We establish two maximal inequalities for functions in the Lorentz space \(L^{p,q}(\Omega ,\mathbb {R})\) in order to prove a Caccioppoli-type inequality for Lorentz solutions to the nonlinear elliptic equation \(\begin{aligned} \textrm{div}\left( \mathcal {A}(x,Du)\right) =\textrm{div}\left( \vert G\vert ^{s-2}G\right) \qquad x\in \Omega , \end{aligned}\) where \(\Omega \subset \mathbb {R}^n\) is a domain, \(2\le s\le n\) , \(u\in L^{p,q}(\Omega ,\mathbb {R})\) with \(Du\in L^{p,q}(\Omega ,\mathbb {R}^n)\) , \(G\in L^{p,q}(\Omega ,\mathbb {R}^n)\) and \(\mathcal {A}:\Omega \times \mathbb {R}^n\rightarrow \mathbb {R}^n\) is a Carathéodory function of growth \(s-1\) .
