Let $\{u_n\} \subset H^1_0(\Omega)$ be a Palais-Smale sequence, and $\Omega$ a smooth and bounded domain in $\mathbb{R}^3$.
Consider $H^1_0(\Omega)$ with the norm $$||u|| = ||\nabla u||_2$$
My question is, there exists some result that gives me the relation: $$||u_n||^2 \leq C||u_n||^2_2.$$
I think the Poincare and the immersions theorems gives me the "conversely", but not this one! Any help will be appreciated!