RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 67Observe that, if f 1 (u) = u|u| q−1 and k = 1, the functions J ± 1(t) defined in (3.2)coincide with the functions J ± (t) defined in section 3.1. As we did in section 3.1, ifφ i ∈ C 1 we can rewrite the functions Ji± in a form similar to (15) from which we canmore easily guess the sign. So we find the analogous of (16):H p ∗(x p ∗(t), t) + lim H p ∗(x p∗(t), t)t→−∞=N∑i=1[Ji + (t) uq i(e t )−q iH p ∗(x p ∗(t), t) − lim H p ∗(x p∗(t), t)t→∞N∑ [=i=1J −i(t) xq ip ∗(t)q i+∫ t−∞∫ ∞t]Ji+ (s)u ′ (e s )u qi−1 (e s )dsJ −i](s)ẋ p∗ (t)x q i−1p ∗(s) dsTherefore we have a result analogous to Remark 3 and repeating the argument of theproof of Theorem 1, we obtain the following generalization.THEOREM 2. Assume that either Ji + (t) ≥ 0 for any i and ∑ Ni=1 Ji+ (t) ̸≡ 0 orJi− (t) ≥ 0 for any i and ∑ Ni=1 Ji− (t) ̸≡ 0. Then all the regular solutions are crossingsolutions; moreover if q 1 > p ∗ , and k i (r) is uniformly positive for r large, there areuncountably many S.G.S. with fast decay.Assume that either Ji + (t) ≤ 0 for any i and ∑ Ni=1 Ji+ (t) ̸≡ 0 or Ji− (t) ≤ 0 forany i and ∑ Ni=1 Ji− (t) ̸≡ 0. Then all the regular solutions are G.S. with slow decay.Moreover there are uncountably many solutions u(r) of the Dirichlet problem in theexterior of a ball.This result is proved in [13] for the case 1 < p ≤ 2. However it can be easilyextended to the case p > 2 putting together the construction of a stable set ˜W s (τ)developed in [17] (and quoted in Theorem 2), and the argument of [13] concerning thefunction H p ∗ (that we have sketched in this section). In fact the minimal requirementfor the fast decay solution to exist, is that there are c > 0 and m > p ∗ such thatg(x m (t), t) > cx m (t)|x m (t)| m−2 for t large.Repeating the argument in Corollary 2 we easily obtain also this result:COROLLARY 3. Assume that all the functions Ji− (t) ≥ 0 for any r but∑ Ni=1Ji − (t) ̸≡ 0, and that there is p ∗ < m ≤ p ∗ such that the limit lim t→∞ g(x m (t), t)/|x m (t)| m−1 = k m (∞) exists, is positive and finite. Then positive solutions have astructure of type A.Analogously assume that all the functions Ji + (t) ≤ 0 for any r but ∑ Ni=1 Ji+ (t) ̸≡0, and that there is s ≥ p ∗ such that the limit g(x s (t), t)/|x s (t)| s−1 = k s (0) exists, isis positive and finite. Then positive solutions have a structure of type B.Once again if m ̸= p ∗ and s ̸= p ∗ respectively, and a further technical conditionis satisfied the S.G.S. with slow decay is unique, see [13]. From Theorem 2 and