一类非线性方程的死角问题

Dead Core Problems for a Class of Nonlinear Equations

应用初等方法 ,证明了边值问题u″ =λh(u′) ,u≥ 0 ,x∈ (- 1,1) ,u(± 1) =1存在唯一非平凡解 (在C2 [- 1,1]中 )的充分必要条件是∫101h(s) ds<∞ .而且非平凡解有死角的充分必要条件是λ<∫λ0 G- 1(t)dt.这里 ,λ>0 ,h∈C(R) ,h(0 ) =0 ,h(s) >0 , s≠ 0 ,G- 1表示G(t) =∫t01h(s) ds的反函数 ,死角是 [-r ,r],r满足λ=∫λ( 1-r)0 G- 1(t)dt .特别 ,若h(s) =｜s｜p,则存在唯一非平凡解的充分必要条件是 0 <p <1,且非平凡解有死角 [-r ,r],r=1- (1- p) - 1λ- 12 - p(2 - p) - p2 - p.

This paper is concerned with the boundary value problems u″=λh(u′) and u(±1)=1, where h∈C(R),h(0)=0 and h(s)>0 for s≠0.In the proofs the elementary methods are used only. The main results are as follows:there exists the unique nontrivial solution if and only if∫ 1 01h(s) d s<∞, and there exists a dead core for this nontrivial solution if and only if λ<∫ λ 0G -1 (t) d t for some λ>0, where G(t)=∫ t 01h(s) d s.