Give a subharmonic function that is not real analytic

Question

Is there a function $f(x)$ defined on an open set of $\mathbb{R}^{2}$ ($ k\geq2 $) such that

1) $f$ is $C^{2}$ smooth,

2) $f$ is subharmonic, i.e. the laplacian $\Delta f$ of $f$ is positive,

3) $f$ is not real analytic?

Answer 1

Define $f(x,y)= |x|^3 + x^2.$ Then $f$ is $C^2$ and $\Delta f(x,y) = 6|x|+2.$ If $f$ were real analytic, then its Laplacian would be real analytic. But clearly this fails.