Splitting theorem, Poincaré–Hopf theorem and jumping nonlinear problems

In this paper, we establish a Gromoll–Meyer splitting theorem and a shifting theorem for J ∈ C2−0 (E,R) and by using the finite-dimensional approximation, mollifiers and Morse theory we generalize the Poincaré–Hopf theorem to J ∈ C1 (E,R) case. By combining the Poincaré–Hopf theorem and the splitting theorem, we study the existence of multiple solutions for jumping nonlinear elliptic equations. © 2004 Elsevier Inc. All rights reserved.