一类带小参数的双调和方程边值问题的可解性

Solvability of Boundary Value Problems for a Class of Biharmonic Equations with Small Parameters

非线性椭圆型偏微分方程是偏微分方程中的一个热门课题。本文讨论了其中一类带小参数的双调和方程边值问题的可解性。通过变量代换的方法将双调和方程边值问题转换为椭圆型方程组边值问题,再利用上、下解方法和不动点定理证明所讨论问题解的存在性,并利用Green恒等式和Poincare不等式证明了解的唯一性。

Nonlinear elliptic partial differential equation is a hot topic in the field of partial differential equations. The solvability of a class of boundary value problems of biharmonic equations with small parameters is discussed. The boundary value problem of the biharmonic equation is transformed into the boundary value problem of the elliptic system of equations by the method of variable substitution. The existence of the solution of the problem is proved by using the upper and lower solution methods and the fixed point theorem. Meanwhile, the uniqueness of the solution is proved by using Green identity and Poincare inequality.