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