The index of an isolated critical point for a class of non-differentiable elliptic operators in reflexive Banach spaces

This paper is motivated by the problem of computing the index of an isolated critical point for a certain class of nonlinear elliptic operators. Such a class includes operators A which are defined by〈Au,ϕ〉=∑|α|⩽1∫Ωaα(x,u,D1u)Dαϕ(x)dx,u,φ∈W˚m1(Ω)and are associated with boundary value problems of the type(*)∑|α|⩽1(-1)|α|Dαaα(x,u,D1u)+λb(x,u,D1u)=0,x∈Ω,(**)u(x)=0,x∈∂Ω,where Ω⊂Rn,n>2, is open, bounded and has a C2-boundary ∂Ω. As an application of our new abstract theory, we obtain an index formula for such an operator A acting from the Sobolev space W˚m1(Ω) into its dual space [W˚m1(Ω)]* for 1