Existence results for fully nonlinear equations in radial domains

We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & $\text{in }\mathrm{\Omega }$\ u=0 & $\text{on }\mathrm{\partial }\mathrm{\Omega }$ \end{cases} \end{equation*} where $F$ is uniformly elliptic, $p>1$ and $\mathrm{\Omega }$ is either an annulus or a ball in $\text{Rn}$, $n\ge 2$. \ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent $p>1$, if $\mathrm{\Omega }$ is an annulus; \item[ii)] existence of infinitely many sign changing radial solutions for every $p>1$, characterized by the number of nodal regions, if $\mathrm{\Omega }$ is an annulus; \item[iii)] existence of infinitely many sign changing radial solutions characterized by the number of nodal regions, if $F$ is one of the Pucci’s operator, $\mathrm{\Omega }$ is a ball and $p$ is subcritical.