  We prove a Brezis--Kato regularity type results for solutions of the higher
order nonlinear elliptic equation \[
  L u = g(x,u)\qquad\text{in }\Omega \] with an elliptic operator $L$ of $2m$
order with variable coefficients and a Carath\'eodory function $g:\Omega\times
\mathbb{C}\to\mathbb{C}$, where $\Omega\subset\mathbb{R}^N$ is an open set with
$N > 2m$.
