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

Abstract

AbstractIn this paper we discuss the ordering properties of positive radial solutions of the equation $$\begin{aligned} \Delta _p u(x)+ k |x|^{\delta } u^{q-1}(x)=0 \end{aligned}$$ Δ p u ( x ) + k | x | δ u q - 1 ( x ) = 0 where $$x \in {\mathbb {R}}^n, n>p>1, k>0, \delta>-p, q>p$$ x ∈ R n , 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 $${\mathbb {R}}^n$$ R n , and in singular ground states v (SGS), defined and positive in $${\mathbb {R}}^n \setminus \{0\}$$ R n \ { 0 } and such that $$\lim _{|x| \rightarrow 0} v(x)=+\infty $$ lim | x | → 0 v ( x ) = + ∞ . A key role in this analysis is played by two bifurcation parameters $$p^{JL}(\delta )$$ p JL ( δ ) and $$p_{jl}(\delta )$$ p jl ( δ ) , such that $$p^{JL}(\delta )>p^*(\delta )>p_{jl}(\delta )>p$$ p JL ( δ ) > p ∗ ( δ ) > p jl ( δ ) > p : $$p^{JL}(\delta )$$ p JL ( δ ) generalizes the classical Joseph–Lundgren exponent, and $$p_{jl}(\delta )$$ p jl ( δ ) its dual. We show that GS are well ordered, i.e. they cannot cross each other if and only if $$q \ge p^{JL}(\delta )$$ q ≥ p JL ( δ ) ; this way we extend to the $$p>1$$ 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 $$p \ge 2$$ p ≥ 2 case. Analogously we show that SGS are well ordered, if and only if $$q \le p_{jl}(\delta )$$ q ≤ p jl ( δ ) ; this latter result seems to be known just in the classical $$p=2$$ p = 2 and $$\delta =0$$ δ = 0 case, and also the expression of $$p_{jl}(\delta )$$ p jl ( δ ) has not appeared in literature previously.