变分方法与反向上下解

Calculus of Variations and Super-and Sub-solutions in Reverse Order

本文在非线性方程的下解不小于上解这一条件（即反向上下解条件）下研究了其解的存在性．我们证明了，如果非线性算子方程有变分结构，有一对反向上下解，对应的算子映某个锥入锥并且满足一定的辅助条件，那么这一方程在锥中至少有两个解．另外，本文还研究了反向上下解条件下非线性椭圆边值问题正解的存在性．

In this paper we study the existence of solutions for nonlinear equations under the condition that the sub-solutions are not less than the super-solutions (i.e., the sub- and supersolutions are in reverse order). We prove that if a nonlinear operator equation with variational structure possesses a pair of super-and sub-solutions in reverse order, and the corresponding nonlinear operator maps some cone into itself, then the equation has two solutions in the cone under some additional conditions. We also study the existence of positive solutions for nonlinear elliptic BVPs.