Positive solutions of superlinear semipositone singular Dirichlet boundary value problems (Q819729)

This interesting paper is devoted to the existence of positive solutions for the semipositone Dirichlet boundary value problem \[ u''+f(t,u)+q(t)=0, \quad t\in (0,1),\qquad u(0)=u(1)=0, \] where \(f:(0,1)\times[0,\infty)\to [0,\infty)\) is continuous \(q:(0,1)\to (-\infty, \infty)\) is Lebesgue integrable, \(f\) may be singular at \(t=0,1\) and \(q\) can have finitely many singularities. The authors obtain sufficient conditions which imply that the considered BVP has at least one \(C[0,1]\cap C^2(0,1)\) positive solution. The proof is based on the properties of the fixed-point index for positive completely continuous operators in a cone.