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.