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

Abstract

In this paper, we consider the existence (and nonexistence) of solutions to −Mλ,Λ ±(u″)+V(x)u=f(u)inR where Mλ,Λ + and Mλ,Λ − denote the Pucci operators with 0<λ≤Λ<∞ 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−1s (p>1). In particular, we are interested in positive solutions which decay at infinity, and the existence (and nonexistence) of such solutions is proved.