Lipschitz Continuity Of Minimizers In A Problem With Nonstandard Growth

In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional J(v) = integral(Omega) (F( x, v, del v) + lambda(x)x({v>0}))dx, under nonstandard growth conditions of the energy function F(x, s, eta) and 0 < lambda(min) <= lambda(x) <= lambda(max) < infinity. This is the optimal regularity for the problem. Our results generalize the ones we obtained in the case of the inhomogeneous p(x)-Laplacian in our previous work. Nonnegative local minimizers u satisfy in their positivity set a general nonlinear degenerate/singular equation divA(x, u, del u) = B(x, u, del u) of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.