A necessary and sufficient condition for singular nonlinear second-order boundary value problems to have positive solutions (Q1387512)

A necessary and sufficient condition is established for the existence of a nonzero solution to the problem (1) \(u''+ f(x,u,u')= 0\), (2) \(\alpha u(a)- \beta u'(a)= 0\), \(\gamma u(b)+ \delta u'(b)= 0\), where \(f(x,u,v)\) is nonnegative, continuous in \((a,b)\times \mathbb{R}^+\times \mathbb{R}^+\), \(f\) may be singular at \(x= a\) and/or \(x=b\), and \(v=0\); \(f\) is nondecreasing in \(u\) for each \(x\), \(v\), nonincreasing in \(v\) for each \(x\), \(u\); and there exists a constant \(q\in(0, 1)\) such that \[ t^qf(x, t^{-1}u, tu)\leq f(x,u,u)\leq \lambda^qf(x,\lambda^{-1}u, \lambda u), \] for each \(0<t<1<\lambda\), \(u\in\mathbb{R}^+\). If problem (1), (2) has a nonzero solution, then it is the unique solution to the problem and it is positive on \((a,b)\).