LIMIT RELATIVE CATEGORY THEORY APPLIED TO THE CRITICAL POINT THEORY

Let H be a Hilbert space which is the direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.)⁄ condition and f|X0'X4 has no critical point with level c. For the proof of main theorem we use the nonsmooth version of the classical deformation lemma and the limit relative category theory. 1. Introduction and statement of main result Let H be a Hilbert space which is a direct sum of five closed subspaces X0, X1, X2, X3 and X4 with X1, X2, X3 of finite dimension. Let J be a C 1,1 functional defined on H with J(0) = 0. In this paper we investigate the number of nontrivial critical points of the C 1,1 functional J under some conditions on the sublevels of J and the shape of J. We show the existence of at least four nontrivial critical points when the sublevels of J (the torus with three holes and sphere) link and the functional J satisfies the sup-inf variational inequality on the linking subspaces, and the functional J satisfies (P.S.) ⁄ condition and J|X0'X4 has no critical point with level c. Micheletti and Saccon prove in (13) that the functional J has at least two nontrivial critical points under the same conditions on J except the condition that the sublevel sets are the torus with one hole and the sphere. In this paper we improve this result to the case that the sublevel sets are the torus with three holes and sphere. Now, we state the main result