A solution of the Laplace equation that is compactly supported or vanishes at $\infty$ or near the boundary

Question

Consider the Laplace equation \begin{equation}\label{1} \Delta u =0 \end{equation} on a bounded smooth domain (or the whole space). Does there exist a solution for the Laplace equation that is compactly supported or vanishes near the boundary (or at $\infty$) ?

Answer 1

Surprisingly, the answer is yes, at least for $n=2$ and on the whole space $\Bbb R^2\equiv \Bbb C$. Consider the following function indeed $$ u(x,y)= \Re \exp\big({z^{-4}}\big)=\Re \exp\big({(re^{i\theta})}^{-4}\big).$$

As the real part of a holomorphic function on a connected domain, it is harmonic: nevertheless

1. it vanishes at the limit for $(x,y)$ going to $\infty$ along any path, and
2. it does not even define a distribution in any neighborhood of $0\in\Bbb C$

Notes

• This example is taken from the exercise book [1], chapter 2, p. 191.
• The reason that fooled the authors of the comments to the OP (and many more) is that they implicitly assumed that all harmonic functions are locally bounded and/or continuous and/or integrable (as weak solutions are).
• Perhaps $u(x,y)$ maybe used to produce a compactly supported harmonic function by taking its trace on the boundary of the unit disk $\Bbb D=\{z\in\Bbb C\mid |z|\le 1\}$ and then solve the exterior Dirichet problem for the Laplace equation on the plane. Said $u_e(x,y)$ the harmonic function found in this way, the function $$ u_c(x,y)= u(x,y)-u_e(x,y) $$ should be harmonic, discontinuous and compactly supported (obviously it is necessary to check this).

Reference

[1] Biler, P., Nadzieja, T. Problems and Examples in Differential Equations, Pure and Applied Mathematics 164, New York:Marcel Dekker, Inc., ISBN: 0-8247-8637-8, viii+244 (1992), MR1198886, Zbl 0760.34001.