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 $J(v)=\int_\Omega\big(F(x, v, \nabla v)+\lambda(x)\chi_{\{v>0\}}\big)\, dx$, under nonstandard growth conditions of the energy function $F(x, s, \eta)$ and $0<\lambda_{\min}\le \lambda(x)\le \lambda_{\max}<\infty$. 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 ${\rm div}A(x, u, \nabla u)=B(x, u, \nabla u)$ of nonstandard growth type. As a by-product of our study, we obtain several results for this equation that are of independent interest.