Half-linear equations and characteristic properties of the principal solution

The characteristic properties of the principal solution for half-linear differential equation (a(t)Φ(x′))′+b(t)Φ(x)=0, where the functions a,b are positive and continuous for t⩾0 and Φ(u)=|u|p−2u, p>1, are investigated. In the linear case it is well-known that the principal solution is the “smallest one” in a neighbourhood of infinity; we show that this property continues to hold in the half-linear case. In addition, it is proved that the principal solutions can be fully characterized by means of two different integral criteria, which reduce to that one well known in the linear case.