We consider the problem \begin{ equation}(1)\;\;\; \begin{ cases} S_k(D^2u)= \lambda |x|^{ \sigma} (1-u)^q &\mbox{ in }\;\; B,\\ u <0 & \mbox{ in }\;\; B,\\ u=0 &\mbox{ on }\partial B, \end{ cases} \end{ equation} where $B$ denotes the unit ball in $\mathbb{ R}^n$, $n>2k$ ($k\in \mathbb{ N}$), $\lambda>0$, $q > k$ and $\sigma\geq 0$. We study the existence, uniqueness and multiplicity of negative bounded radially symmetric solutions of (1). The methodology to obtain our results is based on a dynamical system approach. For this, we introduce a new transformation which reduces problem (1) to an autonomous two dimensional generalized Lotka-Volterra system.
Categories: math.AP