In this paper, we consider a class of non-uniformly elliptic equations whose model is given by \(\begin{aligned}&-\operatorname {div}(a(x)|D u|^{p-2} D u+b(x)|D u|^{q-2} D u)\\&\qquad \qquad \qquad =-\operatorname {div}(a(x)|F|^{p-2} F+b(x)|F|^{q-2} F), \end{aligned}\) where \(1<p<q\) , \(a(\cdot ),b(\cdot )\ge 0\) and \(0<\mu \le a(\cdot )+b(\cdot )\) . Here, the modulating coefficient \(b(\cdot )\) is assumed to be Hölder continuous and the modulating coefficient \(a(\cdot )\) is assumed to be uniformly continuous. We prove the following regularity result with non-standard growth conditions: \(\begin{aligned} \left( a(x)|F|^p+b(x)|F|^q\right) \in L_{\textrm{loc}}^\gamma \Longrightarrow \left( a(x)|D u|^p+b(x)|D u|^q\right) \in L_{\textrm{loc}}^\gamma , \quad \forall \gamma \ge 1, \end{aligned}\) and establish the corresponding gradient estimates.
