Systems with Hamiltonian Potentials.

Let E be a real Hilbert space with an inner product 〈·, ·〉 and the associated norm ‖ · ‖. Let A and B be two bounded subsets of E such that A links B. We describe the situation in which there are two linear, bounded and invertible operators B1, B2 : E → E and a functional H ∈ C1 (E, R) whose values are separated by B1B and B2A, i.e., $$ \mathop {\sup }\limits_{B_2 A} H \leqslant \mathop {\inf }\limits_{B_1 B} H $$. Note that B1B and B2A become much more uncontrollable. We prove the existence of a critical point of H without assuming the (PS) type conditions. This theory fits some special elliptic systems. [ABSTRACT FROM AUTHOR]