Uniqueness of minimizers of constrained or unconstrained energy functionals is a very subtle issue that has attracted many mathematicians in the last decades. This interest is motivated by several reasons: The uniqueness implies that the critical point inherits all the symmetry and monotonicity properties of the problem. For example if the functional and its domain are radially decreasing, then is the critical point. This considerably reduces the difficulty of the study of quantitative properties of the underlying PDE, which is reduced to an ODE. Uniqueness also “guarantees” the stability, and simplifies the dynamics of the gradient flow induced by the functional.

There are very few general results dealing with uniqueness of critical points in the literature. The purpose of this paper is to provide a counterexample of a general result obtained by B. Dacorogna in his book “Introduction to the calculus of variations, Edition 2004”.

受约束或不受约束的能量函数的极小化子的唯一性是一个非常微妙的问题，在过去的几十年中吸引了许多数学家。引起这种兴趣的原因有几个：唯一性意味着临界点继承了问题的所有对称性和单调性。例如，如果功能及其域在径向上减小，那么临界点就是。这大大降低了研究潜在的PDE定量特性的难度，后者被简化为ODE。唯一性还“保证”了稳定性，并简化了由功能引起的梯度流的动力学。

关于文献中关键点的唯一性的一般结果很少。本文的目的是提供一个反例，说明B. Dacorogna在他的书《 2004年变异学概论》中获得的一般结果。