We shall establish a comprehensive variational principle that allows one to apply critical point theory on closed proper subsets of a given Banach space and yet, to obtain critical points with respect to the whole space. This variational principle has many applications in partial differential equations while unifying and generalizing several results in nonlinear analysis such as the fixed point theory, critical point theory on convex sets and the principle of symmetric criticality. As a consequence, several substantial new results are emerged. We shall also provide concrete applications in local and nonlocal partial differential equations, including the symmetry properties of the Allen-Cahn equation on bounded domains, for which the standard methodologies have major limitations to be applied.

我们将建立一种综合的变分原理，该原理允许将临界点理论应用于给定Banach空间的闭合固有子集，而获得整个空间的临界点。这种变分原理在偏微分方程中有许多应用，同时可以统一和归纳非线性分析中的一些结果，例如定点理论，凸集上的临界点理论和对称临界原理。结果，出现了一些实质性的新结果。我们还将在局部和非局部偏微分方程中提供具体应用，包括有界域上的Allen-Cahn方程的对称性质，对此标准方法有很大的限制。