$u = 0$ is the only solution to the homogeneous Helmholtz equation $\Delta u + k^2 u = 0$?

Question

We have that the solution to the inhomogeneous Helmholtz equation

$$\Delta u + k^2 u = f$$

can be represented by

$$u(x) = \int_{\mathbb{R}^3}G(x - y) f(y) dy$$

where $G$ is the fundamental solution of the Helmholtz equation.

But doesn't this imply that

$$\Delta u + k^2 u = 0$$

only has the solution $u = 0$? But this doesn't make sense as

Edit: The Sommerfeld radiation condition applies.

Answer 1

To construct $G$, you needed to impose Sommerfeld's radiation condition which restricts the class of solutions that you could consider. Hence I believe $u=0$ is the only solution to the homogeneous problem in this restricted class.