Eigenvalue problems for nonlinear elliptic variational inequalities on a cone

Eigenvalue problems for variational inequalities on a closed convex cone C in a real Banach space V, of the form 〈g′(v) ? λh′(v), w ? v〉 ? 0 for all w in C, are considered with the normalization g(v) = r, where g and h are real-valued C¹ functions on V. Theorems are proved on the existence of solutions λ(r) and v(r), and on their dependence upon the normalization constant r > 0. In particular, the relation, as r → 0, of λ(r), v(r) to solutions of the analogous problem with g″(0) and h″(0) in place of g′ and h′, is discussed. The theorems are applied to elliptic inequalities for Euler-Lagrange operators corresponding to multiple integral functionals on closed subspaces of Sobolev spaces, and to the inequality arising from the von Karman equations for the buckling of a thin elastic plate which is constrained to buckle in only one direction.