I have encountered the following PDE, and am asked to determine the minimal and maximal smoothness of solutions to this PDE. Unfortunately, I've never studied PDE at all, and don't even begin to know what my tools are.
$$\Delta u(x,y) = |x|^3 + e^y$$
The question is to determine the minimal and maximal $r$ so that a solution can be found in $C^r(\mathbb D)$ (the unit disk). If I wrote this as just $\frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} =|x|^3 + e^y$, I can observe, for example, that $\frac{x^5}{5} \operatorname{sgn}(x) + e^y$ satisfies this, purely by inspection. But this doesn't help me figure out what the maximal and minimal smoothness class is - I can probably come up with a bunch of examples similarly, but I don't know how to work on the general case.
It'd be really great if someone could at least point me in the direction of the tools, if not explain to me how to work this out.
Edit: I changed my engineered function and the title, because it occurs to me that this PDE does not involve the gradient, but the Laplacian, and I am easily confused by these two.
Let $f$ be a function of class $C^k$ - no more and no less. Then if $\Delta g = f$ as you describe, I can tell you two things about $g$. (Note that the RHS in your equation is $C^2$.)
1) $g$ is of class at most $C^{k+2}$. That's because if $g$ is $C^\ell$, then $\Delta g \in C^{\ell-2}$ (this is completely straightforward - you're taking two derivatives!).
2) $g$ is of class at least $C^{k+2}$. This is a remarkable theorem, known as the elliptic regularity theorem. It is quite difficult to prove, and I'm not going to provide a sketch here. It applies in great generality to many more operators than just the Laplacian (including, say, the Cauchy-Riemann operator $\bar \partial$, which implies that complex differentiable functions are smooth.)
So the maximal $r$ is easy to find: write down a solution and use 1)! The minimal $r$ requires some work. One possible simplification is if you know that all harmonic functions are smooth (again, a form of the elliptic regularity theorem); then if $g,g'$ are two solutions, you see that $\Delta(g-g') = 0$, so the difference between your two solutions is smooth; because you've found that one is $C^k$, all of them must be $C^k$, since they differ by a smooth function.