On W1,γ(.)-regularity for nonlinear non-uniformly elliptic equations

We prove a global W-1,W-gamma(.)-estimate for nonlinear non-uniformly elliptic problems on bounded smooth domains. More precisely, we consider the following zero-Dirichlet problem of non-uniformly elliptic equations div A(Du, x) = div G(F, x) x is an element of Omega with the model case of A(Du, x) approximate to vertical bar Du vertical bar(p-2)Du + a(x)vertical bar Du vertical bar(q-2)Du and G(F, x) approximate to vertical bar F vertical bar Fp-2 + a(x)vertical bar F vertical bar Fq-2. Let H(xi, x) = vertical bar xi vertical bar(p) + a(x)vertical bar xi vertical bar(q), we obtain its global variable Calderon-Zygmund estimate with that H(F, x) is an element of L-gamma(x)(Omega) double right arrow H(Du, x) is an element of L-gamma(x)(Omega) under the sharp assumptions that a(.) is C-0,C-alpha-Holder continuous, 1 < p < q < p + alpha p/n, the boundary of domain is of class C-1,C-beta with beta is an element of [alpha, 1], and the variable exponents gamma (x) >= 1 satisfy the log-Holder continuity.