# Give a subharmonic function that is not real analytic

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?

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.
In the whole plane, take the standard (smooth, radial, positive, supported on $$B_1(0)$$) mollifier $$\rho$$ and the fundamental solution of the Laplace operator $$\Gamma$$, s.t. $$\Delta\Gamma=-\delta$$, where $$\delta$$ is the Dirac delta (you can do this in all dimensions). The convolution $$u=-\rho *\Gamma$$ is smooth and subharmonic: $$-\Delta (\rho *\Gamma)= -(\Delta \Gamma)*\rho=\delta*\rho\geq 0$$. It is not analytic because $$\rho *\Gamma(x)=\Gamma(x)$$ outside the unitary ball, as $$\Gamma$$ is harmonic away from $$0$$ (standard regularization preserves harmonic functions) and thanks to unique analytic continuation.
You can do it more generally taking $$u=-\Gamma*\phi$$ for, say, a positive test function $$\phi$$