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For homogeneous Dirichlet boundary problem $$ \left\{\!\! \begin{aligned} &-\Delta u+c(x)u=f(x),~\text{in}~\Omega,~~~~~~~~~~~~~~(I)\\ &u=0,~\text{on}~ \partial\Omega \end{aligned} \right. $$ I know that the strong (classical) solution is defined as the function $u\in C^2(\bar{\Omega})$ satisfying the problem (I) above in the usual sense. On the other hand, the weak solution is the function $u\in H^1_0(\Omega)$ satisfying the following:
$$ \int_\Omega\left(\sum_{i=1}^n\frac{\partial u}{\partial x_i} \frac{\partial v}{\partial x_i}+c(x)uv\right)\,dx=\int_\Omega fv\,dx~~~~~~~(II) $$ for every $v\in H_0^1(\Omega)$.

My question are the following:

(a) How do I prove/show that such a strong solution $u\in C^2(\Omega)$ really exists? In other words, how do I solve for this $u$?

(b) If I want to obtain a weak solution, do I have to solve (II) explicitly to obtain $u$? How is it easily solved? Any reference, guide?

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This is a question of regularity i.e. you know that a weak solution $u \in H^1_0(\Omega)$ exists so now you must prove that any weak solution of (I) is actually an element of $ C^2(\Omega) \cap C(\overline{\Omega})$. (Note: you didn't mention it but for $u$ to be a classical solution you also need $u \in C(\overline{\Omega})$ for the boundary condition to make sense).

Your PDE is elliptic and the regularity theory of elliptic PDE is very well understood. A classical reference is Partial Differential Equations by Lawrence C. Evans (see Chapter 6) or a more in depth reference is Elliptic Partial Differential Equations of Second Order by Gilbarg and Trudinger. In fact, just googling 'regularity theory elliptic PDE' will likely return hundreds of references.

To answer your second question about 'explicitly solving' (II) it is practically impossible. To put it into perspective if $u$ satisfies (II) with $c=0$, $f \in C^\infty(\Omega)$ and $\Omega$ with $C^\infty$ boundary then standard regularity theory implies that $u \in C^\infty(\overline{\Omega})$. Hence, $u$ satisfies $-\Delta u =f $ in $\Omega$ and $u=0$ on $\partial \Omega$. We know then that we can write $u$ in terms of a Green's function, but only in very very specific cases of $\Omega$ can we actually work out that Green's function. If $c \neq 0$ there are still 'Green's functions' but these are even more challenging to calculate than before! This is why mathematicians gave up on trying to 'solve' PDE a long time ago - it is much easier (and often more useful) to understand the underlying theory.

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