Existence of positive solutions for nonlinear singular boundary value problems (Q1362235)

Nonlinear singular two-point boundary value problems of the form \[ u''(t)+\lambda f(t,u(t))= 0\quad\text{in }(0,1),\tag{1} \] \[ u(0)= u(1)= 0\tag{2} \] are considered. It is assumed that \(f(t,u)\) is positive for \(u\geq 0\) and increasing with respect to \(u\) for each fixed \(t\in(0,1)\) and that there exist constants \(\alpha\in[0,2)\) and \(\beta\in(0,1/2)\) such that \(0<f(t,u)\leq M_0(1/t^\alpha)\) on \((0,\beta]\times [0,u]\) for some \(M_0>0\) depending on the interval \([0,u]\). The latter condition means that \(f(t,u)\) may be singular at \(t=0\). The main result concerns the existence of a positive solution for this boundary value problem for \(\lambda\) belonging to a suitable interval. More exactly, it is shown that there exists a \(\lambda^*\leq 8L^2\) such that the above boundary value problem has a positive solution in \(C^2(0,1]\cap C[0,1]\) for \(0<\lambda\leq\lambda^*\), while there is no such solution for \(\lambda>\lambda^*\). Here \(L:= \int^1_{1/2} \int^1_s f(u,0)du ds\in(0,\infty]\). This result generalizes previous theorems by Choi, and by Fink, Gatica and Hernandez.