一类非线性悬臂梁方程正解的存在性与多解性

研究了非线性四阶常微分方程u(4)(t)=f(t,u(t),u'(t)),t ∈0,1]\E在边界条件u(0)=u'(0)=u"(1)=u"'(1)=0下的正解,其中E(∩)0,1]是一个零测度的闭集,而非线性项,(t,u,u)可以在t∈E时奇异.通过构造适当的积分方程并利用锥上的不动点定理证明了这个方程在满足与n有关的条件下存在n个正解,其中n是某个自然数.

Existence and Multiplicity of Positive Solutions to a Class of Nonlinear Cantilever Beam Equations

The positive solutions are studied for the nonlinear fourth-order ordinary differential equation $u^{(4)}(t)=f(t,u(t),u'(t)),~t\in 0,1]\backslash E$, subject tothe boundary conditions $u(0)=u'(0)=u'(1)=u'(1)=0$, where $E\subset 0,1]$ is a closed set with measure zero and nonlinear term $f(t,u,v)$ may be singular for $t\in E$. With some conditions, by constructing suitable integral equation and applying fixed point theorems on cone, the existence of $n$ positive solutions is proved for the equations, where $n$ is a positive integral number.