Definition of weak solutions

Definition of weak solutions

Why are weak solutions defined like:
A function $u \in H^1(\Omega)$ is a weak solution of $$ Lu=div(A\nabla
u)+b\cdot\nabla u+cu=f+divF, in \Omega $$ if $$ \int_\Omega \nabla
\phi\cdot (A\nabla u-F)dx=\int_\Omega\phi(b\cdot \nabla u+cu-f) dx $$
holds for every $\phi \in C_0^\infty(\Omega).$
I mean, we define this new notion "weak solutions" in order to generalize
classical solutions. Then we just need one purpose which is "classical
solution $\Rightarrow$ weak solution" and if everything is sufficiently
smooth, then it follows that a weak solution is automatically a classical
one. If this is our motivation to define weak solutions, then there would
have tons of methods to do.
My point is why this type of definition of weak solutions is so important.
Is there any other motivations like from geometry or other subjects to
make this definition so outstanding and is there any other type of
definitions of generalized solutions?