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Published in:
2020 | OriginalPaper | Chapter
4. The Monotonicity Trick
Author : Martin Schechter
Published in: Critical Point Theory
Publisher: Springer International Publishing
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Abstract
The use of linking or sandwich pairs cannot produce critical points by themselves. The most they can produce are sequences satisfying If such a sequence has a convergent subsequence, we obtain a critical point. Lacking such information, we cannot eliminate the possibility that which destroys any hope of obtaining a critical point from this sequence. On the other hand, knowing that the sequence is bounded does not guarantee a critical point either. But there is a difference. In many applications, knowing that a sequence satisfying (4.1) is bounded allows one to obtain a convergent subsequence. This is just what is needed. For such applications it would be very helpful if we could obtain a bounded sequence satisfying (4.1). This leads to the question: Is there anything we can do to obtain such a sequence?
$$\displaystyle G(u_k)\to a,\quad (1+\|u_k\|)G'(u_k)\to 0. $$
$$\displaystyle \|u_k\| \to \infty , $$