Multiple solutions of nonlinear elliptic functional differential equations

We shall consider weak solutions of boundary value problems for elliptic functional differential equations of the form -Sigma(n)(j=1) D-j[a(j)(x, u, Du; u)] + a(0)(x, u, Du; u) - F, x is an element of Omega with homogeneous boundary conditions, where Omega subset of R-n is a bounded domain and; u denotes nonlocal dependence on u (e.g. integral operators applied to u). By using the theory of pseudomonotone operators, one can prove existence of solutions. However, in certain particular cases it is possible to find theorems on the number of solutions. These statements are based on arguments for fixed points of certain real functions and operators, respectively.