变分方法及其在非线性偏微分方程应用方面的进展和未决问题
Results and Open Problems of the Variational Method with Applications to Nonlinear Partial Differential Equations

先介绍变分法发展的简单历史以及将来的发展趋势. 然后综述变分法应用于非线性偏微分方程的基本思想和最新成果. 通俗介绍环绕理论、变号临界点理论及应用,其中包括对称扰动方程和Rabinowitz公开问题、Brezis-Nirenberg临界指数方程、Li-Lin公开问题、Bose-Einstein凝聚、Berestycki-Caffarelli-Nirenberg猜测和Lane-Emden方程及猜想.
The brief history and the development trend of the variational method are introduced.Then the fundamental ideas and the latest achievements of the variational method with applications to nonlinear partial differential equations are summarized.The critical point theory and its applications are briefly reviewed,including the perturbed equation from symmetry,Rabinowitz's open problem,Brezis-Nirenberg's critical exponent equation,Li-Lin's open problem,Bose-Einstein condensation,Berestycki-Caffarelli-Nirenberg's conjecture and Lane-Emden's equation and conjecture