一类Dirichlet问题正解的精确个数

Exact number of positive solutions to a class of Dirichlet problems

研究Dirichlet问题-■=λ(u^p+u^q),u(0)=u(1)=0,其中1<p<q<+∞,参数λ>0,得到了在1<p<q<p+1条件下,存在λ*>0,当λ≤λ*时,此方程无正解;当λ>λ_*时,此方程恰好有一个正解.

We Study the exact number of positive solutions to the problem for the one-dimensional prescribed mean curvature equation -（u＇/（1＋u＇2）（1/2））＇=λ（u^p＋u^q）,u（0）=u（1）=0,with 1pqp ＋1.We find aλ＊0,so that ifλ≤λ＊,this problem has no positive solution,and ifλλ＊,then this problem has one positive solution.