Introduction

Consider the nonlinear ordinary differential equation

$$(\vert y'\vert^\alpha)' + q(t) \vert y\vert^\beta = 0 \leqno {\rm (A)}$$

where $\alpha, \beta \in R, \alpha \not= 0$, are constants and $q: [a, \infty) \to (0, \infty), a \geq 0$, is a continuous function.

By a solution of (A) on an interval $J = [t_0,T), a \leq t_0 < T \leq \infty$, we mean a function $y \in C^1(J)$ which has the property $\vert y'\vert^\alpha \in C^1(J)$ and satisfies (A) at each point of $J$. If we denote by $T_y$ the maximal existence time of $y$, then we say that $y(t)$ is proper if $T_y = \infty$ and

$$\sup \{ \vert y(t)\vert: t \in [\tau, \infty) \} > 0 \quad {\rm for \quad any} \quad \tau \geq t_0.$$

A solution $y(t)$ is called singular if either $T_y < \infty$ or $T_y = \infty$ and there exists $T \in [t_0, \infty)$ such that

$$\max \{ y(s): t \leq s \leq T \} > 0 \quad {\rm for} \quad t \in (t_0, T)$$

and $y(t) = 0$ for $t \geq T$. In the later case, the interval $[t_0,T)$ is called the support of the solution $y(t)$.

Our main objective here is to investigate the structure of the solution set of (A) in the case $\alpha > 0$ and to show that nonlinear equations of the form (A) may have singular solutions of a new type satisfying

$$\lim_{t \to T_y-0} y(t) = {\rm const} \not= 0 \quad {\rm and} \quad \lim_{t \to T_y-0} y'(t) = 0 \leqno {\rm (1)}$$

at the (finite) right end-point of the maximal interval of existence. By analogy with the concept of ``black hole'' solutions, that is, singular solutions defined on $[t_0, T_y)$ and satisfying

$$\lim_{t \to T_y-0} y(t) = {\rm const} \not= 0 \quad {\rm and}... ... \lim_{t \to T_y-0} \vert y'(t)\vert = \infty, \leqno {\rm (2)}$$

introduced by the present authors in [3] (see also [4], [7] and [8]), positive solutions of (A) satisfying (1) as $t$ approaches the maximal existence time $T_y < \infty$ are called white hole (singular) solutions.

An example of a nonlinear equation of the form (A) (with $\beta = 0$) which possesses singular solutions of this new type is

$$(\vert y'\vert^\alpha)' + \left ( {\alpha +1 \over \alpha} \right )^\alpha = 0, \leqno {\rm (3)}$$

where $\alpha > 0$. Indeed, for any given $T > a$ and $c > 0$, the function $y(t) = c+(T-t)^{\alpha+1 \over \alpha}$ defined and positive on $[a,T)$ is a decreasing solution of (3) with a singularity of white hole type at $T$.

Similarly, for any $c > 0$, the function $y(t) = c-(T-t)^ {\alpha+1 \over \alpha}$ provides an example of a `local' increasing white hole solution of (3) which is defined and positive in some sufficiently small left neighborhood of the maximal existence time $T$.

Another simple example of an equation of the form (A) having white hole singular solutions is the following ``almost linear'' equation

$$(\vert y'\vert)' + \vert y\vert = 0. \leqno {\rm (4)}$$

As easily seen, for any real $T$ and any $c > 0$, the function $y(t) = c \cos (T-t)$ defined and positive on $[t_0,T), t_0 \geq T-{\pi \over 2}$, is an increasing singular solution of (4) which is of the white hole type.

While the existence and asymptotic theory for quasilinear second-order differential equations of the form

$$(p(t)\vert y'\vert^\alpha {\rm sgn} y')' + q(t) \vert y\vert^\beta {\rm sgn} y = 0, \quad t \geq a, \leqno {\rm (B)}$$

and for singular equations

$$(p(t) \vert y'\vert^\alpha {\rm sgn} y')' + q(t) y^{-\beta} = 0, \quad t \geq a, \leqno {\rm (C)}$$

where $\alpha > 0$ and $\beta > 0$ are constants and $p(t)$ and $q(t)$ are positive continuous functions on $[a, \infty)$, is well developed (see, for example, the papers [1], [5-6] and [9-11] ), according to our knowledge there are no papers concerning the existence of singular and proper solutions for the nonlinear equation (A) (with $\alpha > 0$) in our setting. This observation was one of the motivations for the present paper.

The plan of this paper is as follows. In Section 2 we first show that, regardless of positivity or negativity of $\beta$, Eq. (A) always has singular solutions of white hole type iff $\alpha > 0$. Our procedure for establishing the existence of such solutions for (A) is based on the solution of appropriate nonlinear integral equation via the Schauder fixed point theorem. Secondly, we investigate the existence of another type of singular solutions of (A) named ``extinct solutions". More specifically, it is shown that if either $0 < \alpha \leq 1$ and $\vert\beta\vert < \alpha$ or $\alpha > 1$ and $-1 < \beta < \alpha$, then Eq. (A) possesses a singular solution which extincts (together with its first derivative) at an arbitrarily prescribed extinction point $T > a$.

In Sections 3-5 our consideration is focused on the set of proper solutions of (A), i.e., solutions which exist on some interval $[t_0, \infty) \subset [a, \infty)$ and are not identically zero in any neighborhood of infinity. Although the equation (A) has a relativelly simple form, the totality of proper solutions of (A) has surprisingly rich structure. This is demonstrated in Section 3 where the set of all possible proper solutions is classified into eight different types according to their asymptotic behavior as $t \to \infty$.

In Section 4 we establish conditions guaranteeing the existence of increasing proper solutions of each of the types (IV)-(VI) appearing in the general classification scheme given in Section 3. We prove in particular that if $0 < \beta < \alpha$, then for any given $y_0 > 0$, Eq. (A) has a `global' solution $y$ (i.e., a proper solution existing on the whole interval $[a, \infty)$) satisfying $y(a) = y_0$ and growing to infinity as fast as a constant multiple of $t$ as $t \to \infty$ if and only if the function $t^\beta q(t)$ is integrable on $[a, \infty)$. The next theorem in Section 4 presents sufficient conditions under which Eq. (A) possesses an increasing proper solution which grows at infinity like a positive constant multiple of $t^{(\alpha+\sigma+1)/(\alpha-\beta)}$ for some $\sigma \in R$ with $0 < (\alpha+\sigma+1)/(\alpha-\beta) < 1$. As regards increasing proper solutions of (A) which remain bounded as $t \to \infty$, the (`local') existence of such solutions satisfying $\lim_{t \to \infty} y(t) = \omega_0$ for arbitrarily prescribed terminal value $\omega_0 > 0$ is characterized by the integral condition (38) below.

Finally, in Section 5, the existence of decreasing proper solutions of (A) is discussed. First, it is shown that the necessary and sufficient condition for the existence of a decreasing proper solution of (A) which remains positive as long as it exists and tends to a given positive constant $\omega_0$ as $t \to \infty$ is the same as the condition characterizing the existence of bounded increasing proper solutions for (A). Next, we establish conditions guaranteeing the existence of positive proper solutions which decay to zero at infinity like a positive constant multiple of the function $t^{(\alpha+\sigma+1)/(\alpha-\beta)}$ for some $\sigma < -1$ where $(\alpha+\sigma+1)/(\alpha-\beta) < 0$. We end Section 5 with the existence result characterizing the situation in which Eq. (A) (with $0 < \beta < \alpha$) possesses an eventually negative decreasing proper solution emanating from a given point $(a, y_0), y_0 > 0$, with specific asymptotic behavior as $t \to \infty$.