Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications

Abstract

In this manuscript, we investigate regularity estimates for a class of quasilinear elliptic equations in the non-divergence form that may exhibit degenerate behavior at critical points of their gradient. The prototype equation under consideration is $$ |\nabla u(x)|^{\theta}\left( \Delta_{p}^{\mathrm{N}} u(x) + \langle\mathfrak{B}(x), \nabla u\rangle \right) + \varrho(x) |\nabla u(x)|^{\sigma} = f(x) \quad \text{in} \quad B_1, $$ where $\theta > 0$, $\sigma \in (\theta, \theta + 1)$, and $p \in (1, \infty)$. The coefficients $\mathfrak{B}$ and $\varrho$ are bounded continuous functions, and the source term $f \in \mathrm{C}^{0}(B_1) \cap L^{\infty}(B_1)$. We establish interior $\mathrm{C}^{1,\alpha}_{\text{loc}}$ regularity for some $\alpha \in (0,1)$, along with sharp quantitative estimates at critical points of existing solutions. Additionally, we prove a non-degeneracy property and establish both a Strong Maximum Principle and a Hopf-type lemma. In the final part, we apply our analytical framework to study existence, uniqueness, improved regularity, and non-degeneracy estimates for H\'{e}non-type models in the non-divergence form. These models incorporate strong absorption terms and linear/sublinear Hamiltonian terms and are of independent mathematical interest. Our results partially extend (for the non-variational quasilinear setting) the recent work by the second author in collaboration with Nornberg [Calc. Var. Partial Differential Equations 60 (2021), no. 6, Paper No. 202, 40 pp.], where sharp quantitative estimates were established for the fully nonlinear uniformly elliptic setting with Hamiltonian terms.