Mem. Differential Equations Math. Phys. 21(2000), 146–150

V. A. Rabtsevich

ON NONDECREASING SINGULAR SOLUTIONS OF THE EMDEN–FOWLER EQUATION

(Reported on May 22, 2000)

Consider the Emden–Fowler equation

u^{(n)} = p(t)|u|^{λ}signu, p(t) ≥ 0, λ > 1, n ≥ 2,  (1)

with a locally Lebesgue integrable on (a, b) function p(t), which differs 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 nonoscillatory singular solution of the second kind, if

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

Problems on the existence of such solutions and on its asymptotic estimation were studied for the equation (1) in [1, p. 323–325], where they are reduced to the similar ones for a proper strongly increasing solutions, which are more investigated (see [1, 2] and the bibliography therein). In particular, this approach allows to give the sufficient condition

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

for the equation (1) to have a solution (2).

In this paper new necessary conditions of solvability of the problem (1)–(2) and twosided asymptotic estimates of its solutions are obtained. Here also it is established the necessity of the condition (3) in certain cases.

Begin with a simple assertion which presents some important properties of solutions of the problem under consideration.

Lemma 1. Let u(t) be a solution of the problem (1), (2) and φ(t) > 0 be a non-

decreasing function on (a, b). Then v_{i}(t) = ∑_{l=0}^{i} u^{(l)}(t)(b − t)^{l}/l! (i = 0, …, n − 1) are

nondecreasing unbounded functions on (a, b), satisfying there conditions

\frac{v̇_{φ,n−1}(t)}{v_{φ,n−1}(t)} = \frac{p(t)(b − t)^{n−1}v_{φ,0}^{λ}(t)}{(n − 1)!φ^{λ−1}(t)v_{φ,n−1}(t)} + \frac{φ̇(t)}{φ(t)} , \frac{v̇_{φ,i}(t)}{v_{φ,i+1}(t)} ≥ φ_{M} (t),  (4)

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

The lower asymptotic estimate of solutions of the problem (1), (2) gives

2000 Mathematics Subject Classification. 34C05 Key words and phrases. Emden-Fowler equation, nondecreasing singular solution.

Theorem 1. Every solution u of the problem (1), (2) admits the lower estimate

v_{n−1}(t) > ((n − 1)!(λ − 1)^{−1}J(t, b))^{1/(1−λ)} , t ∈ (a, b),  (5)

provided the condition (3) holds.

Proof. Let u be a solution of the problem (1), (2). From (1) we deduce v̇_{n−1}(t) = u^{(n)}(t)(b − t)^{n−1}/(n − 1)! ≥ p(t)(b − t)^{n−1}v_{n−1}^{λ}(t)/(n − 1)! for t ∈ (a, b). Integrate this inequality taking into acount (2):

\frac{v_{n−1}^{1−λ}(t)}{λ − 1} = ∫_{t}^{b} \frac{v̇_{n−1}(τ)dτ}{v_{n−1}^{λ}(τ)} < ∫_{t}^{b} \frac{p(τ)(b − τ)^{n−1}dτ}{(n − 1)!} .

The last estimate is equivalent to (5). Thus the theorem is proved.

The main result of this article is contained in the following

Theorem 2. Let u(t) 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}, (1 − ν)/n), M > 0 and σ > 0 the equality

lim_{t→b} F_{ν,μ,σ,M} (φ)(t) = 0  (6)

is true and the upper estimate

u(t) < γ[F_{ν,μ,σ,M} (φ)(t)]^{1/((1−λ)μ)},  (7)

is fulfilled, where

F_{ν,μ,σ,M} (φ)(t) = φ^{σ}(t) ∫_{t}^{b} \frac{(p(τ)(b − τ)^{n−1})^{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. Let u be a solution of the problem (1), (2) and φ(t) > 0 be a nondecreasing function on (a, b). By lemma 1 and the 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,

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

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

where the numbers μ_{i} are defined by the equalities μ_{n} = μ, μ_{n+1} = ν,

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

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

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

Dividing (8) by ω_{φ}^{ε}(t)φ^{δ}(t) with ε < min{ε_{1}, σ/n}, δ = σ − nε and integrating the result, we get

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

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

Hence by Lemma 1 and (9) it follows (7) and (8). The theorem is proved.

The following part of the report contains some applications of the above results, where they are applied to analysis of the condition (3) and its natural extensions.

Theorem 3. If the equation (1) with the function p(t) sutisfying

p(t)(b − t)^{n} < cJ(a, t) for t ∈ [t_{c}, b) ⊂ (a, b), c > 0,  (10)

has a solution of the type (2), then the condition (3) holds and in some left neighborhood of b

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

If along with (3)

p(t)(b − t)^{n} < cJ(t, b) for t ∈ [t_{c}, b) ⊂ (a, b), c > 0  (12)

holds then there takes place the two-sided estimate

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

where γ_{1} and γ_{2} depend on n, λ.

Proof. Let the equation (1) with the function p(t) satisfying (10) has a solution of the type (2). Suppose to the contrary that (3) is not true. Then the function φ(t) = J(a, t) increases with no bound on (a, b) and by (10) (b−t)φ_{c}(t) = p(t)(b−t)^{n}/J(a, t) < 1 holds.

Therefore, for any σ > μ we have the equality F_{0,μ,σ,M} (φ)(t) = J^{σ}(a, t) ∫_{t}^{b}(p(x)(b −

x)^{n−1}J^{μ−σ−1}(a, x)dx = J^{μ}(a, t)/(μ − σ), which contradicts the conclusion of Theorem 2. This means the validity of (3) and the boundness of φ(t). In view of this fact we obtain F_{0,μ,σ,M} (φ)(t) > γJ(t, b) for t > t_{γ} > a, which by Theorem 2 yields (11).

Now assume that (3) and (12) are fulfilled. Then φ(t) = 1/J(t, b) increases with no bound on (a, b) and for any μ, σ > 0

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

holds from which by Theorem 2 there immediatelly follows the estimate (13). The theorem is proved.

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

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

In the general case it is useful to introduce into consideration the nonnegative functions p_{f∗}(t) = min{p(t), f(t)(b − t)^{−n}}, 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}}{f(x)} dx, J_{f}^{∗}(s, t) = ∫_{s}^{t} \frac{p_{f}^{∗}(x)(b − x)^{n−1}}{f(x)} dx,

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

Theorem 4. If the equation (1) has a soluttion u of the type (2), then for an arbitrary nondecreasing positive function f(t) and all μ ∈ (0, 1/n) J_{f∗}(a, b) < +∞, lim_{t→b} f^{μ}(t)J_{f,∗}(t, b) = 0 and in some neighborhood of b the estimate

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

is true.

If in addition p_{f∗}(t)(b−t)^{n} < cf(t)J_{f∗}(t, b) for t ∈ (a, b), holds then there takes place the estimate u(t) < γ(f(t)J_{f∗}(t, b))^{1/(1−λ)}.

Corollary 2. If the problem (1), (2) with the function p(t), satisfying

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

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

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

Theorem 5. Let the equation (1) with the function p(t) satisfying the condition

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

holds, where f(x) increases on (a, b), have a solution u of the type (2). Then the estimate

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

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

Theorem 6. If the equation (1) with the function p(t) satisfying the condition

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

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

u(t) < γ(f(t)J_{f}^{∗}(a, t)( ∫_{t}^{b} τ^{−1}sgnp_{f}^{∗}(τ)dτ)^{1/μ})^{1/(1−λ)},

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

To prove the Theorem it is sufficient to apply Theorem 2 with the function φ(t) = J_{f}^{∗}(a, t), which is unbounded in the case where (3) is not fulfilled.

References

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

2. N. A. Izobov and V. A. Rabtsevich, On the two Kiguradze problems for the Emden — Fowler equations. (Russian) Proceedings of the Institute of Mathematics. Mn. 2(1999). 73–91.

Author’s address:

Institute of Mathematics

Belorussian Academy of Sciences

11, Surganova St., Minsk 220072

Belarus