## International Journal of Differential Equations

### On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds

#### Abstract

Given that $(M,g)$ is a smooth compact and symmetric Riemannian $n$-manifold, $n\ge 2$, we prove a multiplicity result for antisymmetric sign changing solutions of the problem $-{\epsilon }^{2}{\Delta }_{g}u+u={|u|}^{p-2}u$ in $M$. Here $p>2$ if $n=2$ and $2 if $n\ge 3$.

#### Article information

Source
Int. J. Differ. Equ., Volume 2010, Special Issue (2010), Article ID 432759, 11 pages.

Dates
Accepted: 7 December 2009
First available in Project Euclid: 26 January 2017

https://projecteuclid.org/euclid.ijde/1485399890

Digital Object Identifier
doi:10.1155/2010/432759

Mathematical Reviews number (MathSciNet)
MR2592739

Zentralblatt MATH identifier
1204.58015

#### Citation

Micheletti, Anna Maria; Pistoia, Angela. On the Existence of Nodal Solutions for a Nonlinear Elliptic Problem on Symmetric Riemannian Manifolds. Int. J. Differ. Equ. 2010, Special Issue (2010), Article ID 432759, 11 pages. doi:10.1155/2010/432759. https://projecteuclid.org/euclid.ijde/1485399890

#### 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.MR2180862
• 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.MR2360924
• 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.MR2468411
• 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.MR2455770
• 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.MR2471314
• 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.MR2448651
• 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.MR2536955
• M. Ghimenti and A. M. Micheletti, “On the number of nodal čommentComment on ref. [16?]: Please update the information of this reference, if possible. 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.MR898712
• 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.MR1382508
• 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.MR1758878
• 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.MR1671254
• 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.MR1659466
• 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.MR1750734
• 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.MR1737546
• 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.MR1785679
• 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.MR1639159 (99g:35014)
• 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.MR1342381
• 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.MR1635448
• 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.MR1635869
• 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.MR1696122
• 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.MR1742305
• 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.MR1782991
• C. Gui, “Multipeak solutions for a semilinear Neumann problem,” Duke Mathematical Journal, vol. 84, no. 3, pp. 739–769, 1996.MR1408543
• 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.MR1721719
• 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.MR1758231
• 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.MR1743431
• 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.MR1620632
• 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.MR1219814
• 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.MR1115095
• 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.MR1429093
• 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.MR1622042
• 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.MR1384393
• 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.MR1322324
• G. W. Whitehead, Elements of Homotopy Theory, vol. 61 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1978.MR516508