4-harmonic and ES-4-harmonic maps are two generalizations of the well-studied harmonic map equation which are both given by a nonlinear elliptic partial differential equation of order eight. Due to the large number of derivatives it is very difficult to find any difference in the qualitative behavior of these two variational problems. In this article we prove that finite energy solutions of both 4-harmonic and ES-4-harmonic maps from Euclidean space must be trivial. However, the energy that we require to be finite is different for 4-harmonic and ES-4-harmonic maps pointing out a first difference between these two variational problems.

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关于4-谐波和ES-4-谐波映射的有限能量解

4-谐波和 ES-4-谐波映射是经过充分研究的谐波映射方程的两种推广，它们都由八阶非线性椭圆偏微分方程给出。由于大量的导数，很难找到这两个变分问题的定性行为的任何差异。在本文中，我们证明了来自欧几里得空间的 4-harmonic 和 ES-4-harmonic 映射的有限能量解一定是微不足道的。然而，对于 4-谐波和 ES-4-谐波映射，我们要求为有限的能量是不同的，指出了这两个变分问题之间的第一个区别。