Weak Harnack Estimates for Quasiminimizers with Non-Standard Growth and General Structure

Abstract

We establish the weak Harnack estimates for locally bounded sub- and superquasiminimizers u of $${\int}_{\Omega} f(x,u,\nabla u)\,dx $$ with f subject to the general structural conditions $$|z|^{p(x)} - b(x)|y|^{p(x)}-g(x) \leq f(x,y,z) \leq \mu|z|^{p(x)} + b(x)|y|^{p(x)} + g(x), $$ where p : Ω →] 1, ∞[ is a variable exponent. The upper weak Harnack estimate is proved under the assumption that b, g ∈ Lt(Ω) for some t > n/p−, and the lower weak Harnack estimate is proved under the stronger assumption that b, g ∈ L∞(Ω). As applications we obtain the Harnack inequality for quasiminimizers and the fact that locally bounded quasisuperminimizers have Lebesgue points everywhere whenever b, g ∈ L∞(Ω). Throughout the paper, we make the standard assumption that the variable exponent p is logarithmically Holder-continuous.