Liouville type theorems; linear PDE with decaying potential

Question

Dear Mathoverflowers,

I am interested in the following pde:

$$ -\Delta u(x) + C(x) u(x) = 0 $$ in $ R^N$. Lets assume that $ C(x)$ is bounded and (smooth if you like) and satisfies the following:
$$ \sup_{|x| \ge R} |x|^2 |C(x)| \rightarrow 0$$ as $ R \rightarrow \infty$.

Question. Does there exists non-zero smooth solutions $u(x)$ to the above pde which decay to zero at $|x| =\infty$? Note I am not putting any sign conditions on $u(x)$ or $C(x)$.

thanks for you responses. Craig

Answer 1

For some $C$ satisfying the decay properties, such solutions do exist. Let $v$ be the function which is the fundamental solution $|x|^{2-n}$ in $\mathbb{R}^n - B_1$ (with $n \geq 3$) and $1$ in $B_1$, and let $u = \rho \ast v$ for some smooth mollifier $\rho$ supported in $B_{1/4}$.

Then $\Delta u = 0$ outside $B_{2}$ and $u > c(n)$ in $B_{2}$ so $C(x) = \Delta u / u$ is smooth, bounded and vanishes outside $B_2$.