Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations
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Nonlinear Differential Equations and Applications NoDEA
Ordering properties of radial ground states and singular ground states of quasilinear elliptic equations R. Colucci and M. Franca Abstract. In this paper we discuss the ordering properties of positive radial solutions of the equation Δp u(x) + k|x|δ uq−1 (x) = 0 where 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|→0 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. Mathematics Subject Classification. 35J62, 35J92, 35B08, 35B09, 34C45, 70K05. Keywords. Quasilinear elliptic equation, Radial solutions, Singular solutions, Intersection number, Separation properties, Invariant manifold.
R. Colucci: Partially supported by G.N.A.M.P.A. - INdAM (Italy) and MURST (Italy) M. Franca: Partially supported by G.N.A.M.P.A. - INdAM (Italy) and MURST (Italy). 0123456789().: V,-vol
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NoDEA
1. Introduction In this paper we continue the discussion started by Miyamoto [25] and Miyamoto et al [26] concerning the ordering properties of radial solutions for a class of quasilinear elliptic equations, including p-Laplacian and k-Hessian. Let us start our discussion from the p-Laplace setting, i.e. we consider radial solutions u(|x|) = u(r) for the following equation Δp u(x) + k|x|δ u(x)q−1 = 0,
x ∈ Rn ,
(1.1)
where Δp (u) = div(∇u|∇u|p−2 ). In the whole paper we always assume the following relations for the parameters H k > 0, n > p, δ > −p and q > p. Since we are just interested in radial solutions we restrict to consider the following singular ODE: (u |u |p−2 rn−1 ) + rδ+n−1 kuq−1 = 0,
(1.2)
∂ ∂r .
where = Let us introduce some terminology. We say that a definitively positive solution u(r) is regular if u(0) = d > 0 and that it is singular if limr→0 u(r) = +∞; analogously we say that u(r) has fast decay if limr→∞ u(r)r(n−p)/(p−1) = L > 0 and that it has slow decay if limr→+∞ u(r)r(n−p)/(p−1) = +∞. We denote by u(r; d) a regular solution to (1.2) such that u(0; d) = d > 0 and by v(r; L) a fast decay solution to (1.2) such that limr→+∞ v(r)r(n−p)/(p−1) = L > 0. In fact for any d > 0 and any L > 0 there is a unique regular solution u(r; d) and a unique fast decay solution
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