Energy functional in Sobolev Space

Question

Let $E[u]:=\int_W |\nabla u|^2 +V(x)u^2 dx$ be the energy functional on functions $u\in H^1_0(W)$ and $V\in L^\infty(W)$.

Can someone help me show that $E$ is bounded below for all $u\in H^1_0(W)$ with $||u||_{L^2}=1$ but not bounded above?

Answer 1

For unboundedness from above, just construct a sequence $(u_n)$ that oscillates (with increasingly "stronger" oscillations) around 1 (i.e. $\|\nabla u_n\|_{L^2}$ is going to infinity, and $\|u_n\|_{L^2}\equiv 1$).

For the lower bound, note that $$\int Vu^2 dx\geq -\|V\|_{L^\infty}\|u\|_{L^2}=-\|V\|_{L^\infty}.$$

Edit: for the unboundedness from above, to construct $u_k$, let $W\subset \mathbb{R}^n$ be a bounded box, and let $W'\subset W$ be a sub-box with $W'=[a_1,b_1]\times\cdots\times [a_n,b_n]$. Wlog (upon domain scaling and translation) assume $a_i=0$ for any $i$, $b_1=1$. Let $v_k:=(\cos (k\pi x),0,\cdots,0)$ be defined on $W'$. Let $u_k=v_k$ on $W'$, and whatever you want on $W''\backslash W'$ (just make sure $u_k\in H_0^1$ and $\|u_k\|_{L^2}=1$).