Young differential equations with power type nonlinearities

In this note we give several methods to construct nontrivial solutions to the equation dyt=σ(yt)dxt, where x is a γ-Hölder Rd-valued signal with γ∈(1/2,1) and σ is a function behaving like a power function |ξ|κ, with κ∈(0,1). In this situation, classical Young integration techniques allow to get existence and uniqueness results whenever γ(κ+1)>1, while we focus on cases where γ(κ+1)≤1. Our analysis then relies on Zähle's extension (Zähle, 1998) of Young's integral allowing to cover the situation at hand.