Existence of Multiple Solutions of Higher-Order Nonlinear Elliptic Equations
Matematičeskie zametki, Tome 89 (2011) no. 2, pp. 260-271.

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

We study the existence of multiple solutions for a class of even-order nonlinear elliptic equations with the Dirichlet boundary conditions. We also study the corresponding nonlinear Hamiltonian system of higher-order linear equations.
Keywords: higher-order elliptic equation, convex-concave nonlinearity, fibering method, variational method, Hamiltonian system, Dirichlet problem.
Mots-clés : multiple solutions
@article{MZM_2011_89_2_a8,
     author = {V. F. Lubyshev},
     title = {Existence of {Multiple} {Solutions} of {Higher-Order} {Nonlinear} {Elliptic} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {260--271},
     publisher = {mathdoc},
     volume = {89},
     number = {2},
     year = {2011},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2011_89_2_a8/}
}
TY  - JOUR
AU  - V. F. Lubyshev
TI  - Existence of Multiple Solutions of Higher-Order Nonlinear Elliptic Equations
JO  - Matematičeskie zametki
PY  - 2011
SP  - 260
EP  - 271
VL  - 89
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2011_89_2_a8/
LA  - ru
ID  - MZM_2011_89_2_a8
ER  - 
%0 Journal Article
%A V. F. Lubyshev
%T Existence of Multiple Solutions of Higher-Order Nonlinear Elliptic Equations
%J Matematičeskie zametki
%D 2011
%P 260-271
%V 89
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2011_89_2_a8/
%G ru
%F MZM_2011_89_2_a8
V. F. Lubyshev. Existence of Multiple Solutions of Higher-Order Nonlinear Elliptic Equations. Matematičeskie zametki, Tome 89 (2011) no. 2, pp. 260-271. http://geodesic.mathdoc.fr/item/MZM_2011_89_2_a8/

[1] A. Ambrosetti, H. Brezis, G. Cerami, “Combined effects of concave and convex nonlinearities in some elliptic problems”, J. Funct. Anal., 122:2 (1994), 519–543 | DOI | MR | Zbl

[2] A. Ambrosetti, J. Garcia Azorero, I. Peral, “Multiplicity results for some nonlinear elliptic equations”, J. Funct. Anal., 137:1 (1996), 219–242 | DOI | MR | Zbl

[3] J. García Azorero, I. Peral Alonso, “Some results about existence of a second positive solution in a quasilinear critical problem”, Indiana Univ. Math. J., 43:3 (1994), 941–957 | DOI | MR | Zbl

[4] J. P. García Azorero, J. Manfredi, I. Peral Alonso, “Sobolev versus Hölder local minimizers and global multiplicity for some quasilinear elliptic equations”, Commun. Contemp. Math., 2:3 (2000), 385–404 | MR | Zbl

[5] Ya. Il'yasov, “On nonlocal existence results for elliptic equations with convex-concave nonlinearities”, Nonlinear Anal., 6:1-2 (2005), 211–236 | DOI | MR | Zbl

[6] S. T. Kyritsi, N. S. Papageorgiou, “Pairs of positive solutions of $p$-Laplacian equations with combined nonlinearities”, Commun. Pure Appl. Anal., 8:3 (2009), 1031–1051 | DOI | MR | Zbl

[7] S. Li, S. Wu, H.-S. Zhou, “Solutions to semilinear elliptic problems with combined nonlinearities”, J. Differential Equations, 185:1 (2002), 200–224 | DOI | MR | Zbl

[8] G. Li, G. Zhang, “Multiple solutions for the $p{\}q$-Laplacian problem with critical exponent”, Acta Math. Sci. Ser. B Engl. Ed., 29:4 (2009), 903–918 | MR | Zbl

[9] S. I. Pokhozhaev, “O metode rassloeniya resheniya nelineinykh kraevykh zadach”, Differentsialnye uravneniya i funktsionalnye prostranstva, Sbornik statei. Posvyaschaetsya pamyati akademika Sergeya Lvovicha Soboleva, Tr. MIAN SSSR, 192, Nauka, M., 1990, 146–163 | MR | Zbl

[10] S. I. Pohozaev, “Nonlinear variational problems via the fibering method”, Handbook of Differential Equations: Stationary Partial Differential Equations, v. V, Handb. Differ. Equ., North-Holland, Amsterdam, 2008, 49–209 | MR | Zbl

[11] Yu. Bozhkov, E. Mitidieri, “Existence of multiple solutions for quasilinear equations via fibering method”, CContributions to Nonlinear Analysis, Progr. Nonlinear Differential Equations Appl., 66, Birkhäuser Verlag, Basel, 2006, 115–134 | DOI | MR | Zbl

[12] Yu. Bozhkov, E. Mitidieri, “Existence of multiple solutions for quasilinear systems via fibering method”, J. Differential Equations, 190:1 (2003), 239–267 | DOI | MR | Zbl

[13] Ph. Clément, D. G. de Figueiredo, E. Mitidieri, “Positive splutions of semilinear elliptic systems”, Comm. Partial Differential Equations, 17:5-6 (1992), 923–940 | DOI | MR | Zbl

[14] A. Salvatore, “Multiple solutions for elliptic systems with nonlinearities of arbitrary growth”, J. Differential Equations, 244:10 (2008), 2529–2544 | DOI | MR | Zbl

[15] M. Chhetri, P. Girg, “Existence and nonexistence of positive solutions for a class of superlinear semipositone systems”, Nonlinear Anal., 71:10 (2009), 4984–4996 | DOI | MR | Zbl

[16] J. Hulshof, R. van der Vorst, “Differential systems with strongly indefinite variational structure”, J. Funct. Anal., 114:1 (1993), 32–58 | DOI | MR | Zbl

[17] J. Hulshof, E. Mitidieri, R. C. A. M. van der Vorst, “Strongly indefinite systems with critical Sobolev exponents”, Trans. Amer. Math. Soc., 350:6 (1998), 2349–2365 | DOI | MR | Zbl

[18] D. G. de Figueiredo, B. Ruf, “Elliptic systems with nonlinearities of arbitrary growth”, Mediterr. J. Math., 1:4 (2004), 417–431 | DOI | MR | Zbl

[19] E. S. Tsitlanadze, “Teoremy suschestvovaniya tochek minimaksa v prostranstvakh Banakha i ikh prilozheniya”, Tr. MMO, 2, Izd-vo Mosk. un-ta, M., 1953, 235–274 | MR | Zbl

[20] M. M. Vainberg, Variatsionnye metody issledovaniya nelineinykh operatorov, Gostekhizdat, M., 1956 | MR | Zbl