The index of a critical point for nonlinear elliptic operators with strongcoefficient growth

This paper is devoted to the computation of the index of a critical point f or nonlinear operators with strong coefficient growth. These operators are associated with boundary value problems of the type [GRAPHICS] u(x) = 0, x is an element of partial derivative Omega, where Omega subset of R-n is open, bounded and such that partial derivative Omega is an element of C-2, while rho : R --> R+ can have exponential grow th. An index formula is given for such densely defined operators acting fro m the Sobolev space W-0(1,m)(Omega) into its dual space. We consider differ ent 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 impo rtant for various problems concerning nonlinear equations: solvability, est imates for the number of solutions, branching of solutions, etc. The result s 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 o perators of type (S+). The latter are based in turn upon a new degree theor y for densely defined (S+)-mappings, which has also been developed by the a uthors in a recent paper. Applications of the index formula to the relevant bifurcation problems are also included.