I'm given the following variational integral

$F(u) = \int_{0}^1 \frac{[u'(x)]^2}{2} - u(x) + u^4(x) dx$

defined on the set

$C = \{ u \in C^1([0,1]): u(0) = u(1) = 0 \}$

I have to show that

$ -\infty < \inf\{F(u): u \in C\} < 0 $

I've already proven the first inequality by noticing that the function $g(u) = -u + u^4$ is bounded below. Now, to prove that $\inf\{F(u): u \in C\} < 0 $, I am given the suggestion to consider a positive function $v \in C$, and to show that, for any positive $\lambda$ close to zero, $F(\lambda v) < 0$. Any hints?

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  • 1
    $\begingroup$ Consider, for instance, $v(x)=x(1-x)$. Then $F(\lambda v)=-\frac{\lambda}{6}+\frac{\lambda^2}{6}+\frac{\lambda^4}{630}$. $\endgroup$
    – Gonçalo
    Sep 18 at 21:27


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