Three Types of Solutions of Some Singular Nonlinear
Differential Equation
Irena Rachůnková
Palacký University, Olomouc, Czech Republic
E-mail: irena.rachunkova@upol.cz
The differential equation
(
p(t)u′
)′
= p(t)f (u)
(1)
is investigated on the positive half line under the assumptions (2)–(6):
f ∈ Liploc (R), ∃ L ∈ (0, ∞) : f (L) = 0,
∃ L0 ∈ [−∞, 0) : xf (x) < 0, x ∈ (L0 , 0) ∪ (0, L),
∫x
∃ B̄ ∈ (L0 , 0) : F (B̄) = F (L), where F (x) = − f (z) dz,
(2)
(3)
x ∈ R,
(4)
0
p ∈ C[0, ∞) ∩ C (0, ∞),
1
p′ (t) > 0, t ∈ (0, ∞),
p(0) = 0,
p′ (t)
lim
= 0.
t→∞ p(t)
(5)
(6)
Due to p(0) = 0, equation (1) has a singularity at t = 0.
The following results are proved.
1. For each B < 0 equation (1) has a unique solution uB ∈ C 1 [0, ∞) ∩ C 2 (0, ∞) which satisfies
the initial conditions
u(0) = B, u′ (0) = 0.
(7)
2. A solution uB of problem (1), (7) satisfying sup{u(t) : t ∈ [0, ∞)} < L is called a damped
solution. If Md is the set of all B < 0 such that uB is a damped solution, then Md is nonempty
and open in (−∞, 0).
3. A solution uB of problem (1), (7) satisfying sup{u(t) : t ∈ [0, ∞)} > L is called an escape
solution. If Me is the set of all B < 0 such that uB is an escape solution, then Me is open in
(−∞, 0). In addition, Me is nonempty provided one of the following additional assumptions (A1),
(A2), (A3), or (A4) is valid:
• (A1): L0 ∈ (−∞, 0),
f (L0 ) = 0.
• (A2): f (x) > 0 for x ∈ (−∞, 0) and
0 ≤ lim sup
x→−∞
f (x)
< ∞.
|x|
• (A3): f (x) > 0 for x ∈ (−∞, 0) and there exists k ≥ 2 such that
p′ (t)
∈ (0, ∞).
t→0+ tk−2
lim
Further, there exists r ∈ (1, k+2
k−2 ) such that f fulfils
f (x)
∈ (0, ∞).
x→−∞ |x|r
lim
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• (A4):
∫1
ds
< ∞.
p(s)
0
4. If Me is nonempty, then problem (1), (7) has a solution u such that
{
}
sup u(t) : t ∈ [0, ∞) = L.
Such solution is called homoclinic. It is increasing and limt→∞ u(t) = L.
5. Some other additional conditions for p and f which give asymptotic formulas for damped
and homoclinic solutions are discussed.
References
[1] I. Rachůnková and J. Tomeček, Bubble-type solutions of nonlinear singular problems. Math.
Comput. Modelling 51 (2010), No. 5-6, 658–669.
[2] I. Rachůnková and J. Tomeček, Strictly increasing solutions of a nonlinear singular differential
equation arising in hydrodynamics. Nonlinear Anal. 72 (2010), No. 3-4, 2114–2118.
[3] I. Rachůnková, L. Rachůnek, and J. Tomeček, Existence of oscillatory solutions of singular
nonlinear differential equations. Abstr. Appl. Anal. 2011, Art. ID 408525, 20 pp.
[4] I. Rachůnková and J. Tomeček, Superlinear singular problems on the half line. Bound. Value
Probl. 2010, Art. ID 429813, 18 pp.
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