RENDICONTI DEL SEMINARIO MATEMATICO

64 M. Franca0, then, from an elementary analysis on the phase portrait either there is t 2 > t 1 suchthat x p∗ (t 2 ) < 0, or lim t→∞ x p∗ (t) = 0. Assume the latter, then lim r→∞ u(r)r n−pp−1 =lim t→∞ x p∗ (t) = 0; but from (3) it follows that u ′ (r) n−1p−1 is decreasing and admits limitλ ≤ 0. Using de l’Hospital rule we find that u ′ (r) n−1p−1u ′ (r) ≡ u(r) ≡ 0 for r > R, so the claim is proved.→ p−1n−pλ; so we get λ = 0 andRecall that, when k(r) is differentiable, the Pohozaev identity can be reformulatedin this dynamical context as (10). Therefore we can think of H p ∗ as an energyfunction, which is increasing along the trajectories when φ(s)e α p ∗(p∗ −q)tis increasingand decreasing when φ(s)e α p ∗(p∗−q)t is decreasing. This observation can berefined combining it with the fact that all the regular solutions u(r) are decreasing,and fast decay solutions are such that x p∗ (t) is increasing, whenever they are positive.For this purpose we define two auxiliary functions, which are closely relatedto the Pohozaev identity, and which were first introduced in [35]. In this subsectionwe will always assume (without mentioning) that e nt φ(t) ∈ L 1( (−∞, 0] ) andn−p(n−qe(14)p−1 )s φ(t) ∈ L 1( [0,+∞) ) , so that we can define the following functions:J + (t) := φ(t)entqJ − (t) :=− n − ppφ(t)e(n−qn−pp−1 )tq−∫ t−∞φ(s)e ns dsn − pp(p − 1)∫ t−∞n−p(n−qφ(s)e p−1 )s dsWe will see, that the sign of these functions play a key role in determining the structureof positive solutions for (3). When φ is differentiable we can rewrite J + and J − in thisform, from which we can more easily guess the sign:(15)J + (t) := 1 qJ − (t) := 1 q∫ t−∞∫ +∞tdds [φ(s)eα p ∗(p∗−q)s ]e α p ∗ qs dsdds [φ(s)eα p ∗(p∗ −q)s ]e − (n−p)qp(p−1) s dsLet u(r) be a solution of (3) and let x(t) be the corresponding trajectory of (8). Using(10) and integrating by parts we easily find the following(16)H p ∗(x p ∗(t), t) + lim H(x p∗(t), t)t→−∞= J + (t) |u|q (e t )q−H p ∗(x p ∗(t), t) − lim H p ∗(x p∗(t), t)t→∞== J − (t) |x p ∗| q (t)q∫ t−∞+J + (s)u ′ (e s )u|u| q−2 (e s )ds∫ +∞From (16) we easily deduce the following useful result.tJ − (s)ẋ p∗ (s)x p∗ |x p∗ | q−2 (s)ds