Let $\Omega\subset \mathbb R^n$ and $f\in L^2(\Omega)$.
We consider $\{u\in C^1(G):||u||_{1,2,G}<\infty \}$,where $||u||_{1,2,G}$ is the "sobolev-norm" with parameter $1,2$ on $G$. Then $H^1(G)$ is the completeness of that set and $H^1_0(G)$ is the closure of $H^1(G)$
Show that there exists a $u\in H_0^1(\Omega)$ with $u-\Delta u=f$ in $\mathfrak{D}'(\Omega)$
I guess we should use the lemma of riesz somewhere, but I dont know where.. I am completely lost.
First of all, I think you mean $H_0^1$ is the closure of $C^1$, not $H^1$.
Secondly, your PDE will solved by using second existence theorem or directly by Lax-Milgram theory by using Bilinear operator.
For more information, I suggest you to read our bible Evans book, chapter 6. Look for First existence theorem and second existence theorem