A fully nonlinear two-phase Alt-Phillips problem with gradient nonlinearities

In this manuscript, we analyze a two-phase free boundary problem of Alt-Phillips-type driven by fully nonlinear elliptic equations with lower-order ingredients. More precisely, we consider equations of the type $$F(x,D^{2}u)+\mathcal{H}(x,Du)=\mathcal{F}(x,u^{+},u^{-})\text{in }{B}_1\subset {\mathbb{R}}^n,$$ where the Hamiltonian term exhibits mixed linear and nonlinear gradient dependence, namely $$\mathcal{H}(x,\xi ):=⟨\mathcal{B}(x),\xi ⟩+\varrho (x)\mathrm{\mid }\xi {\mathrm{\mid }}^{\sigma },0<\sigma \le 2,\sigma \mathrm{\ne }1,$$ and the reaction is governed by a two-phase power-type nonlinearity of semilinear-type $$\mathcal{F}(x,u^{+},u^{-}):=\mathfrak{g}(x)[(u^{+}{)}^m-(u^{-}{)}^m]\text{for}0<m<1.$$ Under suitable structural assumptions on the coefficients and the operator $F$, we develop a robust analytical framework to capture the fine properties of solutions near the free boundary. In particular, we establish improved growth estimates at higher-order singular nodal points, points where both phases meet and the solution exhibits critical degeneracy. The results are based upon a fine blow-up argument, stability and a Liouville-type theorem. We further prove quantitative non-degeneracy results, ensuring that solutions detach from zero at a controlled rate on each phase. As a consequence of our analysis, we derive a Liouville-type theorem for global solutions within this class. Our results extend and unify several previously known scenarios, and remain new even in the presence of gradient-dependent nonlinearities of order ($\sigma \mathrm{\ne }1$), thereby encompassing models with nonlinear drift and absorption effects arising in phase transition and reaction-diffusion phenomena.