Let $D=D_1(0)\subset\mathbb{R}^2$ and $\lambda\in\mathbb{R}$, $\lambda>0$.
Consider the Helmholtz operator $L=(\Delta +\lambda I).$

Let $f\in Ker_0(L)$, that is $f$ solves
$$ Lf=0\quad\text{ in $D$}, $$
$$ f=0\quad\text{ on $\partial D$}.$$

Consider, for a given boundary datum $g$, the problem
$$ Lu=f\quad\text{ in $D$},$$
$$ u=g\quad\text{ on $\partial D$}. $$

Are there necessary and sufficient conditions for the solvability of this problem?
If so, which would be the correct formulation in terms of function spaces?