Abstract
In this paper we apply the (variant) fountain theorems to study the symmetric nonlinear Kirchhoff nonlocal problems. Under the Ambrosetti-Rabinowitz’s 4-superlinearity condition, or no Ambrosetti-Rabinowitz’s 4-superlinearity condition, we present two results of existence of infinitely many large energy solutions, respectively.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alves, C.O., Correa, F.J.S.A. Positive solutions for a quasilinear elliptic equation of Kirchhoff type. Comput. Math. Appl., 49: 85–93 (2005)
Ambrosetti, A., Rabinowitz, P. Dual variational methods in critical point theory and applications. J. Funct. Anal., 14: 349–381 (1973)
Ancona, P.D’, Spagnolo, S. Global Solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math., 108: 247–262 (1992)
Andrade, D., Ma, T.F. An operator equation suggested by a class of nonlinear stationary problems. Comm. Appl. Nonli. Anal., 4: 65–71 (1997)
Arosio, A., Pannizi, S. On the well-posedness of the Kirchhoff string. Trans. Amer. Math. Soc., 348: 305–330 (1996)
Bernstein, S. Sur une classe d’équations fonctionnelles aux dérivées partielles. Izv. Akad. Nauk SSSR Ser. Mat., 4: 17–26 (1940)
Cavalcanti, M.M., Cavacanti, V.N., Soriano, J.A. Global existence and uniform decay rates for the Kirchhoff-Carrier equation with nonlinear dissipation. Adv. Diff. Eqns., 6: 701–730 (2001)
Chipot, M., Lovat, B. Some remarks on nonlocal elliptic and parabolic problems. Nonlinear Analysis, 30: 4619–4627 (1997)
Chipot, M., Rodrigues, J.-F. On a class of nonlocal nonlinear elliptic problems. RAIRO Modél. Math. Anal. Numér., 26: 447–467 (1992)
Jeajean, L. On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer type problem set on ℝN, Proc. Roy. Soc. Edinburgh Sect. A, 129: 787–809 (1999)
Kirchhoff, G. Mechanik, Teubner, Leipzig, 1883
Lions, J.-L. On some quations 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, Rio de Janeiro, 1977, in: North-Holland Math. Stud., vol.30, North-Holland, Amsterdam, 284–346, 1978
Ma, T.F., Muñoz Rivera, J.E. Positive solutions for a nonlinear nonlocal elliptic transmission problem. Appl. Math. Lett., 16: 243–248 (2003)
Mao, A., Zhang, Z. Sign-changing and multiple solutions of Kirchhoff type problems without P. S. condition. Nonlinear Analysis: Theory, Methods and Applications, 70(3): 1275–1287 (2009)
Perera, K., Zhang, Z. Nontrivial solutions of kirchhoff-type problems via the Yang index. J. Diff. Eqns., 221: 246–255 (2006)
Pohožaev, S.I. A certain class of quasilinear hyperbolic equations. Mat. Sb.(N.S.), 96: 152–166, 168 (1975)
Willem, M. Minimax Theorems, Birkhäuser, Boston, 1996
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)
Zou, W. Variant fountain theorem and their applivations. Manuscripta Math., 104: 343–358 (2001)
Zou, W., Li, S. Infinitely many homoclinic orbits for the second-order Hamiltonian systems. Appl. Math. Lett., 16: 1283–1287 (2003)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
He, Xm., Zou, Wm. Multiplicity of solutions for a class of Kirchhoff type problems. Acta Math. Appl. Sin. Engl. Ser. 26, 387–394 (2010). https://doi.org/10.1007/s10255-010-0005-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10255-010-0005-2