Existence of Solutions to Fully Nonlinear Elliptic Equations with Gradient Nonlinearity

Abstract

In this article, we study the existence and multiplicity of nontrivial solutions to the problem \[\begin{cases} -\epsilon^{2} F(x,D^{2}u) = f(x,u) + \psi(Du) &\textrm{in $\Omega$}, \\ u = 0 &\textrm{on $\partial \Omega$},\end{cases}\]where $\Omega$ is a smooth bounded domain in $\mathbb{R}^{n}$, $n \gt 2$. We show that the problem possesses nontrivial solutions for small value of $\epsilon$ provided $f$ and $\psi$ are continuous and $f$ has a positive zero. We employ degree theory arguments and Liouville type theorem for the multiplicity of the solutions.