In this paper we develop a Gidas–Ni–Nirenberg technique for polyharmonic equations and systems of Lane-Emden type. As far as we are concerned with Dirichlet boundary conditions, we prove uniqueness of solutions up to eighth order equations, namely which involve the fourth iteration of the Laplace operator. Then, we can extend the result to arbitrary polyharmonic operators of any order, provided some natural boundary conditions are satisfied but not for Dirichlet’s: the obstruction is apparently a new phenomenon and seems due to some loss of information. When the polyharmonic operator turns out to be a power of the Laplacian, and this is the case of Navier’s boundary conditions, as byproduct uniqueness of solutions holds in a fairly general context. New existence results for systems are also established.

在本文中，我们为连调方程和Lane-Emden型系统开发了Gidas-Ni-Nirenberg技术。就Dirichlet边界条件而言，我们证明了多达八阶方程的解的唯一性，即涉及Laplace算子的第四次迭代。然后，只要满足某些自然边界条件，但不满足Dirichlet的条件，我们就可以将结果扩展到任意阶的任意多调和算子：障碍显然是一种新现象，似乎是由于信息丢失所致。当多谐波算子证明是拉普拉斯算子的幂时，这就是Navier边界条件的情况，这是因为溶液的副产物唯一性在相当普遍的情况下仍然有效。还建立了系统的新存在结果。