Geodesic
Some properties of the spectrum of nonlinear equations of Sturm–Liouville type
Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 1-14 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The question is considered of the number of stationary points of the Rayleigh functional \begin{equation} R(x)=R(r,p,q,\Gamma_0,w_r,w_0,x)=\dfrac{\|x\|_{q(w_0)}}{\|x^{(r)}\|_{p(w_r^{-1})}}, \qquad x\big|_{\partial I}\in \Gamma _0, \end{equation} which make up the spectrum of the nonlinear equation of Sturm–Liouville type $(1 \begin{equation} \begin{gathered} (-1)^{r+1}\biggl(\dfrac{(x^{(r)})_{(p)}(t)}{w_r(t)}\biggr)^{(r)}+ \lambda^q w_{0}(t)x_{(q)}(t)=0, \\ x\big|_{\partial I}\in \Gamma_0, \qquad \frac{(x^{(r)})_{(p)}}{w_r}\bigg|_{\partial I} \in \Gamma_1, \end{gathered} \end{equation} where $\bigl(h(\,\cdot\,)\bigr)_{(s)}=|h(\,\cdot\,)|^{s-1}\operatorname{sgn}(h(\,\cdot\,))$. Under various assumptions on the parameters it is proved that a solution with $n$ sign changes interior to $I=[0,1]$ is unique up to normalization. 
@article{SM_1995_80_1_a0,
     author = {A. P. Buslaev},
     title = {Some properties of the~spectrum of nonlinear equations of {Sturm{\textendash}Liouville} type},
     journal = {Sbornik. Mathematics},
     pages = {1--14},
     year = {1995},
     volume = {80},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1995_80_1_a0/}
}
TY  - JOUR
AU  - A. P. Buslaev
TI  - Some properties of the spectrum of nonlinear equations of Sturm–Liouville type
JO  - Sbornik. Mathematics
PY  - 1995
SP  - 1
EP  - 14
VL  - 80
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/SM_1995_80_1_a0/
LA  - en
ID  - SM_1995_80_1_a0
ER  - 
%0 Journal Article
%A A. P. Buslaev
%T Some properties of the spectrum of nonlinear equations of Sturm–Liouville type
%J Sbornik. Mathematics
%D 1995
%P 1-14
%V 80
%N 1
%U http://geodesic.mathdoc.fr/item/SM_1995_80_1_a0/
%G en
%F SM_1995_80_1_a0
A. P. Buslaev. Some properties of the spectrum of nonlinear equations of Sturm–Liouville type. Sbornik. Mathematics, Tome 80 (1995) no. 1, pp. 1-14. http://geodesic.mathdoc.fr/item/SM_1995_80_1_a0/

[1] Kurant R., Gilbert D., Metody matematicheskoi fiziki, T. 1, GTTI, M., 1933

[2] Gantmakher F. R., Krein M. G., Ostsillyatsionnye matritsy, yadra i malye kolebaniya mekhanicheskikh sistem, Gostekhizdat, M., 1950

[3] Lyusternik L. A., “Sur une classe d'equations differentielles nonlinearities”, Matem. sb., 2(44) (1937), 1143–1168

[4] Lyusternik L. A., “Quelques remarques suplementaires sur les equations nonlineaires du type de Sturm–Liouville”, Matem. sb., 4(46) (1938), 227–232

[5] Strett D. V. (Lord Relei), Teoriya zvuka, T. 1, 2, GITTL, M., 1955

[6] Andronov A. A., Vitt A. A., Khaikin S. E., Teoriya kolebanii, FM, M., 1959

[7] Pokhozhaev S. I., “O sobstvennykh funktsiyakh uravneniya $\Delta u-\lambda f(u)=0$”, DAN SSSR, 165:1 (1965), 36–39 | MR | Zbl

[8] Pinkus A., “$n$-widths of Sobolev spases in $L^p$”, Constructive Approximations, 1:1 (1985), 15–62 | DOI | MR | Zbl

[9] Buslaev A. P., Tikhomirov V. M., “Nekotorye voprosy nelineinogo analiza i teoriya priblizhenii”, DAN SSSR, 283:1 (1985), 13–18 | MR

[10] Tikhomirov V. M., “A. N. Kolmogorov i teoriya priblizhenii”, UMN, 44:1(265) (1989), 83–122 | MR

[11] Buslaev A. P., “Ekstremalnye zadachi teorii priblizhenii i nelineinye kolebaniya”, DAN SSSR, 305:6 (1989), 1289–1294 | MR | Zbl

[12] Buslaev A. P., Tikhomirov V. M., “Spektry nelineinykh uravnenii i poperechniki sobolevskikh klassov”, Matem. sb., 181:12 (1990), 1587–1606 | MR

[13] Chan Tkhi Le, Zadachi vosstanovleniya funktsionalov po konechnoi i beskonechnoi informatsii, Dissertatsiya ...kand. fiz. -mat. nauk, MGU, 1986

[14] Zhensykbaev A. A., “Monosplainy minimalnoi normy i nailuchshie kvadraturnye formuly”, UMN, 36:4 (1981), 109–159 | MR

[15] Arnold V. I., Varchenko A. N., Gusein-Zade S. M., Osobennosti differentsiruemykh otobrazhenii, T. 1, Nauka, M., 1982 ; Т. 2, Наука, М., 1984 | MR

[16] Lavrentev M. M., Lyusternik L. A., Osnovy variatsionnogo ischisleniya, T. 1, 2, ONTI, M.–L., 1935

[17] Akhromeeva T. S., Kurdyumov S. P., Malinetskii G. G., Samarskii A. A., Nestatsionarnye struktury i diffuznyi khaos, Nauka, M., 1992 | MR | Zbl