Twin solutions to singular semipositone problems.
The author investigates the existence of multiple positive solutions to the following singular semipositone problem: \[ y''+\lambda f(t,y)=0,\quad t\in(0,1),\quad y(0)=y(1)=0, \tag{1\(_\lambda\)} \] where \(\lambda>0\), and \(f(t,y)\) has not necessarily a constant sign and may be singular at \(t=0\), \(t=1\) and \(y=0\). More precisely, it is shown that, if appropriate assumptions hold on \(f\), then for any \(r>0\) there exists \(\bar\lambda>0\) such that, for any \(\lambda\in (0,\bar\lambda)\), (1\(_\lambda\)) has at least two positive solutions \(x\) and \(y\) satisfying \(0<\| x\| <r<\| y\| \). The proof is based upon a fixed-point index argument.
scientific article; zbMATH DE number 2003904