共振条件下具p-Laplace算子两点边值问题解的存在性

Existence of solutions to two-point p-Laplacian boundary value problem at resonance

利用紧向量场方程的解集连通理论和反序严格上下解方法,研究了一类共振条件下的具p-Laplace算子微分方程两点边值问题[φp(x′(t))]′+f(t,x(t))=0,0<t<1;x(0)=x(1),x′(0)=0.解的存在性.当非线性项f变号时,得到了边值问题存在非负解,非正解及解的充分条件,当非线性项f恒正或恒负时,得到了边值问题无解的结论.

By means of the connected theory of solutions set to compact vector field defined by equation and strict upper and lower solutions with reverse order, the existence of solutions to two - point p - Laplacian boundary value problem {[φp（x＇（t））]＇＋f（t,x（t））=0,0〈t〈1; x（0）=x（1）,x＇（0）=0 at resonance is studied, some sufficient conditions for the boundary value problem with nonnegative solutions, non- positive solutions and solutions are obtained as nonlinear term f variation sign, the inexistence of solutions are given as nonlinear term f〉0, or f〈0.