Triple positive solutions for a type of second-order singular boundary problems
Using the Leggett-Williams fixed point theorem, the authors establish an existence result guaranteeing at least three positive solutions to the singular (at \(t=0\) and/or \(t=1\)) second order boundary value problems of the form \[ u''(t)+p(t)u'(t)+q(t)u(t)+h(t)f(t,u(t))=0,\;t\in(0,1), \] \[ u(0)=u(1)=0, \] where \(p\in C(0,1)\cap L^1(0,1),\,q\in C((0,1),(-\infty,0))\) with \(\int_0^1t(1-t)|q(t)|dt<\infty,\) \(h\in C((0,1),[0,\infty))\) is such that \(h(t)\not\equiv0\) on any subinterval of \((0,1)\) and \(\int_0^1t(1-t)h(t)dt<\infty\) and \(f\in C([0,1]\times[0,\infty),[0,\infty)).\) It is assumed in addition that for numbers \(a,b,\) and \(c\) with the properties \(0<a<b<b/\gamma_\delta\leq c\) the function \(f\) satisfies the conditions \[ \begin{aligned} f(t,u)&<ma\quad\text{for }t\in[0,1]\text{ and }u\in[0,a],\\ f(t,u)&>b\eta\quad\text{for }t\in[\delta,1-\delta]\text{ and }u\in[b,b/\gamma_\delta],\\ f(t,u)&<mc\quad\text{for }t\in[0,1]\text{ and }u\in[0,c],\end{aligned} \] where \(\delta\in(0,1/2),\,\gamma_\delta>0,\,m\) and \(\eta\) are suitable constants.
scientific article; zbMATH DE number 5791557
scientific article