Existence and nonexistence of positive solutions to some fully nonlinear equation in one dimension

TL;DR

This paper investigates conditions under which positive solutions that decay at infinity exist or do not exist for a class of fully nonlinear one-dimensional equations involving Pucci operators, a potential function, and nonlinearities.

Contribution

It provides new existence and nonexistence results for positive solutions to fully nonlinear equations with Pucci operators in one dimension.

Findings

Established criteria for existence of positive solutions.

Proved nonexistence results under certain conditions.

Analyzed solutions with power-type nonlinearities.

Abstract

In this paper, we consider the existence (and nonexistence) of solutions to \[ -\mathcal{M}_{\lambda,\Lambda}^\pm (u'') + V(x) u = f(u) \quad {\rm in} \ \mathbf{R} \] where $\mathcal{M}_{\lambda,\Lambda}^+$ and $\mathcal{M}_{\lambda,\Lambda}^-$ denote the Pucci operators with $0< \lambda \leq \Lambda < \infty$, $V(x)$ is a bounded function, $f(s)$ is a continuous function and its typical example is a power-type nonlinearity $f(s) =|s|^{p-1}s$ $(p>1)$. In particular, we are interested in positive solutions which decay at infinity, and the existence (and nonexistence) of such solutions is proved.