Existence results for fully nonlinear equations in radial domains

Abstract

We consider the fully nonlinear problem \begin{equation*} \begin{cases} -F(x,D^2u)=|u|^{p-1}u & \text{in \(\Omega\)}\\ u=0 & \text{on \(\partial\Omega\)} \end{cases} \end{equation*} where \(F\) is uniformly elliptic, \(p>1\) and \(\Omega\) is either an annulus or a ball in \(\Rn\), \(n\geq2\). \\ We prove the following results: \begin{itemize} \item[i)] existence of a positive/negative radial solution for every exponent \(p>1\), if \(\Omega\) 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 \(\Omega\) 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, \(\Omega\) is a ball and \(p\) is subcritical.