Abstract
This paper is devoted to the computation of the index of a critical point for nonlinear operators with strong coefficient growth. These operators are associated with boundary value problems of the type \begin{center} \displaystyle ∑_{|α|=1}\mathscr{D}α{ρ2(u)\mathscr{D}αu+aα(x, \mathscr{D}1u)}=λ a0(x, u, \mathscr{D}1u), \ x∈Ω, u(x)=0, \ x∈∂Ω, \end{center} where Ω=\bm{R}n is open, bounded and such that ∂Ω∈ \bm{C}2, while ρ:\bm{R}→ \bm{R}+ can have exponential growth. An index formula is given for such densely defined operators acting from the Sobolev space W01, m(Ω) into its dual space. We consider different sets of assumptions for m>2 (the case of a real Banach space) and m=2 (the case of a real Hilbert space). The computation of the index is important for various problems concerning nonlinear equations: solvability, estimates for the number of solutions, branching of solutions, etc. The results of this paper are based upon recent results of the authors involving the computation of the index of a critical point for densely defined abstract operators of type (S+). The latter are based in turn upon a new degree theory for densely defined (S+)-mappings, which has also been developed by the authors in a recent paper. Applications of the index formula to the relevant bifurcation problems are also included.
