3
votes
1answer
341 views

Failure of Palais-Smale Condition C and the Mini-Max Principle

To get a thorough analysis of the critical point structure of a smooth function $f:M\to\mathbb{R}$ on a smooth Hilbert manifold $M$, a compactness assumption gets us far. That assumption is Condition ...
1
vote
1answer
253 views

How to explain the condition (C) in critical point theory?

Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f. How to see the meaning of " ...
1
vote
1answer
148 views

Critical points in Hilbert space [closed]

Let $f$ be a $C^1$ functional on a Hilbert space $X$, and $Y$ a closed subspace of $X$. Suppose the restriction of $f$ on $Y$ has a critical point $x_0 \in Y$. Q: Is $x_0$ a critical point of $f$?