Oscillating solutions for nonlinear equations involving the Pucci's extremal operators

Abstract

This paper deals with the following nonlinear equations \[ \mathcal{M}_{λ,Λ}^\pm(D^2 u)+g(u)=0 \qquad \hbox{ in }\mathbb{R}^N, \] where ${\mathcal{M}}_{\lambda ,\Lambda }^{\pm }$ are the Pucci's extremal operators, for $N\ge 1$ and under the assumption $g^{\prime }(0)>0$. We show the existence of oscillating solutions, namely with an unbounded sequence of zeros. Moreover these solutions are periodic, if $N=1$, while they are radial symmetric and decay to zero at infinity with their derivatives, if $N\ge 2$.

17 pages