Bounded Saddle Point Methods
Although a critical sequence does not necessarily lead to a critical point, we saw in Chapter III (cf. Theorem 3.4.1) that in some applications, bounded critical sequences do indeed lead to critical points. Thus one might ask if there are criteria that can be imposed which will produce bounded critical sequences. We study this question in Section 5.2. There we require the two linking sets A, B to be contained in a ball of radius R and impose a boundary condition on the sphere comprising the boundary of the ball to prevent deformations of the sets from exiting the ball. We then show that this indeed produces a bounded Palais-Smale sequence. However, the boundary condition is an additional restriction which asserts itself in the applications. As we shall see in Section 5.8, the restriction is not as severe as those used to cause a Palais-Smale sequence to be bounded. Consequently, the boundary condition pays for itself in applications.
KeywordsNontrivial Solution Compactness Condition Point Method Double Resonance Convergent Subsequence
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