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This seems like it should be easy but I can’t seem to figure it out.

Suppose $f$ satisfies $\Delta f + \lambda f = 0$ in $\mathbb{R}^n$ (where $\lambda > 0$ is a constant), and in some open set $U$ we have $f = 0$. Does this mean $f = 0$ in $\mathbb{R}^n$?

My idea for solving it:

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Suppose not; WLOG suppose $U$ is a maximal open set ordered by inclusion in which $f = 0$. Let $x \in \partial U$.

If $x$ has a neighborhood where $f \le 0$, then $\Delta f \ge 0$ there, so $f$ is subharmonic and by the maximum principle, $f = 0$ in the neighborhood, contradicting the maximality of $U$.

If $x$ has a neighborhood where $f \ge 0$ then the same argument applies to $-f$ (or the same argument with “superharmonic” and “minimum principle”).

Here is where I’m stuck: How do I rule out a third possibility, where every neighborhood of $x$ contains positive and negative points?

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The property you are looking for is called Unique Continuation Property, (UCP) for short, and it is satisfied for example, by analytic functions.

As @uvs pointed out, every eigenfunction is analytical in the interior of $\Omega$ where $\Omega\subset\mathbb{R}^n$ is a open set. A proof of this fact can be found here in chapter 5. Also, take a look in this book, page 108 onwards and you will find plenty of nice properties of eigenfunctions of the laplacian.

Here you will find more specialized properties of harmonic functions and eigenfunctions of the Dirichlet problem.

Now let $p\in (1,\infty)$. Define formally $\Delta_pu=\operatorname{div}(|\nabla u|^{p-2}\nabla u)$ and consider the problem $$-\Delta_p u=\lambda |u|^{p-2}u,\ u\in W_0^{1,p}(\Omega)\tag{1}$$

If $p=2$, problem (1) reduces to the classic laplacian eigenvalue problem. It is a open problem to know whether the soltuions of (1) (with $p\neq 2$) satisfies the UCP. Even for $p$-harmonic functions, i.e. functions which satisfies (1) with $\lambda=0$, this is a open problem (at least in my humble knowledge) excluding only the case $n=2$.

In the case $n=2$, i.e. the planar case, $p$-harmonic functions are know to be quasiconformal maps and consequently quasisymetric maps, which implies some good properties, like for example, that the set where $\nabla u =0$ is discrete. For references, take a look on this book. Also, these are interesting papers: Granlund-Marola, Granlund-Marola 2.

Step 1: eigenfunctions of Laplace operator are analytical.

Step 2: analytical functions vanishing on an open set are identically zero.

See, however, this article: http://www.ams.org/notices/201005/rtx100500624p.pdf

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