Singular solutions of a singular differential equation (Q1595130)

The authors study the singular differential equation \[ \bigl( |y'|^{-\alpha} \bigr)'+q(t)|y|^\beta =0\tag{*} \] where \(\alpha\) and \(\beta\) are positive constants and \(q(t)\) is a positive continuous function on \([0,\infty)\). A solution with conditions given at \(t\equiv 0\) is called singular if it ceases to exist at some finite point \(T\in (0,\infty)\). The singular solution \(y(t)\) with the property \[ \lim_{t\to T-0}\bigl|y(t) \bigr|<\infty \quad\text{and}\quad \lim_{t\to T-0}\bigl|y'(t) \bigr |=\infty \] is named a black hole solution in view of its specific behavior at \(t=T\). The results of the research are presented in 4 theorems, 3 examples, 2 remarks and 1 lemma.