赖性。据我们所知，在具有非平凡 Robin 边界条件的外部域中的临界行为研究从未在文献中进行过研究。