Geodesic
Bifurcation Calculus by the Extended Functional Method
Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 1, pp. 23-38.

Voir la notice de l'article provenant de la source Math-Net.Ru

We justify variational principles of a new type corresponding to bifurcations of solutions for families of equations given in variational form. To illustrate the method, we consider elliptic equations with sign-indefinite nonlinearities and prove the existence of pairwise creation-annihilation bifurcations of their positive solutions. The corresponding bifurcation points are expressed via explicitly specified variational principles.
Mots-clés : bifurcation of solutions, elliptic equation
Keywords: minimax problem, sign-indefinite nonlinearity. 
@article{FAA_2007_41_1_a1,
     author = {Ya. Sh. Il'yasov},
     title = {Bifurcation {Calculus} by the {Extended} {Functional} {Method}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {23--38},
     publisher = {mathdoc},
     volume = {41},
     number = {1},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2007_41_1_a1/}
}
TY  - JOUR
AU  - Ya. Sh. Il'yasov
TI  - Bifurcation Calculus by the Extended Functional Method
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2007
SP  - 23
EP  - 38
VL  - 41
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2007_41_1_a1/
LA  - ru
ID  - FAA_2007_41_1_a1
ER  - 
%0 Journal Article
%A Ya. Sh. Il'yasov
%T Bifurcation Calculus by the Extended Functional Method
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2007
%P 23-38
%V 41
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2007_41_1_a1/
%G ru
%F FAA_2007_41_1_a1
Ya. Sh. Il'yasov. Bifurcation Calculus by the Extended Functional Method. Funkcionalʹnyj analiz i ego priloženiâ, Tome 41 (2007) no. 1, pp. 23-38. http://geodesic.mathdoc.fr/item/FAA_2007_41_1_a1/

[1] M. S. Agranovich, A. S. Dynin, “Obschie kraevye zadachi dlya ellipticheskikh sistem v mnogomernoi oblasti”, DAN SSSR, 142:3 (1962), 511–514 | MR

[2] A. Ambrosetti, J. G. Azorero, I. Peral, “Existence and multiplicity results for some nonlinear elliptic equations: a survey. Dedicated to the memory of Gaetano Fichera (Italian)”, Rend. Mat. Appl. (7), 20 (2000), 167–198 | MR | Zbl

[3] J. Appell, E. de Pascale, A. Vignoli, Nonlinear spectral theory, de Gruyter Series in Nonlinear Analysis and Applications, 10, Walter de Gruyter, Berlin, 2004 | DOI | MR

[4] M. G. Crandall, P. H. Rabinowitz, “Bifurcation, perturbation of simple eigenvalues and linearized stability”, Arch. Rational Mech. Anal., 52 (1973), 161–180 | DOI | MR | Zbl

[5] E. N. Dancer, “On the structure of solutions of non-linear eigenvalue problems”, Indiana Univ. Math. J., 23 (1974), 1069–1076 | DOI | MR | Zbl

[6] M. del Pino, “Positive solutions of a semilinear elliptic equation on a compact manifold”, Nonlinear Anal., 22:11 (1994), 1423–1430 | DOI | MR | Zbl

[7] M. M. Vainberg, V. A. Trenogin, Teoriya vetvleniya reshenii nelineinykh uravnenii., Nauka, M., 1969 | MR

[8] D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989 | MR | Zbl

[9] Ya. Sh. Ilyasov, “O globalnykh polozhitelnykh resheniyakh parabolicheskikh uravnenii s neopredelennym znakom nelineinosti”, Differentsialnye uravneniya, 41:4 (2005), 518–526 | MR

[10] Ya. Sh. Ilyasov, “Nelokalnye issledovaniya bifurkatsii reshenii nelineinykh ellipticheskikh uravnenii”, Izv. RAN, ser. mat., 66:6 (2002), 19–48 | DOI | MR

[11] Y. Ilżyasov, “On positive solutions of indefinite elliptic equations”, C. R. Acad. Sci. Paris Sér. I Math., 333:6 (2001), 533–538 | DOI | MR | Zbl

[12] M. A. Krasnoselskii, Topologicheskie metody v teorii nelineinykh integralnykh uravnenii, GITTL, 1956 | MR

[13] J. López-Gómez, Spectral Theory and Nonlinear Functional Analysis, Chapman Hall/CRC Research Notes in Mathematics, 426, Chapman Hall/CRC, 2001 | MR | Zbl

[14] L. Nirenberg, Lektsii po nelineinomu funktsionalnomu analizu, Mir, M., 1977 | MR | Zbl

[15] T. Ouyang, “On the positive solutions of semilinear equations $\Delta u+\lambda u+hu^p=0$ on compact manifolds, Part II”, Indiana Univ. Math. J., 40:3 (1991), 1083–1141 | DOI | MR | Zbl

[16] P. H. Rabinowitz, “Some global results for nonlinear eigenvalue problems”, J. Functional Analysis, 7 (1971), 487–513 | DOI | MR | Zbl

[17] M. Struwe, Variational Methods. Application to Nonlinear Partial Differential Equations and Hamiltonian Systems, Springer-Verlag, Berlin, 1996 | MR | Zbl

[18] P. Tolksdorf, “On the Dirichlet problem for quasilinear equations in domains with conical boundary points”, Comm. Partial Differential Equations, 8:7 (1983), 773–817 | DOI | MR | Zbl

[19] E. Ziedler, Nonlinear functional analysis and its applications, I–IV, Springer-Verlag, Berlin–Heidelberg–New York, 1988–1990 | MR