二阶微分方程边值问题的多重正解

MULTIPLE POSITIVE SOLUTIONS FOR BOUNDARY VALUE PROBLEM OF SECOND ORDER DIFFERENTIAL EQUATIONS

基于Leray-Schauder度理论和上下解方法讨论非线性边值问题(t)+g(t,y)=0,(0)=0,y(1)=b≥0的正解存在性,其中g局部Lipschitz连续,g(t,0)≥0,但是可以是变号函数。主要结论是:如果g(t,y)在y=+∞满足一个超线性增长条件,并且存在使得β(1)>0的非负上解β,则存在正数B使得当0<b<B时,至少存在两个正解;当b=0或b=B时,至少存在一个正解;而当b>B时,不存在正解。

The existence of positive solutions has been discussed for the nonlinear boundary value problem y'(t) + g(t,y) = 0, y'(0) = 0 and y(l) = b > 0, where g is locally Lipschitz continuous, g(t, 0) > 0 and may change sign. The main result as follows: If g(t, y) satisfies a superlinear condition at y = +00 and there exists a nonnegative supersolution B with B(1) > 0, then there exists a positive number B such that this problem has at least two positive solutions for 0 < 6 < B, at least one for 6 = 0 or 6 ?B, and none for b > B. Our approach is based on the Leray-Schauder degree arguments and the method of sub- and supersolutions.