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 $$|\nabla u(x){|}^{\theta }\left({\Delta }_p^{\mathrm{N}}u(x)+⟨\mathfrak{B}(x),\nabla u⟩\right)+\varrho (x)|\nabla u(x){|}^{\sigma }=f(x)\text{in}{B}_1,$$ 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 ${\mathrm{C}}_{\text{loc}}^{1,\alpha }$ 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 Hénon-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.

46 pages