I am trying to show the following:
$\forall u \in H_0^s(\Omega), u \ne 0, \exists \psi \in H_0^s(\Omega)$ s.t. $h'(0) \ne 0$, in which $h(t)=\int_{\Omega} |u+t\psi|^{p+1}$. Here $\Omega \subset \mathbb{R}^n$ is a bounded domain, and $H_0^s(\Omega)$ is the Sobolev space on this domain.
It is not difficult to show that $h'(0)=(p+1) \int_\Omega u^p \psi$, yet I don't know how to proceed. (probably this "not difficult" statement is incorrect) Any solutions or hints will be greatly appreciated. Thanks!