## Kyoto Journal of Mathematics

### Multiplicity of solutions for Neumann problems resonant at any eigenvalue

#### Abstract

We consider a semilinear Neumann problem with a reaction which is resonant both at $\pm\infty$ and at zero with respect to any eigenvalue, possibly the same one. Using the reduction method and Morse theory, we show that the problem has at least two nontrivial smooth solutions.

#### Article information

Source
Kyoto J. Math., Volume 54, Number 2 (2014), 259-269.

Dates
First available in Project Euclid: 2 June 2014

https://projecteuclid.org/euclid.kjm/1401741278

Digital Object Identifier
doi:10.1215/21562261-2642386

Mathematical Reviews number (MathSciNet)
MR3215567

Zentralblatt MATH identifier
1296.35040

#### Citation

Gasiński, Leszek; Papageorgiou, Nikolaos S. Multiplicity of solutions for Neumann problems resonant at any eigenvalue. Kyoto J. Math. 54 (2014), no. 2, 259--269. doi:10.1215/21562261-2642386. https://projecteuclid.org/euclid.kjm/1401741278

#### References

• [1] S. Aizicovici, N. S. Papageorgiou, and V. Staicu, Degree Theory for Operators of Monotone Type and Nonlinear Elliptic Equations with Inequality Constraints, Mem. Amer. Math. Soc. 196, Amer. Math. Soc., Providence, 2008.
• [2] H. Amann, Saddle points and multiple solutions of differential equations, Math. Z. 169 (1979), 127–166.
• [3] K.-C. Chang, Infinite-Dimensional Morse Theory and Multiple Solution Problems, Progr. Nonlinear Differential Equations Appl. 6, Birkhäuser, Boston, 1993.
• [4] J. Dugundji, Topology, Allyn and Bacon, Boston, 1966.
• [5] M. Filippakis and N. S. Papageorgiou, Multiple nontrivial solutions for resonant Neumann problems, Math. Nachr. 283 (2010), 1000–1014.
• [6] L. Gasiński and N. S. Papageorgiou, Neumann problems resonant at zero and infinity, Ann. Mat. Pura Appl. (4) 191 (2012), 395–430.
• [7] A. Granas and J. Dugundji, Fixed Point Theory, Springer Monogr. Math., Springer, New York, 2003.
• [8] S. Li, K. Perera, and J. Su, Computation of critical groups in elliptic boundary value problems where the asymptotic limits may not exist, Proc. Roy. Soc. Edinburgh Sect. A 131 (2001), 721–732.
• [9] S. Liu, Remarks on multiple solutions for elliptic resonant problems, J. Math. Anal. Appl. 336 (2007), 498–505.
• [10] C. R. F. Maunder, Algebraic Topology, Cambridge Univ. Press, Cambridge, 1980.
• [11] N. S. Papageorgiou and S. T. Kyritsi-Yiallourou, Handbook of Applied Analysis, Adv. Mech. Math. 19, Springer, New York, 2009.
• [12] C.-L. Tang, Multiple solutions of Neumann problems for elliptic equations, Nonlinear Anal. 54 (2003), 637–650.
• [13] C.-L. Tang and X.-P. Wu, Existence and multiplicity for solutions of Neumann problem for elliptic equations, J. Math. Anal. Appl. 288 (2003), 660–670.