Local solvability of the $k$-Hessian equations

TL;DR

This paper investigates the local solvability of the $k$-Hessian equations in $R^n$, allowing for sign-changing or non-negative right-hand sides, and classifies polynomial solutions to aid in constructing solutions.

Contribution

It provides new existence results for local solutions with sign-changing data and classifies polynomial solutions, advancing understanding of $k$-Hessian equations.

Findings

Existence of local solutions with sign-changing $f$

Classification of second order polynomial solutions

Construction of solutions with uniform ellipticity

Abstract

In this work, we study the existence of local solutions in $\mathbb{R}^{n}$ to $k$-Hessian equation,for which the nonhomogeneous term $f$ is permitted to change the sign or be non negative; if $f$ is $C^\infty,$ so is the local solution. We also give a classification for the second order polynomial solutions to the $k-$Hessian equation, it is the basis to construct the local solutions and obtain the uniform ellipticity of the linearized operators at such constructed local solutions.