带有特殊非线性项的Dirichlet问题正解的确切个数

Exact Multiplicity of Positive Solutions of Dirichlet Problemswith Special Nonlinear Terms

用时间映像原理研究一维Minkowski空间给定平均曲率方程Dirichlet问题-u′1-u′2′=λf(u),x∈(-L,L),u(-L)=0=u(L)正解的确切个数及分歧图,其中参数λ>0,L>0.在λ满足一定的条件下,分别得到了非线性项为f(u)=u(e u-1)和f(u)=e u-1时该问题没有正解、恰有一个正解和恰有两个正解的结果.

By using the principle of time maps,we study the exact multiplicity and bifurcation diagrams of positive solutions of Dirichlet problem for a prescried mean curvature equation in the one-dimensional Minkowski space:-u′1-u′2′=λf(u),x∈(-L,L),u(-L)=0=u(L),whereλ>0 and L>0 are parameters.We obtain the results that the above problem has no positive solution,exactly one positive solution and exactly two positive solutions when the nonlinear terms f(u)=u(e u-1)and f(u)=e u-1,respectively.