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

In this work we prove that the non-negative functions u ∈ Lˢloc(Ω), for some $s>0$, belonging to the De Giorgi classes {equation}{eq0.1} _{Br(1-σ)(x₀)} |∇ (u-k)₋|ᵖ\, dx {c}{σq} \,Λ(x₀, r, k)(k/r)ᵖ({|Bᵣ(x₀)∩\{u k\}|}{|Bᵣ(x₀)|})1-δ, {equation} under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the Lˢ-norm of u. Under suitable assumptions on Λ, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying {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.