Journal Menu
- About this Journal
- Abstracting and Indexing
- Aims and Scope
- Annual Issues
- Article Processing Charges
- Articles in Press
- Author Guidelines
- Bibliographic Information
- Citations to this Journal
- Contact Information
- Editorial Board
- Editorial Workflow
- Free eTOC Alerts
- Publication Ethics
- Reviewers Acknowledgment
- Submit a Manuscript
- Subscription Information
- Table of Contents
International Journal of Differential Equations
Volume 2010 (2010), Article ID 432759, 11 pages
doi:10.1155/2010/432759
Research Article
On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds
1Dipartimento di Matematica Applicata “U.Dini”, Università di Pisa, via F. Buonarroti 1/c, 56100 Pisa, Italy
2Dipartimento di Metodi e Modelli Matematici, Università di Roma “La Sapienza”, via Antonio Scarpa 16, 00161 Roma, Italy
Received 1 October 2009; Accepted 7 December 2009
Academic Editor: Thomas Bartsch
Copyright © 2010 Anna Maria Micheletti and Angela Pistoia. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Linked References
- J. Byeon and J. Park, “Singularly perturbed nonlinear elliptic problems on manifolds,” Calculus of Variations and Partial Differential Equations, vol. 24, no. 4, pp. 459–477, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. Benci, C. Bonanno, and A. M. Micheletti, “On the multiplicity of solutions of a nonlinear elliptic problem on Riemannian manifolds,” Journal of Functional Analysis, vol. 252, no. 2, pp. 464–489, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- N. Hirano, “Multiple existence of solutions for a nonlinear elliptic problem on a Riemannian manifold,” Nonlinear Analysis. Theory, Methods & Applications, vol. 70, no. 2, pp. 671–692, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- D. Visetti, “Multiplicity of solutions of a zero mass nonlinear equation on a Riemannian manifold,” Journal of Differential Equations, vol. 245, no. 9, pp. 2397–2439, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- E. N. Dancer, A. M. Micheletti, and A. Pistoia, “Multipeak solutions for some singularly perturbed nonlinear elliptic problems on Riemannian manifolds,” Manuscripta Mathematica, vol. 128, no. 2, pp. 163–193, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. M. Micheletti and A. Pistoia, “The role of the scalar curvature in a nonlinear elliptic problem on Riemannian manifolds,” Calculus of Variations and Partial Differential Equations, vol. 34, no. 2, pp. 233–265, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- A. M. Micheletti and A. Pistoia, “Nodal solutions for a singularly perturbed nonlinear elliptic problem on Riemannian manifolds,” Advanced Nonlinear Studies, vol. 9, no. 3, pp. 565–577, 2009. View at MathSciNet
- M. Ghimenti and A. M. Micheletti, “On the number of nodal solutions for a nonlinear elliptic problem on symmetric Riemannian manifolds,” to appear in Electronic Journal of Differential Equations.
- V. Benci and G. Cerami, “Positive solutions of some nonlinear elliptic problems in exterior domains,” Archive for Rational Mechanics and Analysis, vol. 99, no. 4, pp. 283–300, 1987. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- D. Cao, N. E. Dancer, E. S. Noussair, and S. Yan, “On the existence and profile of multi-peaked solutions to singularly perturbed semilinear Dirichlet problems,” Discrete and Continuous Dynamical Systems, vol. 2, no. 2, pp. 221–236, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- E. N. Dancer and S. Yan, “Effect of the domain geometry on the existence of multipeak solutions for an elliptic problem,” Topological Methods in Nonlinear Analysis, vol. 14, no. 1, pp. 1–38, 1999. View at Zentralblatt MATH · View at MathSciNet
- E. N. Dancer and S. Yan, “A singularly perturbed elliptic problem in bounded domains with nontrivial topology,” Advances in Differential Equations, vol. 4, no. 3, pp. 347–368, 1999. View at Zentralblatt MATH · View at MathSciNet
- E. N. Dancer and J. Wei, “On the effect of domain topology in a singular perturbation problem,” Topological Methods in Nonlinear Analysis, vol. 11, no. 2, pp. 227–248, 1998. View at Zentralblatt MATH · View at MathSciNet
- M. del Pino, P. L. Felmer, and J. Wei, “Multi-peak solutions for some singular perturbation problems,” Calculus of Variations and Partial Differential Equations, vol. 10, no. 2, pp. 119–134, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. del Pino, P. L. Felmer, and J. Wei, “On the role of distance function in some singular perturbation problems,” Communications in Partial Differential Equations, vol. 25, no. 1-2, pp. 155–177, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. Grossi and A. Pistoia, “On the effect of critical points of distance function in superlinear elliptic problems,” Advances in Differential Equations, vol. 5, no. 10–12, pp. 1397–1420, 2000. View at Zentralblatt MATH · View at MathSciNet
- Y. Y. Li and L. Nirenberg, “The Dirichlet problem for singularly perturbed elliptic equations,” Communications on Pure and Applied Mathematics, vol. 51, no. 11-12, pp. 1445–1490, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- W.-M. Ni and J. Wei, “On the location and profile of spike-layer solutions to singularly perturbed semilinear Dirichlet problems,” Communications on Pure and Applied Mathematics, vol. 48, no. 7, pp. 731–768, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Wei, “On the interior spike solutions for some singular perturbation problems,” Proceedings of the Royal Society of Edinburgh. Section A, vol. 128, no. 4, pp. 849–874, 1998. View at Zentralblatt MATH · View at MathSciNet
- G. Cerami and J. Wei, “Multiplicity of multiple interior peak solutions for some singularly perturbed Neumann problems,” International Mathematics Research Notices, no. 12, pp. 601–626, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- E. N. Dancer and S. Yan, “Multipeak solutions for a singularly perturbed Neumann problem,” Pacific Journal of Mathematics, vol. 189, no. 2, pp. 241–262, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. del Pino, P. L. Felmer, and J. Wei, “On the role of mean curvature in some singularly perturbed Neumann problems,” SIAM Journal on Mathematical Analysis, vol. 31, no. 1, pp. 63–79, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- M. Grossi, A. Pistoia, and J. Wei, “Existence of multipeak solutions for a semilinear Neumann problem via nonsmooth critical point theory,” Calculus of Variations and Partial Differential Equations, vol. 11, no. 2, pp. 143–175, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- C. Gui, “Multipeak solutions for a semilinear Neumann problem,” Duke Mathematical Journal, vol. 84, no. 3, pp. 739–769, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- C. Gui and J. Wei, “Multiple interior peak solutions for some singularly perturbed Neumann problems,” Journal of Differential Equations, vol. 158, no. 1, pp. 1–27, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- C. Gui and J. Wei, “On multiple mixed interior and boundary peak solutions for some singularly perturbed Neumann problems,” Canadian Journal of Mathematics, vol. 52, no. 3, pp. 522–538, 2000. View at Zentralblatt MATH · View at MathSciNet
- C. Gui, J. Wei, and M. Winter, “Multiple boundary peak solutions for some singularly perturbed Neumann problems,” Annales de l'Institut Henri Poincaré, vol. 17, no. 1, pp. 47–82, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- Y. Y. Li, “On a singularly perturbed equation with Neumann boundary condition,” Communications in Partial Differential Equations, vol. 23, no. 3-4, pp. 487–545, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- W.-M. Ni and I. Takagi, “Locating the peaks of least-energy solutions to a semilinear Neumann problem,” Duke Mathematical Journal, vol. 70, no. 2, pp. 247–281, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- W.-M. Ni and I. Takagi, “On the shape of least-energy solutions to a semilinear Neumann problem,” Communications on Pure and Applied Mathematics, vol. 44, no. 7, pp. 819–851, 1991. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Wei, “On the boundary spike layer solutions to a singularly perturbed Neumann problem,” Journal of Differential Equations, vol. 134, no. 1, pp. 104–133, 1997. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- J. Wei, “On the interior spike layer solutions to a singularly perturbed Neumann problem,” The Tôhoku Mathematical Journal, vol. 50, no. 2, pp. 159–178, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. Benci and G. Cerami, “Multiple positive solutions of some elliptic problems via the Morse theory and the domain topology,” Calculus of Variations and Partial Differential Equations, vol. 2, no. 1, pp. 29–48, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
- V. Benci, “Introduction to Morse theory: a new approach,” in Topological Nonlinear Analysis, vol. 15 of Progress in Nonlinear Differential Equations and Their Applications, pp. 37–177, Birkhäuser, Boston, Mass, USA, 1995. View at Zentralblatt MATH · View at MathSciNet
- G. W. Whitehead, Elements of Homotopy Theory, vol. 61 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1978. View at MathSciNet