A note on regularity for a class of quasilinear elliptic equations

Abstract

The following regularity of weak solutions of a class of elliptic equations of the form are investigated,

$$divA\left( {x,u,Du} \right) + B\left( {x,u,Du} \right) = 0 in \Omega $$

(*)

. Here Ω⊂R ⁿ is a bounded domain, A(x,z,p)=(A¹(x,z,p), A²(x,z,p),...,Aⁿ(x,z,p)) and B(x,z,p) satisfy

$$\begin{gathered} \frac{1}{\Lambda }\left( {\kappa + \left| p \right|} \right)^m \left| \xi \right|^2 \leqslant \frac{{\partial A'}}{{\partial p,}}\left( {x,z,p} \right)\xi _1 \xi , \leqslant \Lambda \left( {\kappa + \left| p \right|} \right)^m \left| \xi \right|^2 , \hfill \\ \left| {A\left( {x,z,p} \right) - A\left( {y,w,p} \right)} \right. \leqslant \left( {1 - \left. p \right|} \right)^{m + 1} \varphi \left( {\left| {x - y} \right| + \left| {z - w} \right|} \right) \hfill \\ \end{gathered} $$

and

$$\left| {B\left( {x,z,p} \right)} \right| \leqslant \Lambda \left( {1 + \left| p \right|} \right)^{m + 2} $$

for all (x,z,p), (y,m,p)∈Ω×R×R ⁿ and all ξ∈R ⁿ, where m≥0, κ≥0, Λ>0 and ϕ(r) is a bounded increasing function in [0, ∞).

The results of the paper are: a) if lim _r→0+ϕ(r)=ϕ(0)=0, then any bounded solution of (*) belongs to C ^β_loc (Ω) for any β∈(0, 1); b) if m=0, and ϕ(r) is Dini continuous, that is, lim _r→0+ϕ(r)=ϕ(0)=0, $\int_0^1 {\frac{{\varphi \left( r \right)}}{r} } dr < + \infty$, then any bounded solution of (*)∈C ^l_loc (Ω).