Properties of the set of positive solutions to Dirichlet boundary value problems with time singularities (Q1935086)

The authors study the structure and properties of the set \(\mathcal {S}\) of all positive solutions to the singular Dirichlet boundary value problem \[ u''(t)+\frac{a}{t}u'(t)-\frac{a}{t^2}u(t)=f(t,u(t),u'(t)),\;\;u(0)=u(T)=0 \] with \(a\in (-\infty,-1)\). They obtain that, for each \(c\geq0\), the set \(\mathcal {S}_{c}=\{u\in \mathcal {S}, u'(T)=-c\}\) is nonempty and compact and \(\mathcal {S}=\bigcup_{c\geq 0} \mathcal {S}_{c}\). They also give an order in \(\mathcal {S}\) and show that the difference of any two solutions in \(\mathcal {S}_{c}\) keeps its sign on \([0,T]\). As an application, the equation \[ v''(t)+k \frac{v'(t)}{t}=\psi(t)+g(t,v(t))\quad (k\in (1,\infty)) \] is considered.