$p$-Laplacian boundary value problem with jumping nonlinearities

We investigate multiplicity of solutions for one dimensional $p$-Laplacian Dirichlet boundary value problem with jumping nonlinearites. We obtain three theorems: The first one is that there exists exactly one solution when nonlinearities cross no eigenvalue. The second one is that there exist exactly two solutions, exactly one solutions and no solution depending on the source term when nonlinearities cross one first eigenvalue. The third one is that there exist at least three solutions, exactly one solutions and no solution depending on the source term when nonlinearities cross the first and second eigenvalues. We obtain the first theorem and the second one by eigenvalues and the corresponding normalized eigenfunctions of the $p$-Laplacian Dirichlet eigenvalue problem, and the contraction mapping principle on $p$-Lebesgue space (when $p \geqslant 2$). We obtain the third result by Leray-Schauder degree theory.
This is a joint work with Q-Heung Choi (Inha University, Incheon, South Korea).