The Monotonicity Trick

Martin Schechter

Chapter Chapter 4 in Critical Point Theory, 2020, pp 47-60 from Springer

Abstract: Abstract The use of linking or sandwich pairs cannot produce critical points by themselves. The most they can produce are sequences satisfying G ( u k ) → a , ( 1 + ∥ u k ∥ ) G ′ ( u k ) → 0. $$\displaystyle G(u_k)\to a,\quad (1+\|u_k\|)G'(u_k)\to 0. $$ If such a sequence has a convergent subsequence, we obtain a critical point. Lacking such information, we cannot eliminate the possibility that ∥ u k ∥ → ∞ , $$\displaystyle \|u_k\| \to \infty , $$ 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?

Date: 2020
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DOI: 10.1007/978-3-030-45603-0_4

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