Weak solutions for PDEs with Radon measures as right-hand sides can be obtained by a duality technique, which roughly proceeds as follows. Assume that the adjoint operator $A^*$ acts as isomorphism from $W^{1,q}_0(\Omega)$ to $W^{-1,q}(\Omega)$ for some $q>n$ with $\Omega\subset\mathbb{R}^n$. (This is the case if the coefficients of $A$ and the domain $\Omega$ is sufficiently smooth.) The closed range theorem and reflexivity of the spaces then implies that $A$ is an isomorphism from $W^{1,q'}_0(\Omega)$ to $W^{-1,q'}(\Omega)$ for $1/q'+1/q = 1$, i.e., $q'< n/(n-1)$. Since $W^{1,q}_0(\Omega)$ is continuously and densely embedded into $C_0(\Omega)$ for $q>n$, we have the dual embedding of $C_0(\Omega)^*$ (which can be identified by the Riesz theorem with the space of Radon measures) into $W^{-1,q'}(\Omega)$. Together, this yields for every Radon measure $f\in C_0(\Omega)^*\hookrightarrow W^{-1,q'}(\Omega)$ a unique solution $u\in W^{1,q'}(\Omega)$ to $Au=f$.
If $A^*$ lacks this maximal regularity, the solution is no longer unique, and you need to either extend the space of test functions (leading to solutions in the sense of Stampacchia) or restrict the space of solutions (leading to solutions in the sense of Boccardo–Gallouët); both approaches are equivalent. All this is very nicely explained in
Meyer, C., Panizzi, L., and Schiela, A. (2011). Uniqueness criteria for solutions of the adjoint equation in state-constrained optimal control. Numerical Functional Analysis and Optimization 32.9, pp. 983–1007.
For parabolic and nonlinear equations, see, e.g.,
Boccardo, L., and Gallouët, T. (1989). Non-linear elliptic and parabolic equations involving measure data. Journal of Functional Analysis 87, pp. 149–169.
