Lipschitz continuity of minimizers in a problem with nonstandard growth

Abstract

In this paper we obtain the Lipschitz continuity of nonnegative local minimizers of the functional J(v) = ∫ Ω - F(x; v; ∇v) + (x)νfv>0) dx, under nonstandard growth conditions of the energy function F(x; s; η) and 0 < λmin ≤ λ (x) ≤ λmax < 1. 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; ∇u) = B(x; u; ru) of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.