I am no expert on PDE and analysis but I am looking for certain technique from PDE. Let $D_2$ be the Laplace operator and $f$ is an eigenfunction, i.e., $D_2 f=\lambda f$ for some $\lambda>1$. (or even weakly $\lambda\gg 1$)

Let $D_3$ be a degree 3 partial differential operator, which may be explicitly written down. We assume that $D_3 f=\lambda_3 f$. Is there any technique which may prove $$|\lambda_3|\leq c\lambda ^{3/2}$$ for some $c>0$.