Abstract and Applied Analysis

Infinitely Many Solutions of Superlinear Elliptic Equation

Anmin Mao and Yang Li

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Via the Fountain theorem, we obtain the existence of infinitely many solutions of the following superlinear elliptic boundary value problem: Δ u = f ( x , u ) in Ω, u = 0 on , where N ( N > 2 ) is a bounded domain with smooth boundary and f is odd in u and continuous. There is no assumption near zero on the behavior of the nonlinearity f , and f does not satisfy the Ambrosetti-Rabinowitz type technical condition near infinity.

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Abstr. Appl. Anal., Volume 2013 (2013), Article ID 769620, 6 pages.

First available in Project Euclid: 27 February 2014

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Mao, Anmin; Li, Yang. Infinitely Many Solutions of Superlinear Elliptic Equation. Abstr. Appl. Anal. 2013 (2013), Article ID 769620, 6 pages. doi:10.1155/2013/769620. https://projecteuclid.org/euclid.aaa/1393511896

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  • D. G. Costa and C. A. Magalhães, “Variational elliptic problems which are nonquadratic at infinity,” Nonlinear Analysis. Theory, Methods & Applications, vol. 23, no. 11, pp. 1401–1412, 1994.
  • O. H. Miyagaki and M. A. S. Souto, “Superlinear problems without Ambrosetti and Rabinowitz growth condition,” Journal of Differential Equations, vol. 245, no. 12, pp. 3628–3638, 2008.
  • W. Zou, “Variant fountain theorems and their applications,” Manuscripta Mathematica, vol. 104, no. 3, pp. 343–358, 2001.
  • T. Bartsch, “Infinitely many solutions of a symmetric Dirichlet problem,” Nonlinear Analysis. Theory, Methods & Applications, vol. 20, no. 10, pp. 1205–1216, 1993.
  • T. Bartsch and M. Willem, “On an elliptic equation with concave and convex nonlinearities,” Proceedings of the American Mathematical Society, vol. 123, no. 11, pp. 3555–3561, 1995.
  • P. H. Rabinowitz, J. Su, and Z.-Q. Wang, “Multiple solutions of superlinear elliptic equations,” Rendiconti Lincei. Serie IX. Matematica e Applicazioni, vol. 18, no. 1, pp. 97–108, 2007.
  • L. Jeanjean, “On the existence of bounded Palais-Smale sequences and application to a Landesman-Lazer-type problem set on ${\mathbf{R}}^{N}$,” Proceedings of the Royal Society of Edinburgh A, vol. 129, no. 4, pp. 787–809, 1999.
  • Y. Ding and C. Lee, “Multiple solutions of Schrödinger equations with indefinite linear part and super or asymptotically linear terms,” Journal of Differential Equations, vol. 222, no. 1, pp. 137–163, 2006.
  • H.-S. Zhou, “Positive solution for a semilinear elliptic equation which is almost linear at infinity,” Zeitschrift für Angewandte Mathematik und Physik, vol. 49, no. 6, pp. 896–906, 1998.
  • M. Schechter, “Superlinear elliptic boundary value problems,” Manuscripta Mathematica, vol. 86, no. 3, pp. 253–265, 1995.
  • A. Ambrosetti and P. H. Rabinowitz, “Dual variational methods in critical point theory and applications,” Journal of Functional Analysis, vol. 14, pp. 349–381, 1973.
  • M. Willem and W. Zou, On a Semilinear Dirichlet Problem and a Nonlinear Schrödinger Equation with Periodic Potential, vol. 305 of Séminaire de Mathématique, Université Catholique de Louvain, 2000.
  • M. Schechter and W. Zou, “Superlinear problems,” Pacific Journal of Mathematics, vol. 214, no. 1, pp. 145–160, 2004.
  • Q. Y. Zhang and C. G. Liu, “Infinitely many periodic solutions for second order Hamiltonian systems,” Journal of Differential Equations, vol. 251, no. 4-5, pp. 816–833, 2011.
  • V. Benci and P. H. Rabinowitz, “Critical point theorems for indefinite functionals,” Inventiones Mathematicae, vol. 52, no. 3, pp. 241–273, 1979.
  • P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS, 1986.
  • M. Willem, Minimax Theorems, Birkhäuser, Boston, Mass, USA, 1996.