Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
6415085 | Journal of Functional Analysis | 2014 | 32 Pages |
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,â) and Cq(J(a,b),0) is yielded as (a,b)âΣ, where Cq(J,â) 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)âR2 for which the problem{âÎu+αu=auâ+bu+,xâΩ,âuâν=0,xââΩ, has a nontrivial solution, u+=maxâ¡{u,0}, uâ=minâ¡{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.