Given some $\|\psi \|$ $\in$ $L^2 (\mathbb R^n) $ such that $\| \psi \|_2 =1$ and a function (potential) $V: \mathbb R^n \rightarrow \mathbb R$. The Schorödinger equation tells us that $-\triangle \psi(x) +V(x)\psi(x)= E\psi(x)$, where $E$ is an eigenvalue of the Schroedinger Equation. Out of this equation arises the variation problem as follows: define the functional $\mathcal E(\psi) = T_{\psi} + V_{\psi}$ (the kinetic and potential energies, respectively). Then, $T_{\psi} = \int_{\mathbb R^n} \mid \nabla \psi(x) \mid ^2 dx$ and $V_{\psi} = \int_{\mathbb R^n} V(x) \mid \psi (x) \mid ^2 dx$. The idea here is that we want to find a minimizing function $\psi_0$ of the functional $\mathcal E(\psi)$ under the constraint $\|\psi\|_2 =1$. To first show that such a minimizer exists, we must show that $\mathcal E(\psi)$ is bounded from below.
A little thought shows us that we want the Potential Energy,$V_{\psi}$, (which can be negative), to be dominated by $T_{\psi}$ so that we can make sure that $\mathcal E(\psi)$ is bounded below. To assure this, we appeal to the generalized Sobolev inequality that promise us our desired bounds. Let us appeal to the following Sobolev inequality ( from now on restrict ourselves to case $n \geq 3$: $\| f \|_2 ^2 \geq S_n \|f\|_2 ^2$ where $S_n = n(n-2)/4 \mid \mathcal S^n \mid ^{2/n}$. So, we can say that $ T_\psi \geq S_n {\int_{\mathbb R^n} \mid \psi \mid ^ (2n/(n-2)) dx}^{(n-2)/n}$
Appealing to Hölder's inequality, we have for any potential $V \in L^{n/2} (\mathbb R^n)$ (and as I said above, restricting ourselves to the case $n\geq 3$, $T_{\psi} \geq S_n \langle \psi, V \psi \rangle \| V \|_{n/2} ^{-1}$. We immediately deduce from this fact that whenever $\| V \|_{n/2} \leq S_n$, then $T_\psi \geq V_\psi$. Since $T_{\psi}$ is never negative and dominates $V_\psi$, we found a lower bound for the fucntional: $\mathcal E(\psi) = T_\psi +V_\psi \geq 0$.
Now here is where the problem arises:
Let us extend this to potential $V \in L^{n/2} (\mathbb R^n) + L^{\infty} (\mathbb R^n)$. So suppose $V(x)=v(x) + w(x)$, and $v\in L^{n/2} (\mathbb R^n)$ and $w \in L^{\infty} (\mathbb R^n)$. To find the lower bound for the functional $\mathcal E(\psi)$ in this case, we must first how that there is some constant $\lambda$ such that $h(x)=\min (v(x)-\lambda,0)\leq 0$ satisfies $\| h\|_{n/2} \leq \frac 1 2 S_n$. Is this supposed to be simple to do? I have very little experience with Sobolev inequalities; does a proof of the above stated fact depend on them explicitly? The cool thing about proving this fact is that we can again find another lower bound for $\mathcal E(\psi) \leq \lambda - \|w\|_{\infty}$. Even better, we can confirm out physical intutions and bound the kinetic energy in terms of the total energy.