Upper bound for the number of critical levels for nonlinear operators in Banach spaces of the type of second order nonlinear partial differential operators

Abstract

Let f and g be two nonlinear functionals defined on a real Banach space X. Consider the eigenvalue problem λf′( u) = g′( u), u ϵ M r ( f) = { x ϵ X; f( x) = r} ( r > 0 is a prescribed number, f′ and g′ denote Frechet derivatives of f and g respectively). The value of the functional g at the critical point of the functional g with respect to the manifold M r ( f) is called the critical level. Denote Γ the set of all critical levels. It is known that Γ is at least countable (see, for instance, E. S. Citlanadze, Trudy Mosk. Mat. Obšč. 2 (1953), 235–274). In this paper we give an abstract theory for upper bound for the number of points of Γ and an application to partial differential equations of the second order. The regularity properties of solutions of such equations are of great importance.