How to show that $\Delta u=f$ admits no $C^2$ solution?

Question

Let $D(0,2)$ be the open disk centered at the origin with radius $2$ in $\mathbb{R}^2,$ let $P$ be a homogeneous harmonic polynomial of degree $2$ with $D^{\alpha}P$ not identically zero for some multi index $\alpha(|\alpha|=2),$ that's: $\Delta P=0$ and $P(\lambda x,\lambda y)=\lambda^2P(x,y)$ for any $(x,y)\in\mathbb{R}^2$ and $\lambda>0.$ Choose $\eta\in C^{\infty}_{0}(D(0,2))$ with $\eta=1$ on $D(0,1),$ define $$f(x)=\sum_{n=1}^{\infty}\frac{1}{n}\Delta(\eta P)(2^nx)$$ I have showed that $f$ is continuous, but I actually have little idea to show that any function $u$ satisfies $\Delta u=f$ is not in $C^2,$ can someone help me or give me some hints?