Are there any $L^\infty$ bounds on the eigenfunctions of a first order elliptic dfferential operator on $\mathbb{R}^n$ in terms of eigenvalues?

Question

The question is as in the title. All references I searched for only seem to deal with compact manifolds. So I ask for noncompact cases.

For a first-order elliptic differential operator with "smooth and bounded" coefficients on $\mathbb{R}^n$, three issues bother me:

1. Can we obtain "smooth integrable" eigenfunctions on $\mathbb{R}^n$ for this operator?
2. On a compact $n-$dimensional manifold, the eigenvalues $\{ \lambda_k \}$ have the asymptotic behavior $O(k^{1/n})$ as $k \to \infty$. Does the same result hold for $\mathbb{R}^n$ with my assumptions?
3. Most importantly, do there exist any $L^\infty$ bounds on the eigenfunctions in terms of the eigenvalues for the operator? For example, it would be nice if there were estimates of the form \begin{equation} \lVert \phi_k \rVert_{L^\infty(\mathbb{R}^n)} \leq C \cdot \lambda_k^\alpha \end{equation} for some positive constants $C$ and $\alpha$.

I am trying my best to find references for the above issues, but totally stuck at the moment. Could anyone please help me?