# analytic solution to elliptic PDE in R^n

I am looking for (minimal) conditions, which guarantee that the problem

Lu = 0 in R^n,

where L is a second-order (uniformly) elliptic operator with analytic coefficients, has a unique global analytic solution. Does anybody have any references to relevant work?

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As Liviu indicates, $u = 0$ is already a solution, so your question is equivalent to asking what conditions imply that $u = 0$ is the only solution. I don't know the answer to this, but one possible approach is to write an arbitrary solution as a power series and study whether there are identifiable conditions on $L$ that force the radius of convergence to be finite. – Deane Yang Jul 31 '13 at 21:31

When $L$ has analytic coefficients, any solution of $Lu=0$ in $\mathbb{R}^n$ is automatically analytic. You should be asking about conditions guaranteeing that the only solution of this equation is $u=0$. For example, if you assume that $u(x)\to 0$ as $|x|\to \infty$, then the maximum principle will do the trick. For more refined results you could try a Gooogle search or a MathSciNet search.