The weak Harnack inequality for unbounded minimizers of elliptic functionals with generalized Orlicz growth

Abstract

In this work we prove that the non-negative functions $u \in L^s_{loc}(\Omega)$, for some $s>0$, belonging to the De Giorgi classes \begin{equation}\label{eq0.1} \fint\limits_{B_{r(1-\sigma)}(x_{0})} \big|\nabla \big(u-k\big)_{-}\big|^{p}\, dx \leqslant \frac{c}{\sigma^{q}} \,\Lambda\big(x_{0}, r, k\big)\bigg(\frac{k}{r}\bigg)^{p}\bigg(\frac{\big|B_{r}(x_{0})\cap\big\{u\leqslant k\big\}\big|}{|B_{r}(x_{0})|}\bigg)^{1-\delta}, \end{equation} under proper assumptions on $\Lambda$, satisfy a weak Harnack inequality with a constant depending on the $L^s$-norm of $u$. Under suitable assumptions on $\Lambda$, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying \eqref{eq0.1}; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.