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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

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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 $B_2$.

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  • $\begingroup$ Actually, the OP mentions "Liouville type theorem" in the title, so maybe you are answering the intended question and I'm not. I guess we'll find out eventually... $\endgroup$ Sep 19, 2014 at 19:54
  • $\begingroup$ @ChristianRemling: Yes, I interpreted the question as: Does the decay property of C always imply that decaying $u$ are trivial? Of course you are right that it seems to be true for most C. $\endgroup$ Sep 19, 2014 at 23:02
  • $\begingroup$ @Christian and Connor. Yes my intended question was a Liouville theorem in the sense that I wanted a non-existence result, which appears not to be true by your examples. Thank you very much for the examples. $\endgroup$
    – Craig
    Sep 20, 2014 at 1:08
  • $\begingroup$ @Craig: Thanks for the clarification. I'll delete my answer, which seems pointless now. $\endgroup$ Sep 20, 2014 at 3:27
  • $\begingroup$ @Christian. Sorry, for the wording. It definitly was not clear what i was asking. $\endgroup$
    – Craig
    Sep 20, 2014 at 4:30

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