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

In this paper, we consider the existence (and nonexistence) of solutions to $$-{\mathcal{M}}_{\lambda ,\mathrm{\Lambda }}^{\pm }(u^{\mathrm{\prime }\mathrm{\prime }})+V(x)u=f(u)\mathrm{i}\mathrm{n}\mathbf{R}$$ where ${\mathcal{M}}_{\lambda ,\mathrm{\Lambda }}^{+}$ and ${\mathcal{M}}_{\lambda ,\mathrm{\Lambda }}^{-}$ denote the Pucci operators with $0< \lambda \leq \Lambda < \infty$,$V(x)$isaboundedfunction,$f(s)$ 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.