Multiple solutions for some elliptic boundary value problem with jumping nonlinearities

Abstract

We investigate the multiplicity of solutions for the one-dimensional elliptic Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems depending on the source terms when nonlinearities cross some eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the elliptic eigenvalue problem, and the contraction mapping principle on p-Lebesgue space. We obtain the third result by Leray–Schauder degree theory.

Keywords:

• Elliptic problem
• elliptic eigenvalue problem
• jumping nonlinearity
• contraction mapping principle
• Leray–Schauder degree theory

2000 Mathematics Subject Classifications:

• 35A01
• 35A16
• 35J30
• 35J40
• 35J60

Disclosure statement

No potential conflict of interest was reported by the author(s).

Authors' contributions

Q-Heung Choi participate in applying the method for solving this problem and drafted the manuscript. Tacksun Jung introduced the main ideas of multiplicity study for this problem. All authors contributed equally to read and approved the final manuscript.

Additional information

Funding

Q-Heung Choi was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology [grant number NRF-2017R1D1A1B03030024]. Tacksun Jung was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Science, ICT and Future Planning [grant number NRF-2017R1A2B4005883].