# On some properties of the solution of the differential equation ${u}^{\text{'}\text{'}}+\frac{2{u}^{\text{'}}}{r}=u-{u}^{3}$

Aplikace matematiky (1990)

• Volume: 35, Issue: 4, page 315-336
• ISSN: 0862-7940

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## Abstract

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In the paper it is shown that each solution $u\left(r,\alpha \right)$ ot the initial value problem (2), (3) has a finite limit for $r\to \infty$, and an asymptotic formula for the nontrivial solution $u\left(r,\alpha \right)$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u\left(r,\overline{\alpha }\right)$, $u\left(r,\stackrel{^}{\alpha }\right)$.

## How to cite

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Šeda, Valter, and Pekár, Ján. "On some properties of the solution of the differential equation $u^{\prime \prime }+\frac{2u^{\prime }}{r}=u-u^3$." Aplikace matematiky 35.4 (1990): 315-336. <http://eudml.org/doc/15633>.

@article{Šeda1990,
abstract = {In the paper it is shown that each solution $u(r,\alpha )$ ot the initial value problem (2), (3) has a finite limit for $r\rightarrow \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha )$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\bar\{\alpha \})$, $u(r,\hat\{\alpha \})$.},
author = {Šeda, Valter, Pekár, Ján},
journal = {Aplikace matematiky},
keywords = {spherically symmetric solution; trajectory of the solution; со-limit point of the trajectory; asymptotic formula; antitone and contractive operator; zero of the solution; Klein-Gordon equation; global behavior; Klein-Gordon equation; global behavior; asymptotic formulas; zeros of the solutions},
language = {eng},
number = {4},
pages = {315-336},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On some properties of the solution of the differential equation $u^\{\prime \prime \}+\frac\{2u^\{\prime \}\}\{r\}=u-u^3$},
url = {http://eudml.org/doc/15633},
volume = {35},
year = {1990},
}

TY - JOUR
AU - Šeda, Valter
AU - Pekár, Ján
TI - On some properties of the solution of the differential equation $u^{\prime \prime }+\frac{2u^{\prime }}{r}=u-u^3$
JO - Aplikace matematiky
PY - 1990
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 35
IS - 4
SP - 315
EP - 336
AB - In the paper it is shown that each solution $u(r,\alpha )$ ot the initial value problem (2), (3) has a finite limit for $r\rightarrow \infty$, and an asymptotic formula for the nontrivial solution $u(r,\alpha )$ tending to 0 is given. Further, the existence of such a solutions is established by examining the number of zeros of two different solutions $u(r,\bar{\alpha })$, $u(r,\hat{\alpha })$.
LA - eng
KW - spherically symmetric solution; trajectory of the solution; со-limit point of the trajectory; asymptotic formula; antitone and contractive operator; zero of the solution; Klein-Gordon equation; global behavior; Klein-Gordon equation; global behavior; asymptotic formulas; zeros of the solutions
UR - http://eudml.org/doc/15633
ER -

## References

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1. Ch. V. Coffman, 10.1007/BF00250684, Arch. Rational Mech. Anal. 46 (1972), 81 - 95. (1972) Zbl0249.35029MR0333489DOI10.1007/BF00250684
2. L. Erbe K. Schmitt, On radial solutions of some semilinear elliptic equations, Differential and Integral Equations, Vol. 1 (1988), 71 - 78. (1988) Zbl0721.35020MR0920490
3. J. Chauvette F. Stenger, 10.1016/0022-247X(75)90155-9, J. Math. Anal. Appl. 51 (1975), 229-242. (1975) Zbl0311.65063MR0373320DOI10.1016/0022-247X(75)90155-9
4. G. H. Ryder, 10.2140/pjm.1967.22.477, Pacific J. Math. 22 (1967), 477-503. (1967) Zbl0152.28303MR0219794DOI10.2140/pjm.1967.22.477
5. G. Sansone, Su un'equazione differenziale non lineare della fisica nucleare, Istituto Nazionale di Alta Matem. Sympozia Mathematica, Vol. VI, (1970). (1970) Zbl0249.34038

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