Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications
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arXiv
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| Natura: | Preprint |
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2025
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| _version_ | 1866912372434665472 |
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| author | Bessa, Junior da Silva da Silva, João Vitor |
| author_facet | Bessa, Junior da Silva da Silva, João Vitor |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10723 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications Bessa, Junior da Silva da Silva, João Vitor Analysis of PDEs 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. |
| title | Regularity estimates for quasilinear elliptic PDEs in non-divergence form with Hamiltonian terms and applications |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.10723 |