Existence and a priori bounds for fully nonlinear PDEs with a harmonic map-like structure
Gabrielle Nornberg, Ricardo Ziegele
TL;DR
This work introduces a harmonic-map-like, fully nonlinear elliptic PDE class with gradient-quadratic growth and Pucci operators. It develops an integro-exponential change of variables to tame gradient terms, establishing existence of strong solutions under coefficient smallness, ABP-type bounds, and a comparison principle. It then proves uniform a priori bounds and, in the Laplacian case, a multiplicity result via degree theory and sub/super-solution methods, revealing a global continuum of positive solutions and detailed asymptotic behavior as the noncoercive parameter λ varies. The results significantly extend known coercive and noncoercive theory to a broad nonlinear structure, offering new insights into solvability, regularity, and multiplicity for fully nonlinear PDEs with harmonic-map-like terms.
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}_{λ,Λ}(D^2u) \pm b(x) |Du| \pm β(u)\langle M(x) Du,Du \rangle \pm c(x) u = f(x)\; \textrm{ in } Ω, \end{equation*} where $Ω\subset \mathbb{R}^n$ is a bounded $C^{1,1}$ domain, $\mathcal{M}^{\pm}$ are the Pucci extremal operators, $β(s) = s^k$ for some $k \in \mathbb{N} $ odd, $b \in L^{q}_{+}(Ω)$, $c,f \in L^p(Ω)$, 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.