非齐次边界条件下弹性梁方程正解的多解性

研究带非齐次边界条件的两端简单支撑的弹性梁方程{Y ⁽⁴⁾(x)=f(x,y), x∈(0,1),y(0)=0, y(1)=b, y″(0)=0, y″(1)=0多个正解的存在性,其中f∈C(［0,1］×［0,∞),［0,∞)), b>0,且对给定的x∈［0,1］, f(x,s)关于s单调递增。在适当的条件下,证明存在b^*>0,使得当0*时至少存在两个正解;当b=b^*时至少存在一个正解;当b>b^*时无正解。该结果的证明基于上下解方法和拓扑度理论。

Multiplicity of positive solutions for elastic beam equations under inhomogeneous boundary conditions

This paper studies the existence of multiple positive solutions for elastic beam equations simply supported at both ends with inhomogeneous boundary conditions{Y ⁽⁴⁾(x)=f(x,y), x∈(0,1),y(0)=0, y(1)=b, y″(0)=0, y″(1)=0,where f∈C(［0,1］×［0,∞),［0,∞)), b>0, and f(x,s) is a monotone increasing function with respect to s for a fixed x∈［0,1］. Under appropriate conditions, there exists b^*>0 such that the problem has at least two positive solutions for 0*, at least one positive solution for b=b^*, and no positive solution for b>b^*. The proof of the main results is based on the upper and lower solution method and topological degree theory.