# On quasi-periodic solutions to a higher-order Emden-Fowler type differential equation

- Irina Astashova
^{1, 2}Email author

**2014**:174

https://doi.org/10.1186/s13661-014-0174-7

© Astashova; licensee Springer. 2014

**Received: **5 February 2014

**Accepted: **30 June 2014

**Published: **25 September 2014

## Abstract

The paper is devoted to the existence of oscillatory and non-oscillatory quasi-periodic, in some sense, solutions to a higher-order Emden-Fowler type differential equation.

## Keywords

## 1 Introduction

The fact of the existence of such solutions answers the two questions posed by IT Kiguradze:

### Question 1

Can we describe more precisely qualitative properties of oscillatory solutions to (1)?

### Question 2

Do all blow-up solutions to this equation (and similarly all Kneser solutions) have the power asymptotic behavior?

A lot of results on the asymptotic behavior of solutions to (1) are described in detail in [1]. In particular (see Ch. IV, §15), the existence of oscillatory solutions to a generalization of this equation was proved (see also [2] Ch. I, §6.1). In [3] a result was formulated on non-extensibility of oscillatory solutions to (1) with odd *n* and ${p}_{0}>0$. In the cases $n=3$ and $n=4$ the asymptotic behavior of all oscillatory solutions is described in [4]–[6]. Some results on the existence of blow-up solutions are in [1] (Ch. IV, §16), [2] (Ch. I, §5), [7], [8]. Some results on the existence of some special solutions to this equation are in [2], [4], [5], [7], [9]–[13].

## 2 On existence of quasi-periodic oscillatory solutions

In this section some results will be obtained on the existence of special oscillatory solutions. The main results of this section were formulated in [14].

### Theorem 1

*For any integer*$n>2$

*and real*$k>1$

*there exists a periodic oscillatory function*

*h*

*such that for any*${p}_{0}>0$

*and*${x}^{\ast}\in \mathbb{R}$

*the function*

### Proof

satisfying the initial conditions ${y}^{(j)}(0)={q}_{j}$ with $j=0,\dots ,n-1$.

For $0\le j<n$ put ${B}_{j}=\frac{nk}{n+j(k-1)}>1$ and ${\beta}_{j}=\frac{1}{{B}_{j}}$.

and satisfying the equality $N(\tilde{N}(q))=1$ for all $q\in {\mathbb{R}}^{n}\setminus \{0\}$.

- (1)
${q}_{0}=0$,

- (2)
${q}_{j}\ge 0$ for all $j\in \{1,\dots ,n-1\}$,

- (3)
$N(q)=1$.

*Q*is a homeomorphism of

*Q*onto the convex compact subset

### Lemma 1

*For any*$q\in Q$*there exists*${a}_{q}>0$*such that*${y}_{q}({a}_{q})=0$*and*${y}_{q}^{(j)}({a}_{q})<0$*for all*$j\in \{1,\dots ,n-1\}$.

### Proof

Put $J=max\{j\in \mathbb{Z}:0\le j<n,{q}_{j}>0\}$. This *J* exists and is positive due to the definition of *Q*. On some interval $(0;\epsilon )$ all derivatives ${y}_{q}^{(j)}(x)$ with $0\le j\le J$ are positive. Those with $J<j\le n$, due to (3), are negative on the same interval.

While keeping this sign combination, the function ${y}_{q}$ and its derivatives are bounded, which provides extensibility of ${y}_{q}(x)$ as the solution to (3) outside the interval $(0;\epsilon )$.

On the other hand, this sign combination cannot take place up to +∞. Indeed, in that case ${y}_{q}(x)$ would increase providing ${y}_{q}^{(n)}(x)<-{y}_{q}{(\epsilon )}^{k}<0$ for all $x>\epsilon $, which is impossible for any positive function on the unbounded interval $(0;+\mathrm{\infty})$.

So, ${y}_{q}(x)$ must change the sign combination of its derivatives. The only possible combination to be the next one corresponds to the positive derivatives ${y}_{q}^{(j)}(x)$ with $0\le j\le J-1$ and the negative ones with $J\le j\le n$.

The same arguments show that the new sign combination must also change and finally, after *J* changes, we arrive at the case with ${y}_{q}(x)>0$ and ${y}_{q}^{(j)}(x)<0$ with $1\le j\le n$. Now, contrary to the previous cases, the function ${y}_{q}(x)$ does not increase, but its first derivative is negative and decreases (recall that $n>2$). Hence this sign combination also cannot take place on an unbounded interval and therefore it must change to the case with all negative ${y}_{q}^{(j)}(x)$, $0\le j\le n$. By the way, the function ${y}_{q}(x)$ must vanish at some point ${a}_{q}>0$, which completes the proof of Lemma 1. □

Note that ${a}_{q}$ is not only the first positive zero of ${y}_{q}(x)$, but the only positive one. Indeed, all ${y}_{q}^{(j)}(x)$ with $0<j<n$ are negative at ${a}_{q}$, whence, according to (3), all ${y}_{q}^{(j)}(x)$ with $0\le j<n$ decrease and are negative for all $x>{a}_{q}$ in the domain of ${y}_{q}(x)$.

is the ${C}^{1}$ ‘solution’ mapping defined on a domain including ${\mathbb{R}}^{n}\times \{0\}$. The necessary for the implicit function theorem condition $\frac{\partial {S}_{0}}{\partial X}({q}_{0},\dots ,{q}_{n-1},{a}_{q})\ne 0$ is satisfied since the left-hand side of the last inequality is equal to ${y}_{q}^{\prime}({a}_{q})<0$. Besides, any function $X(q)$ implicitly defined near a point $({q}_{0},{a}_{{q}_{0}})$ must be positive in some its neighborhood. Hence locally $X(q)$ must be equal to $\xi (q)$, but neither to a non-positive zero of ${y}_{q}(x)$ nor to a non-first positive one, which does not exist. Hence the function $\xi (q)$ is continuous as well as $X(q)$.

Now we can consider the mapping $\tilde{S}:q\mapsto \tilde{N}(-S(q,\xi (q)))$, which maps *Q* into itself. Since $\tilde{S}$ is continuous and *Q* is homeomorphic to a convex compact subset of ${\mathbb{R}}^{n-2}$, the Brouwer fixed-point theorem can be applied. Thus, there exists $\stackrel{\u02c6}{q}\in Q$ such that $\tilde{S}(\stackrel{\u02c6}{q})=\stackrel{\u02c6}{q}$.

*S*, and

*ξ*, this yields the result that there exists a non-negative solution $\stackrel{\u02c6}{y}(x)={y}_{\stackrel{\u02c6}{q}}(x)$ to (3) defined on a segment $[0;{a}_{1}]$ with ${a}_{1}={a}_{\stackrel{\u02c6}{q}}$, positive on the open interval $(0;{a}_{1})$, and such that

*c*is also a solution to (7). Indeed, we have $\alpha +n=k\alpha $ and ${y}_{2}^{(j)}(x)=-{b}^{\alpha +j}{y}_{1}^{(j)}(bx+c)$ for all $j=0,\dots ,n$, whence

So, the function $z(x)=-{b}^{\alpha}\stackrel{\u02c6}{y}(bx-{a}_{1}b)$ is a solution to (7) and is defined on the segment $[{a}_{1};{a}_{2}]$ with ${a}_{2}={a}_{1}+\frac{{a}_{1}}{b}$.

*λ*defined by (6). Then

where $x\in [{a}_{s};{a}_{s+1}]$ and hence $b(x-{a}_{s})+{a}_{s-1}\in [{a}_{s-1};{a}_{s}]$.

Now we will investigate whether *b* is greater or less than 1.

### Lemma 2

*In the above notation the solution*$y(x)=\stackrel{\u02c6}{y}(x)$

*satisfies the following inequalities*:

### Proof

since ${y}^{\prime}({a}_{1,s})={y}^{(n-1)}({a}_{n-1,s+1})=0$ and ${y}^{\u2033}(x){y}^{(n-1)}(x)>0$ on the interval $({a}_{1,s};{a}_{n-1,s+1})$, where only $y(x)$ itself changes its sign, while all other ${y}^{(j)}(x)$ with $0<j<n$ keep the same one. Recall that $n>2$, which makes ${y}^{\u2033}(x)$ to be one of these others. Inequality (9) is proved.

Inequality (10) follows from $y(x){y}^{\prime}(x)>0$ on the interval $({a}_{j+1,s},{a}_{j,s})$, where the derivatives ${y}^{(j)}(x)$ and ${y}^{(j+1)}(x)$ with $0<j<n-1$ keep different signs, while all lower-order derivatives keep the same sign as ${y}^{(j)}(x)$. □

From the lemma proved it follows that $|\stackrel{\u02c6}{y}({a}_{1,s})|<|\stackrel{\u02c6}{y}({a}_{1,s+1})|={b}^{\alpha}|\stackrel{\u02c6}{y}({a}_{1,s})|$, whence it follows that $b>1$ and ${a}_{s}-{a}_{s-1}=b({a}_{s+1}-{a}_{s})>{a}_{s+1}-{a}_{s}$.

So, $h(t+logb)=-h(t)$ for all $t\in \mathbb{R}$ and hence the function $h(t)$ is periodic with period $2logb$.

Now, according to (11), we can express the solution $\stackrel{\u02c6}{y}(x)$ to (7) just as $\stackrel{\u02c6}{y}(x)={({a}^{\ast}-x)}^{-\alpha}h(log({a}^{\ast}-x))$. Multiplying it by ${p}_{0}^{\frac{1}{k-1}}$ we obtain a solution to (3) having the form needed. It still will be a solution to (3) after replacing ${a}^{\ast}$ by arbitrary ${x}^{\ast}\in \mathbb{R}$. □

The substitution $x\mapsto -x$ produces the following.

### Corollary 1

*For any integer*$n>2$

*and real*$k>1$

*there exists a periodic oscillatory function*

*h*

*such that for any*${p}_{0}\in \mathbb{R}$

*satisfying*${(-1)}^{n}{p}_{0}>0$

*and any*${x}^{\ast}\in \mathbb{R}$

*the function*

*is a solution to* (1).

Note that the following theorem was earlier proved in [4], [5].

### Theorem 2

*For*$n=3$,

*there exists a constant*$B\in (0,1)$

*such that any oscillatory solution*$y(x)$

*to*(1)

*with*${p}_{0}<0$

*satisfies the conditions*

*for some*$M>0$*and*${x}_{\ast}$, *where*${x}_{1}<{x}_{2}<\cdots <{x}_{i}<\cdots $*and*${x}_{1}^{\prime}<{x}_{2}^{\prime}<\cdots <{x}_{i}^{\prime}<\cdots $*are sequences satisfying*$y({x}_{j})=0$, ${y}^{\prime}({x}_{j}^{\prime})=0$, $y(x)\ne 0$*if*$x\in ({x}_{i},{x}_{i+1})$, ${y}^{\prime}(x)\ne 0$*if*$x\in ({x}_{i}^{\prime},{x}_{i+1}^{\prime})$.

With the help of this theorem, another one can be proved, namely the following.

### Theorem 3

*For*$n=3$*and any real*$k>1$*there exists a periodic oscillatory function* *h* *such that the functions*$y(x)={p}_{0}^{\frac{1}{k-1}}{|x-{x}_{\ast}|}^{-\alpha}h(log|x-{x}_{\ast}|)$*with*$\alpha =\frac{n}{k-1}$*and arbitrary*${x}_{\ast}$*are solutions*, *respectively*, *to* (1) *with*${p}_{0}<0$*if defined on*$(-\mathrm{\infty};{x}_{\ast})$*and to* (1) *with*${p}_{0}>0$*if defined on*$({x}_{\ast};+\mathrm{\infty})$.

## 3 On existence of positive solutions with non-power asymptotic behavior

*N*and $K>1$ there exist an integer $n>N$ and $k\in \mathbf{R}$ such that $1<k<K$ and (1) has a solution of the form

where $\alpha =\frac{n}{k-1}$ and *h* is a positive periodic non-constant function on **R**.

*i.e.*those satisfying $y(x)\to 0$ as $x\to \mathrm{\infty}$ and ${(-1)}^{j}{y}^{(j)}(x)>0$ for $0\le j<n$. Namely, if ${p}_{0}={(-1)}^{n-1}$, then for any

*N*and $K>1$ there exist an integer $n>N$ and $k\in \mathbf{R}$ such that $1<k<K$ and (1) has a solution of the form

where *h* is a positive periodic non-constant function on **R**.

Still it was not clear how large *n* should be for the existence of that type of positive solutions.

### Theorem 4

[13]

*If*$12\le n\le 14$,

*then there exists*$k>1$

*such that*(1)

*with*${p}_{0}=-1$

*has a solution*$y(x)$

*such that*

*where*$\alpha =\frac{n}{k-1}$*and*${h}_{j}$*are periodic positive non*-*constant functions on* **R**.

### Remark

Computer calculations give approximate values of *α*. They are, with the corresponding values of *k*, as follows:

if $n=12$, then $\alpha \approx 0.56$, $k\approx 22.4$;

if $n=13$, then $\alpha \approx 1.44$, $k\approx 10.0$;

if $n=14$, then $\alpha \approx 2.37$, $k\approx 6.9$.

### Corollary 2

*If*$12\le n\le 14$,

*then there exists*$k>1$

*such that*(1)

*with*${p}_{0}={(-1)}^{n-1}$

*has a Kneser solution*$y(x)$

*satisfying*

*with periodic positive non*-*constant functions*${h}_{j}$*on* **R**.

## 4 Conclusions, concluding remarks, and open problems

- 1.
So, we give the negative answer to Question 1 and prove the existence of oscillatory solutions with special qualitative properties for Question 2.

- 2.
It would be interesting to know if positive solutions like (12) exist for $n\ge 15$ and for $5\le n\le 11$.

- 3.
If a positive solution like (12) exists for some ${k}_{0}>1$, does it follow, for the same

*n*, that such solutions exist for all $k>{k}_{0}$?

## Declarations

### Acknowledgements

The research was supported by RFBR (grant 11-01-00989).

## Authors’ Affiliations

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