Multiple positive solutions of singular eigenvalue type problems involving the one-dimensional \(p\)-Laplacian (Q1827129)

The author studies existence and multiplicity of positive solutions for the singular Dirichlet eigenvalue-type problem \[ -(\varphi_p(u'(t)))'= \lambda h(t) f(u(t)) \quad \text{in }(0,1), \qquad u(0)= u(1)= 0, \] where \(\varphi_p(\xi)=| \xi|^{p-2}\xi\), \(p>1\), and \(\lambda\) is a positive parameter, \(h(t)\) is a nonnegative measurable function on \((0,1)\) that may be singular at \(t=0\) and/or \(t=1\), \(f(u)\) is a nonnegative continuous function on \([0,+\infty)\), moreover \(f\) is either sublinear or superlinear at zero and/or \(+\infty\). To this end, the author uses the fixed-point index theory as well as a fixed-point theorem in cones.