[also asked here:http://mathoverflow.net/questions/122191/existence-of-a-function]
All arguments are in $\mathbb{R}^3$.
Suppose $n(x)$ is a smooth function where $\mathbf{supp}(n(x)-1)$ is a compact set $\Omega$. i.e. $n(x) = 1$ when $x$ is outside $\Omega$.
Assume there are some points $x_j\in\Omega$, where $j=1,2,\cdots.m$.
Consider Helmholtz equation
$\Delta u + k^2 n(x) u = 0$
And I want to know if there is a complex-valued function $u:\mathbb{R}^3\rightarrow\mathbb{C}$ satisfies the equation and also vanishes at $x_j$. i.e. $u(x_j) = 0$. [Certainly $u$ $\text{can have more zeros}$ than $x_j$]
Here you may try to give a method to construct $u$, or prove the existence.
And you may take $m=1$ here.
Thanks.
[UPDATE]. Here is a case that this function can exist with some freedom to choose the points.
If we assume the points are on $xy$ plane, i.e. with the corrdinate $(x,y,0)$ in $\mathbb{R}^3$, and $n=1$ as a constant.
Then for $x_j = (a_j,b_j,0)$ in $\mathbb{R}^3$,
we construct function
$\phi(x,y) = \prod_{i=1}^m(x+iy -a_j-ib_j)$
thus $\Delta \phi = 0$, since it is analytic.
My function $u(x,y,z) = \phi(x,y)e^{-ikz}$ satisfies that $u(x_j) = 0$, and satisfies Helmholtz equation.
What I am still thinking about is, can I choose the points randomly, or the points should satisfy some condition?
[UPDATE] For $n(x)$ is not a constant. I have found a way to construct the function for the case that $x_j$ sits on the $xy$ plane.
Take the form of solution as $u(x,y,z) = \phi(x,y)e^{-ik\rho(x,y,z)}$, then
$\rho:\mathbb{R}^3\rightarrow\mathbb{C}$ satisfies that $\Delta \rho + 2\dfrac{\nabla\phi}{\phi}\cdot \nabla\rho + ik(n(x)-\nabla\rho\cdot\nabla\rho) = 0$
Here we cannot say the existence is assured by PDE theories, since $\dfrac{\nabla\phi}{\phi}$ is unbounded, we shall either have $\dfrac{\nabla\phi}{\phi}\cdot \nabla\rho=0$ or $\nabla\rho(x_j) =0$.
I have found a way to construct $\rho$ to saitisfy this, but I still want to know if there is a function such that can have zeros randomly. My example requires the points are on the same plane.
[UPDATE] I have come up with an easier proof. We can prove the existence by induction. The only thing we have to ensure is the $\nabla u\neq\mathbf{0}$ in $\Omega$.