Counterexample for Palais-Smale condition

Question

I have trouble proving that functional $I:H\to\mathbb{R}$ given by $$I(u)=\frac{1}{2}\|u\|^2-\frac{1}{2}(u,f)^2$$ does not satisfy Palais-Smale condition if $\|f\|=1$. I managed to prove that when $\|f\|\not =1$ the condition is satisfied, but I cannot find any suitable counterexample for $\|f\|=1$.

I think it will be helpful if I put here the derivative of this functional: $I'(u)=u-(u,f)f$.

Thanks a lot, K.

Answer 1

I am assuming that $H$ is a Hilbert space and $(\cdot,\cdot)$ the associated inner product.

Let $e_1,e_2,...$ be a Hilbert basis and write $u=\sum_{i=1}^\infty a_ie_i$. Take $f=e_1$ and note that $$I(u)=\frac{1}{2}\sum _{i=2}^\infty a_i^2\ \mbox{and}\ \ I'(u)=\sum _{i=2}^\infty a_ie_i.$$

Take any sequence of the form $u_n=a_{1n}e_1$, where $|a_{1n}|\to \infty$ as $n\to \infty$. Can you conclude?