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

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.