The existence of countably many positive solutions for singular multipoint boundary value problems (Q1049057)

Sufficient conditions are given for the existence of countably many positive solutions of a singular multipoint boundary value problem of the form \[ \phi_p(u'(t))'+a(t)f(u(t))=0, t \in (0,1), \] \[ u'(0)- \sum_{i=1}^{m-2}\alpha_i u(\xi_i)=0, u'(1)=\sum_{i=1}^{m-2}\alpha_i u(\eta_i)=0, \] where \(\phi_p(s)=|s|^{p-2}s, p>1, \xi_i, \eta_i \in (0,1), \alpha_i >0\). It is assumed that \(\sum_{i=1}^{m-2}\alpha_i u(\xi_i) < 1, \sum_{i=1}^{m-2}\alpha_i u(1-\eta_i) < 1\) and that \(f \in C([0,+\infty),(0,+\infty))\). Moreover, it is required the existence of a decreasing convergent sequence \(\{t_i\}\) and a real non-negative function \(a\) such that \(\lim_{t\to t_i} a(t)=\infty\) and \(0<\int_0^1a(t) dt < +\infty\). A concrete (though quite artificial) example of a function \(a\) satisfying this condition is given at the end of the paper. For the proof, the Leggett-Williams fixed point theorem in cones is applied.