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

## TL;DR

This paper investigates oscillating solutions to nonlinear equations involving Pucci's extremal operators, demonstrating existence, periodicity in one dimension, and radial decay in higher dimensions.

## Contribution

It establishes the existence of oscillating solutions for nonlinear equations with Pucci's operators, including their periodicity and decay properties.

## Key findings

- Existence of oscillating solutions with infinitely many zeros.
- Solutions are periodic in one dimension.
- Solutions are radial, symmetric, and decay at infinity in higher dimensions.

## Abstract

This paper deals with the following nonlinear equations \[ \mathcal{M}_{\lambda,\Lambda}^\pm(D^2 u)+g(u)=0 \qquad \hbox{ in }\mathbb{R}^N, \] where $\mathcal{M}_{\lambda,\Lambda}^\pm$ are the Pucci's extremal operators, for $N \ge 1$ and under the assumption $g'(0)>0$. 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 $N\ge 2$.