Generalized Poincaré-Hopf theorem and application to nonlinear elliptic problem

In this paper we get an extended version of Poincaré-Hopf theorem. Without the assumption that critical point set between two level sets of energy functional is finite, this result actually generalizes Morse inequality. And the isomorphism between Cq(J,∞) and Cq(J(a,b),0) is yielded as (a,b)∉Σ, where Cq(J,∞) denotes the critical groups of energetic functional J at infinity and Cq(J(a,b),0) stands for the critical groups of functional J(a,b) at zero, and Σ is the set of points (a,b)∈R2 for which the problem{−Δu+αu=au−+bu+,x∈Ω,∂u∂ν=0,x∈∂Ω, has a nontrivial solution, u+=max⁡{u,0}, u−=min⁡{u,0}. (Concerning the definitions of J and J(a,b), see Section 4.) As to application aspects, we are mainly concerned with nonlinear elliptic problem with Neumann boundary condition provided that the origin is a non-isolated critical point and obtain the existence of multiple solutions.