Dirichlet problem, Poisson equation with extra term Let me state the problem first of all. $f$ is an $L^2(U)$-function for some $U$, a bounded domain of $R^n$ with a smooth boundary. Consider the Dirichlet problem
$$\begin{cases}\Delta u\,+\,a(x)u=f(x)\,\quad \text{in }U\\
u=0\,\quad \text{on }\partial U\,.
\end{cases}
$$
Furthermore, we may assume that
\begin{equation}
||u||_{L^2(U)}^2 \leq C||Du||_{L^2(U)}^2
\end{equation}
holds on $U$ for some $C>0$ and all $u\in C_0^1(U)$. Here are the following questions to answer.
$i$. Find a 'smallness condition' for $||a||_{L^{\infty}(U)}$, such that under this condition a classical solution is unique.
$ii$.   Show the existence of a weak solution in $H_0^1(U)$.
The inequality we are given very much resembles a Sobolev inequality, I have seen a few of those. However, I am not sure how to come up with something similar for $a$, depending only on $C$ maybe even. And how do I translate all this into proving certain solutions exists?
I guess my question is a lot more general. I am doing a course in PDEs right now, following Evans' book. We have been presented some theory on certain solutions (some weak, some classical) to the most common PDEs, and seen some regularity results on these. E.g. the Laplacian, Heat equation, Elliptic etc. I am fairly comfortable with functional analysis and elementary Sobolev space theory. However, everytime I am presented with a PDE problem like this and asked to work questions such as these, I honestly do not even know where to begin. The slight changes in this Dirichlet problem as opposed to similar results from class/the book throws me off, because the proofs in there seem very aimed at the specific problems. I am looking for any kind of help to get started on this. Thanks.
 A: A good starting point with something like this (more item $ii$, but it can help with $i$ as well) is to figure out what the weak formulation is.  Once you have that in hand, the functional analytic tools that you need will begin to become clearer.  It can also sometimes be useful when you're setting up the weak formulation to initially ignore the subtle issues of integrability and differentiability and only begin to consider these once you have the problem written in an appropriate form.
Suppose, then, we have a smooth solution $u$ (translation: let's ignore the subtleties as we said above) to your Dirichlet problem.  We multiply it by another smooth function $v$ and integrate by parts over $\Omega:$
$$
\int_\Omega f v = \int_\Omega a u v + \int_\Omega \Delta u v = \int_\Omega a u v - \int_\Omega \nabla u \cdot \nabla v  + \int_{\partial \Omega} \partial_\nu u v.
$$
Now, we have no idea whatsoever what $\partial_\nu u = \nabla u \cdot \nu$ (where $\nu$ is the outward unit normal) is, so if we want to ignore this term then we should assume that $v =0$ on $\partial \Omega$ as well.  Assuming this, we then find that
$$
\int_\Omega -f v = \int_\Omega \nabla u \cdot \nabla v - a uv
$$
for all smooth $v$ that vanish on the boundary.  On the other hand, say we find a smooth $u$ satisfying this integral identity for all smooth $v$ vanishing on the boundary.  Then we can integrate by parts again (as above) to see that
$$
\int_\Omega (\Delta u + a u - f) v =0
$$
for all such $v$, which then implies that $\Delta u + a u -f =0$ in $\Omega$.
This all tells us that for smooth $u$, the pointwise identity $\Delta u +a u =f$ is equivalent to the above integral identity.  The key observation is then that we can make sense of the integral identity without assuming anything even close to smoothness.  If we want to work in the context of Hilbert spaces, then a natural choice is that all of the $L^2$ pairings make sense, and so we take $a \in L^\infty$, $f \in L^2$, $u \in L^2$, and $\nabla u \in L^2$.  We also want $u=0$ on $\partial \Omega$, and we can make that happen with trace theory.   In other words, we want $u \in H^1_0$.  Note that we could actually go further and replace with $f \in (H^1_0)^\ast = H^{-1}$, but since you ask for $f \in L^2$ we'll stick with that.  The other thing to note is that once we've dropped the requirements on $u$ all the way down to $u \in H^1_0$, it is clear that we can also relax the requirements for $v$.  In particular, we can get away with requiring $v \in H^1_0$ as well.
We have thus arrived at our weak formulation of the PDE.  We want to find $u \in H^1_0$ such that
$$
\int_\Omega -f v = \int_\Omega \nabla u \cdot \nabla v - a uv \;\; \text{ for all } v \in H^1_0.
$$
So now the question is: have we actually gained anything by switching to this formulation?  Is this form of the problem somehow more tractable than what we started with?  The answer is: yes!  We now have the ability to wield the tools of functional analysis to attack this problem.  In particular we have Riesz representation, Lax-Milgram, etc.  Since you just asked for help in getting started I will trail off here.  I hope this helps.
