Regularity of a class of non-uniformly nonlinear elliptic equations

Abstract

In this paper we obtain the interior $C^{1,\alpha }$ regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations

$$\begin{aligned} \text {div} ~\! \left( a_1\left( \left| \nabla u \right| \right) \nabla u + a_2\left( \left| \nabla u \right| \right) \nabla u \right) =0, \end{aligned}$$

including the following special model

$$\begin{aligned} \text {div} ~\! \left( \left| \nabla u \right| ^{p-2} \nabla u + \left| \nabla u \right| ^{q-2} \nabla u \right) =0\quad \ \text{ for } \text{ any } \ p, q>1. \end{aligned}$$

These equations come from variational problems whose model energy functional is given by

$$\begin{aligned} \mathcal {P}(u, \Omega )=: \int _{\Omega } B^1\left( \left| \nabla u \right| \right) + B^{2}\left( \left| \nabla u \right| \right) dx, \end{aligned}$$

where

$$\begin{aligned} B^k(t)=\int _0^t \tau a_k(\tau )~d\tau \quad \text{ for } \quad t\ge 0 \quad \text{ and }\quad k=1,2. \end{aligned}$$

We remark that

$$\begin{aligned} B^k(t)= |t|^{\alpha _k} \log \big ( 1+|t|\big ) \quad \text{ for } ~~ ~~\alpha _k>1~~~ \text{ and }~~k=1,2 \end{aligned}$$

satisfy the given conditions in this work.