Multiplicity results for fully nonlinear elliptic equations with natural gradient growth

TL;DR

This paper establishes the existence of three solutions for a class of fully nonlinear elliptic boundary value problems with gradient growth, using sub and supersolution methods for specific nonlinearities.

Contribution

It proves the existence of three solutions for a nonlinear elliptic PDE with gradient terms, introducing new sub and supersolution constructions for sublinear nonlinearities.

Findings

Existence of three solutions for the boundary value problem.

Construction of ordered sub and supersolutions for specific nonlinearities.

Validation of multiple positive solutions under certain conditions.

Abstract

In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{\lambda,\Lambda}^+(D^2u)-\Gamma|Du|^2=f(u)~~~\text{in}\ \Omega, u=0~~~\text{on}\ \partial\Omega, \end{equation*} where $f:[0,\infty]\to[0,\infty]$ is a $C^{\alpha}$ function and $\Omega$ denotes a bounded, smooth domain in $\mathbb{R}^N$. By constructing two ordered pairs of sub and supersolutions for a specific class of $f$ exhibiting sublinear growth, we further establish the existence of three positive solutions to the aforementioned boundary value problem.