Multiplicity results for p-Laplacian boundary value problem with jumping nonlinearities

• Main Authors: Tacksun Jung, Q-Heung Choi
• Format: Article
• Language: English
• Published: SpringerOpen 2019-03-01
• Series: Boundary Value Problems
• Subjects: p-Laplacian problem; p-Laplacian eigenvalue problem; Jumping nonlinearity; Contraction mapping principle; Leray–Schauder degree theory
• Online Access: http://link.springer.com/article/10.1186/s13661-019-1165-5

Abstract We investigate the multiplicity of solutions for one-dimensional p-Laplacian Dirichlet boundary value problem with jumping nonlinearities. We obtain three theorems: The first states that there exists exactly one solution when nonlinearities cross no eigenvalue. The second guarantees that there exist exactly two solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross just the first eigenvalue. The third claims that there exist at least three solutions, exactly one solution and no solution, depending on the source term, when nonlinearities cross the first and second eigenvalues. We obtain the first and second theorem by considering the eigenvalues and the corresponding normalized eigenfunctions of the p-Laplacian eigenvalue problem, and the contraction mapping principle in the p-Lebesgue space (when p≥2 $p\ge 2$). We obtain the third result by Leray–Schauder degree theory.