Mem. Differential Equations Math. Phys. 20(2000), 149–153

V. A. Rabtsevich

ON NONINCREASING SINGULAR SOLUTIONS OF THE EMDEN–FOWLER EQUATION

(Reported on May 15, 2000)

Consider the Emden–Fowler equation

u^{(n)} = (−1)^{n}p(t)|u|^{λ}sign u, p(t) ≥ 0, λ ∈ (0, 1), n ≥ 2,  (1)

with the coefficient p locally Lebesgue integrable on (a, b) and differing from zero on a set of positive measure in any left neighborhood of b.

A solution u : [a, b) → (0, +∞) of the equation (1) is said to be a Kneser singular solution of the first kind, if

(−1)^{i}u^{(i)}(t) > 0 (i = 0, …, n − 1) for t ∈ (a, b), lim_{t→b} u(t) = 0.  (2)

The problem (1), (2) is studied in [1, 2]. There, in particular, there are obtained the sufficient condition

J(a, b) < +∞, where J(s, t) ≡ ∫_{s}^{t} p(τ)(b − τ)^{(n−1)λ}dτ  (3)

for solvability of the problem and an associated upper asymptotic estimate of its solutions

|u^{(n−1)}(t)| ≤ [(1 − λ)J(t, b)/(n − 1)!]^{1/(1−λ)}.  (4)

The conversion of (1), (2) to a similar problem on proper nonoscillatory Kneser solutions vanishing at infinity, which is more studied [1, 3], allows one to get necessary conditions of solvability and lower asymptotic estimates of solutions. In the same manner it is easy to conclude on the basis of [4, 5], that (3) is not a necessary condition.

In the present report we use a direct approach to the problem under consideration, which helps us to simplify proofs and to find out specific properties of solutions. Here new necessary conditions of solvability of the problem (1), (2) and lower asymptotic estimates of its solutions are obtained. Here also the necessity of the condition (3) is established in certain cases.

Begin with a simple Lemma which is basic in the subsequent discussion.

Lemma 1. Let u(t) be a solution of the problem (1), (2). Then the auxiliary functions

v_{i}(t) = |u^{(i)}(t)|(b − t)^{i−n+1} (i = 0, …, n − 1)

increase on (a, b) and tend to zero as t → b.

2000 Mathematics Subject Classification. 34C11.

Key words and phrases. Emden–Fowler equation, nonincreasing singular solution.

Proof. Let u be a solution of the problem (1), (2). Then for all i = 0, …, n − 1 the equalities v̇_{i}(t) = (b−t)^{i−n}R_{i}(t) hold, where R_{i}(t) = (n−i−1)|u^{(i)}(t)|−(b−t)|u^{(i+1)}(t)| are continuous on (a, b) functions satisfying there R_{i}(b) = 0, R′_{i}(t) = −R_{i+1}(t) (i = 0, …, n − 2), R_{n−1}(t) = −(b − t)|u^{(n)}(t)| ≤ 0. Thus in view of the representations

R_{i}(t) = ∫_{t}^{b} (τ − t)^{n−i−2}R_{n−1}(τ)dτ/(n − i − 2)! (i = 0, …, n − 2)

we see that R_{i}(t) < 0 on (a, b) for all i.

Corollary 1. Every solution of the problem (1), (2) satisfies the inequalities

|u^{(i+1)}(t)|(b − t) > (n − i − 1)|u^{(i)}(t)| (i = 0, …, n − 2), t ∈ (a, b).  (6)

The proof of the next Lemma is a matter of easy calculation.

Lemma 2. Let u be a solution of the problem (1), (2) and let φ(t) > 0 be a nondecreasing function on (a, b). Then for all t ∈ (a, b)

− \frac{v̇_{φ,n−1}(t)}{v_{φ,n−1}(t)} = \frac{p(t)(b − t)^{(n−1)λ}v_{φ,0}^{λ}(t)}{φ^{1−λ}(t)v_{φ,n−1}(t)} + \frac{φ̇(t)}{φ(t)} ,

\frac{v̇_{φ,i}(t)}{v_{φ,i+1}(t)} ≤ − \frac{φ_{M} (t)}{n − i − 1} ,

where v_{φ,i}(t) = v_{i}(t)/φ(t), φ_{M} (t) = min{(b − t)^{−1}, φ̇(t)/(Mφ(t))}.

The main result of the report contains the following

Theorem 1. Let u be a solution of the problem (1), (2) and let φ : (a, b) → (0, +∞) be any nondecreasing function. Then for any numbers ν ∈ [0, 1), μ ∈ ((1−ν)/n, (1−ν)/n_{1}), M > 0 and σ > 0 the equality

lim_{t→b} F_{ν,μ,σ,M} (φ)(t) = 0  (7)

is fulfilled and the upper estimate

|u^{(n−1)}(t)| > γ[F_{ν,μ,σ,M} (φ)(t)]^{1/((1−λ)μ)}  (8)

is true, where

F_{ν,μ,σ,M} (φ)(t) = φ^{σ}(t) ∫_{t}^{b} \frac{(p(τ)(b − τ)^{(n−1)λ})^{μ}}{φ^{σ}(τ)φ_{M}^{μ+ν−1}(τ)} (\frac{φ̇(τ)}{φ(τ)})^{ν}dτ,

n_{1} = 1 + (n − 1)λ and γ > 0 depends only on n, λ, μ, ν.

Proof. In accordance with the given values of μ and ν define the numbers μ_{i} by the equalities μ_{n} = μ, μ_{n+1} = ν,

μ_{i} = \{ \begin{matrix}μ − (n − i)ε_{1}, ε_{1} = \frac{2(ν + nμ − 1)}{n(n − 1)} , & μ ∈ (\frac{1 − ν}{n} , \frac{2(1 − ν)}{n + n_{1}}) \\ λμ + iε_{1}, ε_{1} = \frac{2(1 − ν − n_{1}μ)}{n(n − 1)} , & μ ∈ (\frac{2(1 − ν)}{n + n_{1}} , \frac{1 − ν}{n_{1}} ) \end{matrix}

for i = 1, …, n. It is clear that these numbers satisfy the conditions

μ_{i+1} − μ_{i} ≥ ε_{1} > 0 (i = 1, …, n − 1), μ_{1} − λμ_{n} ≥ ε_{1}, ∑_{i=1}^{n+1} μ_{i} = 1.  (9)

Assume u is a solution of the problem (1), (2) and let φ(t) > 0 be a nondecreasing function on (a, b). Then by Lemma 2 and the well known inequality

∑_{i=1}^{n} β_{i}x_{i} ≥ ∏_{i=1}^{n} x_{i}^{β_{i}}, x_{i} > 0, β_{i} ≥ 0, ∑_{i=1}^{n} β_{i} = 1

we obtain for the derivative of the function ω_{φ}(t) = ∏_{i=0}^{n−1} v_{φ,i}(t)

\frac{|ω̇_{φ}(t)|}{ω_{φ}(t)} = ∑_{i=0}^{n−1} \frac{|v̇_{φ,i}(t)|}{v_{φ,i}(t)} > \frac{p(t)(b − t)^{(n−1)λ}v_{φ,0}^{λ}(t)}{φ^{1−λ}(t)v_{φ,n−1}(t)} + φ_{M} (t) ∑_{i=0}^{n−2} \frac{v_{φ,i+1}(t)}{v_{φ,i}(t)} +

+ \frac{φ̇(t)}{φ(t)} ≥ γ(\frac{p(t)(b − t)^{(n−1)λ}v_{φ,0}^{λ}(t)}{φ^{1−λ}(t)v_{φ,n−1}(t)})^{μ_{n}} ∏_{i=0}^{n−2} (\frac{v_{φ,i+1}(t)}{v_{φ,i}(t)} φ_{M} (t))^{μ_{i+1}} ×

×(\frac{φ̇(t)}{φ(t)})^{ν} = γ(\frac{φ̇(t)}{φ(t)})^{ν} \frac{((p(t)(b − t)^{(n−1)λ})^{μ}v_{φ,0}^{λμ−μ_{1}}(t)}{φ^{1−λ}(t)(φ_{M} (t))^{μ+ν−1}} ∏_{i=1}^{n−1} v_{φ,i}^{μ_{i}−μ_{i+1}}(t).

Multiplying the above inequality by ω_{φ}^{ε}(t)φ^{−δ}(t) with positive ε < min{ε_{1}, σ/n}, δ = σ − nε and integrating the result, we get

(γε)^{−1}w_{φ}^{ε}(t)φ^{−δ}(t) ≥ γ^{−1} ∫_{t}^{b} ω̇_{φ}(s)ω_{φ}^{−1+ε}(s)φ^{−δ}(s)ds ≥

≥ ∫_{t}^{b} \frac{(p(x)(b − x)^{(n−1)λ}φ^{λ−1})^{μ}}{φ^{δ}(x)φ_{M}^{μ+ν−1}(x)} (\frac{φ̇(x)}{φ(x)})^{ν}v_{φ,0}^{λμ_{n}−μ_{1}+ε}(x) ∏_{i=1}^{n−1} v_{φ,i}^{μ_{i}−μ_{i+1}+ε}(x)dx,

where γ depends on n, λ, μ, ν. From here by Lemma 1, (6) and (9) we get

v_{n−1}^{(1−λ)μ}(t) ≥ v_{0}^{μ_{1}−λμ_{n}}(t) ∏_{i=1}^{n−1} v_{i}^{μ_{i+1}−μ_{i}}(t) ≥ F_{ν,μ,σ,M} (φ)(t)

which completes the proof.

By means of this theorem it is easy to find the case where the sufficient condition (3) turns out to be a necessary one as well.

Theorem 2. Let the equation (1) with the coefficient p satisfying

p(t)(b − t)^{n_{1}} < c(J(a, t) + 1), c > 0,  (10)

on (a, b) have a solution of the type (2). Then the condition (3) must hold and for all μ ∈ (0, n_{1}^{−1}) the estimate

|u^{(n−1)}(t)| > γJ^{1/(μ(1−λ))}(t, b), where γ = γ(n, λ, μ),  (11)

is fulfilled.

If along with (3)

p(t)(b − t)^{n_{1}} < cJ(t, b) for t ∈ (a, b), c > 0,  (12)

holds, then there takes place the two-sided estimate

0 < γ_{1} < |u^{n−1}(t)|J^{1/(λ−1)}(t, b) < γ_{2}.  (13)

Here γ_{1} and γ_{2} depend only on n, λ, μ.

Proof. Assume that u is a solution of the problem (1), (2), and let (10) hold. Then, putting φ(t) = J(a, t) + 1, we have for σ > μ

F_{ν,μ,σ,c}(φ)(t) = φ^{σ}(t) ∫_{t}^{b} φ̇(x)φ^{μ−σ−1}(x)dx = φ^{μ}(t)/(σ − μ)

which contradicts the conclusion of Theorem 1. Consequently, (3) must be fulfilled and φ(t) is bounded. Owing to this fact we obtain F_{ν,μ,σ,c}(φ)(t) > γJ(t, b) on (a, b), which by Theorem 1 yields (11).

To prove (13) we can use (4) and (8) with φ(t) = 1/J(t, b) because

F_{ν,μ,σ,c}(φ)(t) = J^{−σ}(t, b) ∫_{t}^{b} p(x)(b − x)^{(n−1)λ}J^{μ+σ−1}(x, b)dx = γJ^{μ}(t, b).

The theorem is proved.

Corollary 2. The problem (1), (2) with the function p satisfying 0 < c_{1} < p(t)(b − t)^{(n−1)λ} ln_{k}^{σ}(1/(b − t))l_{k}(1/(b − t)) < c_{2}, k ≥ 0 on (a, b) has a solution if and only if σ > 0 and every its solution admits the estimate 0<γ_{1} <|u^{(n−1)}(t)| ln_{k}^{σ/(1−λ)}(1/(b − t))

< γ_{2}, where ln_{0} t = t, ln_{j+1} t = ln(ln_{j} t), l_{j}(t) = ∏_{i=0}^{j} ln_{i} t and γ_{1}, γ_{2} depend only on n,

λ, σ.

In the general case where (3) is not assumed to be fulfilled we will introduce into consideration the nonnegative functions p_{f∗}(t) = min{p(t), f(t)(b − t)^{n_{1}}}, p_{f}^{∗}(t) = p(t) − p_{f∗}(t) and the integrals

J_{f∗}(s, t) = ∫_{s}^{t} \frac{p_{f∗}(x)(b − x)^{(n−1)λ}}{f(x)} dx, J_{f}^{∗}(s, t) = ∫_{s}^{t} \frac{p_{f}^{∗}(x)(b − x)^{(n−1)λ}}{f(x)} dx,

where f(x) is an arbitrary nondecreasing positive function.

Theorem 3. If the equation (1) has a solution u of the type (2), then for an arbitrary nondecreasing positive function f and all μ ∈ (0, 1/n_{1}) there hold J_{f∗}(a, b) < +∞, lim_{t→b} f^{μ}(t)J_{f,∗}(t, b) = 0 and in some neighborhood of b the estimate

|u^{(n−1)}(t)| > γ(f(t)J_{f∗}^{1/μ}(t, b))^{1/(1−λ)}

is true.

If in addition p_{f∗}(t)(b − t)^{n_{1}} < cf(t)J_{f∗}(t, b) for t ∈ (a, b), then |u^{(n−1)}(t)| > γ(f(t)J_{f∗}(t, b))^{1/(1−λ)}. The constant γ depends only n, λ, μ.

Corollary 3. If the problem (1), (2) is solvable with p satisfying

J_{f} (t, b) = ∫_{t}^{b} \frac{p(x)(b − x)^{(n−1)λ}}{f(x)} dx = +∞ for t < b,

where f(x) > 0 is an arbitrary nondecreasing function, then

\overline{lim}_{t→b} p_{f}^{∗}(t)(b − t)^{(n−1)λ}/f(t) = +∞, J_{f}^{∗}(t, b) = +∞.

Theorem 4. If the equation (1) with the function p satisfying the condition

p_{f∗}(t)(b − t)^{n_{1}} > cf(t)J_{f∗}(t, b) for t ∈ (a, b),

where f(x) increases on (a, b), has a solution u of the type (2), then the estimate

|u^{(n−1)}(t)| > γ(f(t)J_{f∗}^{(1−ν)/μ}(t, b))^{1/(1−λ)},

holds, where γ depends only on n, λ, μ.

Theorem 5. If the problem (1), (2) with p satisfying p_{f}^{∗}(t)(b − t)^{n_{1}} > cf(t)J_{f}^{∗}(a, t) on (a, b), where f(x) increases on (a, b), has a solution u of the type (2), then

|u^{(n−1)}(t)| > γ(f(t)J_{f∗}(a, t)( ∫_{t}^{b} x^{−1}sgnp(x)dx)^{1/μ})^{1/(1−λ)},

where γ = γ(n, λ, μ) > 0.

References

1. I. T. Kiguradze and T. A. Chanturia, Asymptotic properties of solutions of nonautonomous nonlinear ordinary differential equations. (Russian) Nauka, Moscow. 1990.

2. G. G. Kvinikadze, On singular solutions of the first kind of nonlinear ordinary differential equations. (Russian) Dokl. Rasshir. Zased. Sem. Inst. Prikl. Mat. I. N. Vekua 3(1988), No. 3, 66–70.

3. N. A. Isobov and V. A. Rabtsevich, On two Kiguradze problems for the Emden– Fowler equations. (Russian) Proc. Inst. Mat. Mn. 2(1999), 73–91.

4. N. A. Isobov, On Knezer solutions. (Russian) Differentsial’nye Uravnenija 21(1985), No. 4, 581–585.

5. N. A. Isobov and V. A. Rabtsevich, On vanishing Knezer solutions of the Emden–Fowler equation. (Russian) Differentsial’nye Uravnenija 26(1990), No. 4, 578– 585.

Author’s address:

Institute of Mathematics

Belorussian Academy of Sciences

11, Surganova St., Minsk 220072

Belarus