On very weak solutions of certain elliptic systems with double phase growth

In this paper, we prove a higher integrability result for very weak solutions of higher-order elliptic systems involving a double phase operator as the principal part. As a model case, we consider \begin{equation} \int_{\Omega} \left( |D^m u|^{p-2}D^m u + a(x)|D^m u|^{q-2}D^m u \right) \cdot D^m \varphi = 0 \quad \text{for any } \varphi \in C_c^{\infty}(\Omega), \end{equation} where $n,m \in \mathbb{N},\ n\ge 2,\,1 < p \le q < \infty,\,\Omega \subset \mathbb{R}^n$ is an open set and $a:\Omega \rightarrow [0,\infty)$ is a measurable function. The proof is based on a construction of an appropriate test function by the Lipschitz truncation technique, a deduction of a reverse H\"older inequality and an application of Gehring's lemma. Our contributions include estimates for weighted mean value polynomials and sharp Sobolev--Poincar\'e-type inequalities for the double phase operator. Our result can be viewed as a generalization with respect to the derivative order, the coefficient function and the growth conditions of the recent paper by Baasandorj, Byun and Kim (Trans. Amer. Math. Soc. 376:8733-8768,2023).