Regularity of a class of non-uniformly nonlinear elliptic equations

In this paper we obtain the interior C-1,C- alpha regularity of weak solutions for a class of non-uniformly nonlinear elliptic equations div (alpha(1) (vertical bar del u vertical bar)del u + alpha(2) (vertical bar del u vertical bar)del u = 0, including the following special model div (vertical bar del u vertical bar(p-2) del u + vertical bar del u vertical bar(q-2) del u) = 0 forany p, q > 1. These equations come from variational problems whose model energy functional is given by P(u, Omega) =: integral(Omega) B-1 (vertical bar del u vertical bar) + B-2 (vertical bar del u vertical bar) dx, where B-k(t) = integral(t)(0) tau alpha(k)(t) d tau for t >= 0 and k = 1, 2. We remark that B-k(t) = vertical bar t vertical bar(alpha k) log (1 + vertical bar t vertical bar) for alpha(k) > 1 and k = 1, 2 satisfy the given conditions in this work.