Bifurcation on Fully Nonlinear Elliptic Equations and Systems

Abstract

In this paper, we study the following fully nonlinear elliptic equations \begin{equation*} \left\{\begin{array}{rl} \left(S_{k}(D^{2}u)\right)^{\frac1k}=λf(-u) & in\quadΩ\\ u=0 & on\quad \partialΩ\\ \end{array} \right. \end{equation*} and coupled systems \begin{equation*} \left\{\begin{array}{rl} (S_{k}(D^{2}u))^\frac1k=λg(-u,-v) & in\quadΩ\\ (S_{k}(D^{2}v))^\frac1k=λh(-u,-v) & in\quadΩ\\ u=v=0 & on\quad \partialΩ\\ \end{array} \right. \end{equation*} dominated by $k$-Hessian operators, where $\hat{I}\text{c◯}$ is a $(k$-$1)$-convex bounded domain in ${\mathbb{R}}^N$, $\hat{I}»$ is a non-negative parameter, $f:\left[0,+\infty \right)\to \left[0,+\infty \right)$ is a continuous function with zeros only at $0$ and $g,h:\left[0,+\infty \right)\times \left[0,+\infty \right)\to \left[0,+\infty \right)$ are continuous functions with zeros only at $(\cdot ,0)$ and $(0,\cdot )$. We determine the interval of $\hat{I}»$ about the existence, non-existence, uniqueness and multiplicity of $k$-convex solutions to the above problems according to various cases of $f,g,h$, which is a complete supplement to the known results in previous literature. In particular, the above results are also new for Laplacian and Monge-Ampère operators. We mainly use bifurcation theory, a-priori estimates, various maximum principles and technical strategies in the proof.

comments are welcome!