Does there exist two linearly independent functions $u$ in $\mathbb{R}^{2}$ satsfying $\bar{\partial}^{2}u(x,y) + A(x,y)u(x,y) = 0$?

Question

Does there exist two linearly independent functions $u$ in $\mathbb{R}^{2}$ satsfying $$\frac{\partial^{2}}{\partial \bar{z}^{2}}u(x,y) + A(x,y)u(x,y) = 0$$ where $A\in C^{\infty}(\mathbb{R}^{2})$?

We define

$$\frac{\partial}{\partial\bar{z}}u(x,y) ={\frac {1}{2}}\left({\frac {\partial }{\partial x}u(x,y)} + i{\frac {\partial }{\partial y}}u(x,y)\right).$$

We know that if $A$ is a antiholomorphic function, then our equation has two independent antiholomorphic solutions $u$.

Answer 1

It seems to me that this fails already in your example with $A=0$. All entire functions solve $$(\partial_\overline z)^2u=0, $$ and certainly that is not a 2-dimensional space.

EDIT. It is now clear that you want at least two independent solutions. This may be true and it is not disproved by this answer. This (trivial) answer proves that there may be more than two independent solutions.