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

Abstract

Abstract In this work we prove that the non-negative functions u ∈ L loc s ⁢ ( Ω ) {u\in L^{s}_{\rm loc}(\Omega)} , for some s > 0 {s>0} , belonging to the De Giorgi classes ⨍ B r ⁢ ( 1 - σ ) ⁢ ( x 0 ) | ∇ ( u - k ) - | p d x ⩽ c σ q Λ ( x 0 , r , k ) ( k r ) p ( | B r ( x 0 ) ∩ { u ⩽ k } | | B r ⁢ ( x 0 ) | ) 1 - δ , \barint_{B_{r(1-\sigma)}(x_{0})}|\nabla(u-k)_{-}|^{p}\,dx\leqslant\frac{c}{% \sigma^{q}}\,\Lambda(x_{0},r,k)\bigg{(}\frac{k}{r}\bigg{)}^{p}\bigg{(}\frac{|B% _{r}(x_{0})\cap\{u\leqslant k\}|}{|B_{r}(x_{0})|}\bigg{)}^{1-\delta}, under proper assumptions on Λ, satisfy a weak Harnack inequality with a constant depending on the L s {L^{s}} -norm of u. Under suitable assumptions on Λ, the minimizers of elliptic functionals with generalized Orlicz growth belong to De Giorgi classes satisfying the above condition; thus this study gives a wider interpretation of Harnack-type estimates derived to double-phase, degenerate double-phase functionals and functionals with variable exponents.