Electronic Journal of Qualitative Theory of Differential Equations

We consider the second order differential equation ( a(t)|x′|α sgn x′ )′ + b(t)|x| sgn x = 0 in the super-linear case α < β. We prove the existence of the so-called intermediate solutions and we discuss their coexistence with other types of nonoscillatory and oscillatory solutions. Our results are new even for the Emden–Fowler equation (α = 1).

In this paper, by a solution of (1.1) we mean a function x, defined on some ray [τ x , ∞), τ x ≥ 0, such that its quasiderivative x [1] , i.e. the function x [1] (t) = a(t)|x (t)| α sgn x (t), (1.3) is continuously differentiable and satisfies (1.1) for any t ≥ τ x . Since α < β, the initial value problem associated to (1.1) has a unique local solution, that is, a solution x such that x(t) = x 0 , x (t) = x 1 for arbitrary numbers x 0 , x 1 and any t ≥ 0. Moreover, in view of the regularity of the functions a, b, any local solution of (1.1) is continuable to infinity, see, e.g., [27,Section 3] or [18,Theorem 9.4].
As usual, a solution x of (1.1) is said to be nonoscillatory if x(t) = 0 for large t and oscillatory otherwise. Equation (1.1) is said to be nonoscillatory if any solution is nonoscillatory. Since α < β, nonoscillatory solutions of (1.1) may coexist with oscillatory ones, while this fact is impossible when α = β, see, e.g., [18,Chapter III,Section 10]. Set In view of (1.2), any eventually positive solution of (1.1) is nondecreasing for large t. Moreover, the class S of all eventually positive solutions of (1.1) can be divided into three subclasses, according as their asymptotic growth at infinity, see, e.g., [8]. More precisely, any solution x ∈ S satisfies one of the following asymptotic properties: where c x is a positive constant depending on x. Let x, y, z ∈ S satisfy (1.4), (1.5), (1.6), respectively. Then we have for large t x(t) < y(t) < z(t).
Hence, the interesting question arises: can these three types of nonoscillatory solutions of (1.1) simultaneously coexist? This problem has a long history. For equation it started sixty years ago by Moore-Nehari [21] in which it is proved that this triple coexistence is impossible and intermediate solutions of (1.7) cannot coexist with dominant solutions or subdominant ones. For the more general equation (1.1), this study was continued in the nineties and recently, see, e.g., [3,4,8,22] and completely solved in [7,Corollary 4]. Our goal here is to study the existence of intermediate solutions to equation (1.1) when the function is monotone for large t. To more understand the meaning on this assumption in the theory of oscillation to (1.1), let us consider the prototype Jasný [10] and Kurzweil [16] have proved that if for large t the function then every solution x of (1.10), with x(t 0 ) = 0 and |x (t 0 )| sufficiently large, t 0 ≥ 0, is oscillatory. Observe that for equation (1.10) we have γ = (λ + 3)/2 and the function F in (1.8) reads as F 1 . Moore and Nehari [21] have posed the question as to whether it is possible the coexistence of oscillatory solutions with nonoscillatory solutions having at least one zero. Kiguradze [13] negatively answered this question, by proving that: if F 1 is nondecreasing for t ≥ T and lim t→∞ F 1 (t) = ∞, then every solution of (1.10), with a zero at some τ ≥T, is oscillatory.
Later on, other criteria for the existence of an oscillatory solution to (1.10) under the assumption (1.11) are given by Kiguradze [14], Coffman and Wong [5] and Heidel and Hinton [9]. The sharpness of the monotonicity condition for t (λ+3)/2 b(t) has been noticed by Kiguradze [13], see also Nehari [25], which have shown that (1.10) does not have (nontrivial) oscillatory solutions if b(t)(t ln t) (λ+3)/2 is nonincreasing. Finally, concerning the nonoscillation of (1.10), Kiguradze in [13] proved that: if the function t ε F 1 (t) is nonincreasing for large t and some ε > 0, then (1.10) is nonoscillatory. For more details on these topics, we refer to the monographs [1,15] and to the survey [27].
Sufficient conditions for the existence of intermediate solutions in the case here considered, can be found in [24], where the special equation (1.7) is considered when, roughly speaking, the function b is close to the function t −ν , ν > 0 or in [23], in which the equation (1.1) is considered with a ≡ 1 and b(t) = kt −µ (1 + η(t)) for t ≥ t 0 > 0, where µ, k are positive constants and η is a continuous function such that 1 + η(t) > 0 for t ≥ t 0 .
In this paper, we present two existence results for intermediate solutions, according to the function F is for large t either nonincreasing on nondecreasing, respectively. The proof of the first result is based on certain monotonicity properties of an energy-type function, jointly with a suitable transformation. The second one is proved by using a topological limit process which enables, to obtain intermediate solutions of (1.1) as the limit of a suitable sequence of subdominant solutions. This second criterion improves an analogous one in [7,Theorem 5]. Finally, we study the claimed question in [21] on the possible coexistence between oscillatory and nonoscillatory solutions, too. This study is achieved by using the obtained existence results, jointly with some known results, which are analogue ones of Kurzweil and Kiguradze oscillation criteria. Some examples complete the paper.

Existence of intermediate solutions
Using certain monotonicity properties of an energy-type function, jointly with a suitable transformation we have the following existence results.
and γ > 1 we have Similarly, as we show below in the proof of Theorem 2.2, the assumption (2.3) implies Y = ∞. Remark 2.4. Theorem 2.2 extends Corollary 3 and Theorem 5 of [7] where it is assumed (2.1) and an additional assumption needed for the topological limit process.
First, we prove both theorems for the particular case in which a ≡ 1, that is for the equation Later on, we extend the result to (1.1) by means of a suitable change of the independent variable. Observe that for (2.4), the integral Y is For any solution x of (2.4), define the energy function where x [1] is defined in (1.3) and . (2.6) The following lemmas are needed for proving Theorem 2.1.
Thus, in view of (2.4), we obtain the assertion. Lemma 2.6. Assume that where γ is given by (1.9). Then for any solution x of (2.4) we have for t ≥ T Proof. From (2.4) and (2.8) we obtain or, in view (2.6) and Lemma 2.5 Using the identity and (2.9), the assertion follows. Proof. Let x be an oscillatory solution of (2.4) such that x vanishes at some t * > T, that is Then, the assertion follows from Lemma 2.6. Now, let x be a subdominant solution of (2.4). Since lim t→∞ x(t)x [1] (t) = 0 and tx (t)x [1] (t) = t|x (t)| α+1 , the assertion again follows.
Proof of Theorem 2.1. Step 1. We prove the statement for equation (2.4). As already claimed, any solution of (2.4), which is defined in a neighborhood of T > 0, is continuable to infinity, see, e.g., [27].
We show that (2.4) has solutions y for which E y (t) < 0 at some t ≥ T. Put where µ is a parameter such that . (2.11) A standard calculation shows that the point u such that is a point of minimum for ϕ. Moreover, we have In view of (2.11), we have Step 2. Now, we extend the assertion to the more general equation (1.1). The change of the independent variable [11, Section 3] where t(r) is the inverse function of r(t), the function c is given by c(r) = a 1/α (t(r)))b(t(r)) and the symbol˙denotes the derivative with respect to the variable r.  (2.18) where γ is given by (1.9). Then any subdominant solution x ∈ S satisfies on the whole interval [T, ∞) c is a suitable constant which depends on α, β and Proof. Let x ∈ S be a subdominant solution. By a result in [2, Lemma 5 and Theorem 3], see also [7,Theorem 2], we obtain the boundedness of x by ϕ given by (2.20). It remains to prove the boundedness of x by the constant M ϕ given by (2.21). First, we claim that the function is nondecreasing for t ≥ T.
(2.23) Indeed, following [24], we get from (2.4) for t ≥ T where B is given in (2.22), and using this, Now, using (2.23) we have Since x is decreasing on [T, ∞), that is x (t) < x (T) for t > T, we get Hence there exists a uniformly converging subsequence x n j to a function x such that x n j uniformly converges on every finite subinterval of [T, ∞). Clearly, x is an unbounded solution of (2.4). Now, let us show that Y = ∞. In virtue of (2.18) we have for t ≥ T Since β > α, we have Hence, from (2.25) we get Y = ∞. Hence, in virtue of Theorem A, the solution x is an intermediate solution of (2.4). Reasoning as in the proof of Theorem 2.1 -Step 2, we get the assertion for (1.1).

Oscillatory and nonoscillatory solutions
The properties of the function F, given in (1.8), plays an important role also in studying the existence of oscillatory solutions and their coexistence with nonoscillatory ones. We recall the following two results.
Combining these results with Theorem A, Theorem 2.1 and Theorem 2.2, we get the following.

Corollary 3.3.
Assume that the function F, given in (1.8), is nonincreasing for large t ≥ T > 0. Then the following statements hold.
Since A (t) = a −1/α (t), reasoning as in Remark 2.3 we obtain J < ∞. From Theorem A and Theorem 3.1, equation (1.1) has both oscillatory solutions and subdominant solutions. The second assertion follows from Theorem A and Theorem 2.1.
The following examples illustrate our results.
Example 3.6. Consider for t ≥ 0 the equation In this case, as in Example 3.6, we have γ = 5/3 and A(t) = log(t + 2) − log 2. Thus F(t) = (t + 2) −1 and F(t)A ε (t) = (log(t + 2) − log 2) ε t + 2 . is eventually increasing for any ε > 0, the last statement in Corollary 3.3 cannot be applied and so the existence of an oscillatory solution of (3.5) is an open problem.