Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications
Junior da Silva Bessa, João Vitor da Silva
TL;DR
The paper addresses interior regularity for non-divergence quasilinear elliptic PDEs with Hamiltonian terms that may degenerate at critical points of the gradient, focusing on the prototype equation $| abla u|^{\theta}\Delta_p^{\mathrm{N}} u + \mathscr{H}(\nabla u, x) = f(x)$ with $\mathscr{H}(\xi,x) = \langle \mathfrak{B}(x), \xi \rangle |\xi|^{\theta} + \varrho(x)|\xi|^{\sigma}$ and $\sigma\in(\theta,1+\theta)$. The authors develop a robust compactness/approximation framework based on translated problems and affine approximations, together with Ishii–Lions calculus and a geometric iteration, to obtain $C^{1,\alpha}_{\text{loc}}$ regularity (up to the homogeneous exponent $\alpha_{\mathrm{Hom}}^{(p)}$ or $1/(1+\theta)$ in certain regimes) and sharp estimates at critical points. They also prove a sharp non-degeneracy estimate, a Strong Maximum Principle, and a Hopf-type lemma for equations with borderline first-order terms, and apply the theory to Hénon-type problems with strong absorption, obtaining existence, uniqueness, improved regularity, and dead-core non-degeneracy. These results advance non-variational quasilinear regularity theory, connect to stochastic Tug-of-War PDEs, and provide tools for geometric free boundary analysis in nonlinear elliptic models with Hamiltonian terms.
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 $$ |\nabla u(x)|^θ\left( Δ_{p}^{\mathrm{N}} u(x) + \langle\mathfrak{B}(x), \nabla u\rangle \right) + \varrho(x) |\nabla u(x)|^σ = f(x) \quad \text{in} \quad B_1, $$ where $θ> 0$, $σ\in (θ, θ+ 1)$, 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 $\mathrm{C}^{1,α}_{\text{loc}}$ regularity for some $α\in (0,1)$, 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.