# Nonzero bounded solutions of one class of nonlinear ordinary differential equations

## Abstract

The paper is concerned with an ordinary differential equation of the form -{psi}''(x)+(1+c/x{sup 2}){psi}(x)=1/x{sup {alpha}}|{psi}(x)|{sup k-1}{psi}(x), x>0; (1) where k and {alpha} are positive parameters, k>1, and c is a constant, subject to the boundary condition {psi}(0)=0, {psi}(+{infinity})=0; (2) A variational approach based on finding the eigenvalues of the gradient of the functional F{sub k,{alpha}}(f)={integral}{sub 0}{sup +}{infinity}|f(s)|{sup k+1}s{sup -}{alpha} ds acting on the space of absolutely continuous functions H{sub 0}{sup 1}={l_brace}f:f,f' element of L{sub 2}(0,+{infinity}), f(0)=0{r_brace} is used to show that if c>-1/4, k>1, 0<2{alpha}<k+3, then problem (1), (2) has a countable number of nonzero solutions, at least one of which is positive. For nonzero solutions, asymptotic formulae as x{yields}0 and x{yields}+{infinity} are obtained. Bibliography: 7 titles.

- Authors:

- Vologda State Technical University, Vologda (Russian Federation)

- Publication Date:

- OSTI Identifier:
- 21612603

- Resource Type:
- Journal Article

- Journal Name:
- Sbornik. Mathematics

- Additional Journal Information:
- Journal Volume: 202; Journal Issue: 9; Other Information: DOI: 10.1070/SM2011v202n09ABEH004191; Country of input: International Atomic Energy Agency (IAEA); Journal ID: ISSN 1064-5616

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICAL METHODS AND COMPUTING; ASYMPTOTIC SOLUTIONS; BOUNDARY CONDITIONS; DIFFERENTIAL EQUATIONS; EIGENVALUES; FUNCTIONS; MATHEMATICAL SPACE; NONLINEAR PROBLEMS; VARIATIONAL METHODS; CALCULATION METHODS; EQUATIONS; MATHEMATICAL SOLUTIONS; SPACE

### Citation Formats

```
Muhamadiev, Ergashboy M, and Naimov, Alijon N.
```*Nonzero bounded solutions of one class of nonlinear ordinary differential equations*. United States: N. p., 2011.
Web. doi:10.1070/SM2011V202N09ABEH004191; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Muhamadiev, Ergashboy M, & Naimov, Alijon N.
```*Nonzero bounded solutions of one class of nonlinear ordinary differential equations*. United States. doi:10.1070/SM2011V202N09ABEH004191; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).

```
Muhamadiev, Ergashboy M, and Naimov, Alijon N. Fri .
"Nonzero bounded solutions of one class of nonlinear ordinary differential equations". United States. doi:10.1070/SM2011V202N09ABEH004191; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA).
```

```
@article{osti_21612603,
```

title = {Nonzero bounded solutions of one class of nonlinear ordinary differential equations},

author = {Muhamadiev, Ergashboy M and Naimov, Alijon N},

abstractNote = {The paper is concerned with an ordinary differential equation of the form -{psi}''(x)+(1+c/x{sup 2}){psi}(x)=1/x{sup {alpha}}|{psi}(x)|{sup k-1}{psi}(x), x>0; (1) where k and {alpha} are positive parameters, k>1, and c is a constant, subject to the boundary condition {psi}(0)=0, {psi}(+{infinity})=0; (2) A variational approach based on finding the eigenvalues of the gradient of the functional F{sub k,{alpha}}(f)={integral}{sub 0}{sup +}{infinity}|f(s)|{sup k+1}s{sup -}{alpha} ds acting on the space of absolutely continuous functions H{sub 0}{sup 1}={l_brace}f:f,f' element of L{sub 2}(0,+{infinity}), f(0)=0{r_brace} is used to show that if c>-1/4, k>1, 0<2{alpha}<k+3, then problem (1), (2) has a countable number of nonzero solutions, at least one of which is positive. For nonzero solutions, asymptotic formulae as x{yields}0 and x{yields}+{infinity} are obtained. Bibliography: 7 titles.},

doi = {10.1070/SM2011V202N09ABEH004191; COUNTRY OF INPUT: INTERNATIONAL ATOMIC ENERGY AGENCY (IAEA)},

journal = {Sbornik. Mathematics},

issn = {1064-5616},

number = 9,

volume = 202,

place = {United States},

year = {2011},

month = {9}

}