Liouville theorem for a quasilinear non-uniformly elliptic equation in half-spaces and applications

The purpose of this paper is to study Liouville-type theorems forthe equation 2∆∞u+ ∆u= 0 in half-spaces. Our main result classifiesC1-viscosity solutions that continuously vanish on the flat boundary and growlinearly at infinity. This is somehow equivalent to treating a class of equationsin divergence form in Orlicz space without the ∆2condition. The ingredientsfor the proof involve the regularity theory for solutions; the construction ofbarriers along the boundary; and new up to the boundary gradient estimates.These elements allow to implement a Lipschitz impliesC1,αtype approach.We obtain some applications that encompass the recovery of classical resultslike Rad ́o’s zero-level set removability result and Schwarz reflection principleyielding substantially more regularity