A geometric approach to two-phase problems for quasilinear elliptic equations in nondivergence form

In this manuscript, we investigate a two-phase problem governed by quasilinear elliptic equations in non-divergence form with (linear-sublinear) Hamiltonian terms, given by $${aligned} |Du|θΔ ₚNu +H(Du,x) \!=\! {F(u⁺, u⁻, x)} \! \! |x|μ[ λ _1 · (u^+)^m - λ _2 · (u^-)^m] in B_1 ⊂ Rⁿ, {aligned}$$ | D u | θ Δ p N u + H ( D u , x ) = F ( u + , u - , x ) ≲ | x | μ λ 1 · ( u + ) m - λ 2 · ( u - ) m in B 1 ⊂ R n , for $$θ , μ , λ _1, λ _2 ≥ 0$$ θ , μ , λ 1 , λ 2 ≥ 0 , $$p ∈ (1, ∞ )$$ p ∈ ( 1 , ∞ ) , and $$m ∈ (0, θ +1)$$ m ∈ ( 0 , θ + 1 ) . In this framework,