Existence and a priori bounds for fully nonlinear PDEs with a harmonic map-like structure

TL;DR

This paper investigates a new class of fully nonlinear elliptic PDEs with a harmonic map-like structure, establishing existence, bounds, and qualitative properties under certain conditions, and extending classical results to this novel setting.

Contribution

It introduces a new class of nonlinear PDEs with harmonic map-like features and provides existence, bounds, and multiplicity results, extending classical PDE theory to this context.

Findings

Existence of solutions under small coefficient regimes.

Classical estimates like Aleksandrov–Bakelman–Pucci are established.

New multiplicity and qualitative behavior results for these PDEs.

Abstract

In this paper, we study a new class of fully nonlinear uniformly elliptic equations with a so-called harmonic map-like structure, whose model case is given by \begin{equation*} \mathcal{M}^{\pm}_{\lambda,\Lambda}(D^2u) \pm b(x) |Du| \pm \beta(u)\langle M(x) Du,Du \rangle \pm c(x) u = f(x)\; \textrm{ in } \Omega, \end{equation*} where $\Omega\subset \mathbb{R}^n$ is a bounded $C^{1,1}$ domain, $\mathcal{M}^{\pm}$ are the Pucci extremal operators, $\beta(s) = s^k$ for some $k \in \mathbb{N} $ odd, $b \in L^{q}_{+}(\Omega)$, $c,f \in L^p(\Omega)$, and $n \leq p \leq q$, $q>n$. We obtain existence results under a smallness regime on the coefficients, along with some classical results such as the Aleksandrov--Bakelman--Pucci estimate and the comparison principle, as well as a priori bounds for the respective Dirichlet problem in the noncoercive case. We also establish multiplicity results and qualitative behavior, which seem to be new in the case of the Laplacian operator.