Regularity of a class of non-uniformly nonlinear elliptic equations

In this paper we obtain the interior C1,α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^{1,\alpha }$$\end{document} regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations diva1∇u∇u+a2∇u∇u=0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {div} ~\! \left( a_1\left( \left| \nabla u \right| \right) \nabla u + a_2\left( \left| \nabla u \right| \right) \nabla u \right) =0, \end{aligned}$$\end{document}including the following special model div∇up-2∇u+∇uq-2∇u=0foranyp,q>1.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \text {div} ~\! \left( \left| \nabla u \right| ^{p-2} \nabla u + \left| \nabla u \right| ^{q-2} \nabla u \right) =0\quad \ \text{ for } \text{ any } \ p, q>1. \end{aligned}$$\end{document}These equations come from variational problems whose model energy functional is given by P(u,Ω)=:∫ΩB1∇u+B2∇udx,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \mathcal {P}(u, \Omega )=: \int _{\Omega } B^1\left( \left| \nabla u \right| \right) + B^{2}\left( \left| \nabla u \right| \right) dx, \end{aligned}$$\end{document}where Bk(t)=∫0tτak(τ)dτfort≥0andk=1,2.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B^k(t)=\int _0^t \tau a_k(\tau )~d\tau \quad \text{ for } \quad t\ge 0 \quad \text{ and }\quad k=1,2. \end{aligned}$$\end{document}We remark that Bk(t)=|t|αklog(1+|t|)forαk>1andk=1,2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} B^k(t)= |t|^{\alpha _k} \log \big ( 1+|t|\big ) \quad \text{ for } ~~ ~~\alpha _k>1~~~ \text{ and }~~k=1,2 \end{aligned}$$\end{document}satisfy the given conditions in this work.