A note on the local regularity of distributional solutions and subsolutions of semilinear elliptic systems (Q1679195)

The author considers semilinear systems of elliptic partial differential equations of the form \[ L_k^mu_k=f_k(x,u_1,\ldots,u_N) \quad \text{in } \Omega,\tag{1} \] for \(k=1,\ldots, N\) and \(m\in \mathbb{N}\) where \(\Omega\subseteq \mathbb{R}^n\) is an open set, the operators \(L_k\) are of divergence form and of second order and the right-hand side are certain nonlinearities. The paper presents local regularity results for distributional solutions and subsolutions of (1). In particular, if \(f_k(x,z) \leq C(1+|z|^p)\) for \(1 \leq p <\frac{n}{n-2m}\) and \(k=1,\ldots, N\), then the distributional subsolutions of (1) are locally bounded.