Abstract
An attempt is made to study the problem of existence of singular solutions to singular differential equations of the type
(|y′|−α)′+q(t)|y|β=0,
(∗)
which have never been touched in the literature. Here α and β are positive constants and q(t) is a positive continuous function on [0,∞). A solution with initial conditions given at t=0 is called singular if it ceases to exist at some finite point T∈(0,∞). Remarkably enough, it is observed that the equation (∗) may admit, in addition to a usual blowing-up singular solution, a completely new type of singular solution y(t) with the property that
limt→T→0|y(t)|<∞
and
limt→T→0|y′(t)|=∞.
Such a solution is named a black hole solution in view of its specific behavior at t=T. It is shown in particular that there does exist a situation in which all solutions of (∗) are black hole solutions.