On the dimension of the solution space of elliptic systems in unbounded domains

This article is a study of the Dirichlet problem
$$ \begin{cases} Lu=0&\text{in}\ \Omega, \\ \partial^\alpha u\big|_{\partial \Omega}=0,&|\alpha|\leqslant m-1, \end{cases} $$
where $\Omega\subset R^n$ is an open (possibly unbounded) set, $\alpha=(\alpha_1,\dots,\alpha_n)$ is a multi-index, $|\alpha|=\alpha_1+\dots+\alpha_n$,
$$ L=\sum_{|\alpha|=|\beta|=m}\partial^\alpha \bigl(a_{\alpha\beta}(x)\partial^\beta\bigr), $$
and the coefficients $a_{\alpha\beta}(x)$ are $N\times N$ matrices.