Suppose L is a partial differential operator of arbitrary order with constant coefficients.


If u is in $L^p(\mathbb{R}^n)$ and Lu=0 in distributions, is it necessarily the case that u=0? Does the answer depend on p? 


Also, if u is a compactly supported distribution in $\mathbb{R}^n$ with Lu=0 (in the usual sense, i.e. strongly), is it necessarily the case that u=0?


(Suggested reference material appreciated) 



Uniqueness of weak solution L[u]=0
If $Lu=0$, then the Fourier transform of $u$ must have its support on the manifold where the symbol of $L$ is zero. Hence the Fourier transform of $u$ cannot be a function. This rules out $u\in L^p$ for $p\le 2$; it also rules out a compactly supported distribution. The Bessel function $J\_0(\sqrt{x^2+y^2})$ satisfies $\Delta u+u=0$, and it is in $L^p$ for every $p>4$. You can generalize this example to $n$ dimensions, and you find that $u\in L^p$ for every $p>2n/(n-1)$.


