Multiplicity results for fully nonlinear elliptic equations with natural gradient growth

Abstract

In this paper, we prove a theorem concerning the existence of three solutions for the following boundary value problem: \begin{equation*} -\mathcal{M}_{λ,Λ}^+(D^2u)-Γ|Du|^2=f(u)~~~\text{in}\ Ω, u=0~~~\text{on}\ \partialΩ, \end{equation*} where $f:[0,\infty ]\to [0,\infty ]$ is a $C^{\hat{I}}\pm$ function and $\hat{I}\text{c◯}$ 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.