RENDICONTI DEL SEMINARIO MATEMATICO

60 M. Franca0 ≤ u ≤ U and r ≥ 0. Then u(r) > 0 for r > R and the limit lim r→∞ u(r)r n−pp−1 =λ exists. Moreover, if f (u, r) > 0 for u small and r large, then λ > 0, while iff (u, r) < 0 for u small and r large, then λ < ∞.When Hypothesis B is satisfied we can go a bit further. Now we distinguishbetween the case in which f (u, r) is always positive and the case in which it is negativefor u small.COROLLARY 1. Assume that Hypothesis B of the previous Proposition is satisfied.First assume that f (u, r) > 0 for u small and r large.1 If q 1 ≤ p ∗ , and there are U > 0, c > 0 and Q 1 ∈ (p, q 1 ] such that f (u, r) >cu Q 1−1 for r large and 0 ≤ u < U. Then λ = ∞.Assume now that f (u, r) < 0 for u small and r large.2 If q 1 > p ∗ , then λ > 0.3 If q 1 ≤ p ∗ , and there are U > 0, c > 0 and Q 1 ∈ (p, q 1 ] such that − f (u, r) >cu Q1−1 for r large and 0 ≤ u < U. Then λ = 0 and lim sup r→∞ u(r)r − pQ 1 −p 0, it ispossible to prove the existence of a local stable and unstable manifold also for thenon-autonomous system (5), under suitable hypotheses on g l (x l , t), or equivalently onf (u, r).PROPOSITION 2. Assume that f (u, r) is continuous for r = 0 and considersystem (5) where l > p; then there is a local unstable set˜W u (τ) := {Q ∈ R 2 + | xτ l (Q, t) ∈ R2 +for any t ≤ 0 and limt→−∞ xτ l (Q, t) = O}.This sets contains a closed connected component to which O belongs and whose diameteris positive, uniformly in τ.