Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations

In this paper we discuss the ordering properties of positive radial solutions of the equation Δpu(x)+k|x|δuq-1(x)=0where x∈ Rn, n> p> 1 , k> 0 , δ> - p, q> p. We are interested both in regular ground states u (GS), defined and positive in the whole of Rn, and in singular ground states v (SGS), defined and positive in Rn { 0 } and such that lim |x|→v(x) = + ∞. A key role in this analysis is played by two bifurcation parameters pJL(δ) and pjl(δ) , such that pJL(δ) > p∗(δ) > pjl(δ) > p: pJL(δ) generalizes the classical Joseph–Lundgren exponent, and pjl(δ) its dual. We show that GS are well ordered, i.e. they cannot cross each other if and only if q≥ pJL(δ) ; this way we extend to the p> 1 case the result proved in Miyamoto (Nonlinear Differ Equ Appl 23(2):24, 2016), Miyamoto and Takahashi (Arch Math Basel 108(1):71–83, 2017) for the p≥ 2 case. Analogously we show that SGS are well ordered, if and only if q≤ pjl(δ) ; this latter result seems to be known just in the classical p= 2 and δ= 0 case, and also the expression of pjl(δ) has not appeared in literature previously.