zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Infinitely many radial solutions for Kirchhoff-type problems in $\bbfR^N$. (English) Zbl 1196.35221
Summary: We concern with a class of Kirchhoff-type problems in $\Bbb R^N$. By using the fountain theorem, we obtain three existence results of infinitely many radial solutions for the problem.

35R09Integro-partial differential equations
35D30Weak solutions of PDE
35A15Variational methods (PDE)
45K05Integro-partial differential equations
Full Text: DOI
[1] Alves, C. O.; Corrêa, F. J. S.A.; Ma, T. F.: Positive solutions for a quasilinear elliptic equation of Kirchhoff type, Comput. math. Appl. 49, 85-93 (2005) · Zbl 1130.35045 · doi:10.1016/j.camwa.2005.01.008
[2] Arosio, A.; Panizzi, S.: On the well-posedness of the Kirchhoff string, Trans. amer. Math. soc. 348, 305-330 (1996) · Zbl 0858.35083 · doi:10.1090/S0002-9947-96-01532-2
[3] Bernstein, S.: Sur une class d’équations fonctionnelles aux dérivées partielles, Bull. acad. Sci. URSS, sér. Math. 4, 17-26 (1940) · Zbl 0026.01901
[4] Cavalcanti, M. M.; Cavalcanti, V. N. Domingos; Soriano, J. A.: Global existence and uniform decay rates for the Kirchhoff -- carrier equation with nonlinear dissipation, Adv. differential equations 6, 701-730 (2001) · Zbl 1007.35049
[5] Cheng, B.; Wu, X.: Existence results of positive solutions of Kirchhoff type problems, Nonlinear anal. 71, 4883-4892 (2009) · Zbl 1175.35038 · doi:10.1016/j.na.2009.03.065
[6] Chipot, M.; Lovat, B.: Some remarks on nonlocal elliptic and parabolic problems, Nonlinear anal. 30, No. 7, 4619-4627 (1997) · Zbl 0894.35119 · doi:10.1016/S0362-546X(97)00169-7
[7] D’ancona, P.; Spagnolo, S.: Global solvability for the degenerate Kirchhoff equation with real analytic data, Invent. math. 108, 247-262 (1992) · Zbl 0785.35067 · doi:10.1007/BF02100605
[8] He, X.; Zou, W.: Infinitely many positive solutions for Kirchhoff-type problems, Nonlinear anal. 70, No. 3, 1407-1414 (2009) · Zbl 1157.35382 · doi:10.1016/j.na.2008.02.021
[9] Kirchhoff, G.: Mechanik, (1883)
[10] J.L. Lions, On some questions in boundary value problems of mathematical physics, in: Contemporary Developments in Continuum Mechanics and Partial Differential Equations, Proc. Internat. Sympos. Inst. Mat., Univ. Fed. Rio de Janeiro, 1997, North-Holland Math. Stud., vol. 30, 1978, pp. 284 -- 346 · Zbl 0404.35002
[11] Ma, T. F.; Rivera, J. E. Muñoz: Positive solutions for a nonlinear nonlocal elliptic transmission problem, Appl. math. Lett. 16, 243-248 (2003) · Zbl 1135.35330 · doi:10.1016/S0893-9659(03)80038-1
[12] Mao, A.; Zhang, Z.: Sign-changing and multiple solutions of Kirchhoff type problems without the P.S. Condition, Nonlinear anal. 70, 1275-1287 (2009) · Zbl 1160.35421 · doi:10.1016/j.na.2008.02.011
[13] Perera, K.; Zhang, Z.: Nontrivial solutions of Kirchhoff-type problems via the Yang index, J. differential equations 221, 246-255 (2006) · Zbl 05013580
[14] Pohoz&circ, S. I.; Aev: A certain class of quasilinear hyperbolic equations, Mat. sb. (N.S.) 96, 152-168 (1975)
[15] Willem, M.: Minimax theorem, (1996) · Zbl 0856.49001
[16] Zhang, Z.; Perera, K.: Sign changing solutions of Kirchhoff type problems via invariant sets of descent flow, J. math. Anal. appl. 317, 456-463 (2006) · Zbl 1100.35008 · doi:10.1016/j.jmaa.2005.06.102