Positive solutions of superlinear semipositone nonlinear boundary value problems (Q639105)

The authors investigate the boundary value problem: \[ \begin{cases} -(p(x)u'(x))'+q(x)u(x)=\lambda f(x,u(x)),\;x\in(0,1),\\ \alpha_0u(0)+\beta_0u'(0)=0,\quad\alpha_1u(0)+\beta_1u'(0)=0, \end{cases} \] where \(\alpha_i\), \(\beta_i\;(i=0,1)\) satisfy some non-resonance conditions. The non-linearity \(f\), \(f(x,u)=a(x)u+H(x,u)\), is assumed to be semipositone (bounded from below but may change sign), superlinear, and is allowed to possess singularities at the end-points \(x=0\) and/or \(x=1\). Using global structure and results from bifurcation theory, existence of two positive solutions is proved for large values of the parameter \(\lambda\).