RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 61Assume that there are ν > 0 and q 2 > p ∗ such that, for any r ∈ [0,ν], we havef (u,r)lim sup u→∞ < a(r) where 0 < a(r) < ∞. Moreover assume that one of theu q 2following hypotheses −1are satisfied• f (u, r) > 0 for r large and u > 0; moreover there is q 1 > p ∗ such that f (u,r) isu q 1bounded for u positive and small and r large.−1• f (u, r) < 0 for r large and u > 0.Then there is a local stable set˜W s (τ) := {Q ∈ R 2 + | xτ q 2(Q, t) ∈ R 2 +for any t ≤ 0 and limt→−∞ xτ q 2(Q, t) = O} .This sets contains a closed connected component to which O belongs and whose diameteris positive, uniformly in τ.Proof. Consider first the case f (u, r) = k(r)u|u| q−2 , and assume that k(r) is uniformlycontinuous. Then the existence of these stable and unstable sets follows frominvariant manifold theory for non-autonomous system, see [28], [30]. Moreover insuch a case we also know that these sets are indeed smooth manifolds which dependsmoothly on τ. In the general case the existence of the unstable set easily follows fromthe existence of regular solutions for (3). The existence of the stable set is more complicatedand can be proved through Wazewski’s principle, see [15] and [17]. In [17]the proof is given for the case f (u, r) < 0 for u small and r large, but the argumentcan be easily extended also to the case f (u, r) < ku|u| q 1−2 with q 1 > p ∗ , for u smalland r large.We give now a result proved in [13] and [17] which explains the relationshipbetween stable and unstable sets of (5) and solutions of (3).PROPOSITION 3. Consider system (5) and assume that g l (x l , t) is bounded ast → −∞, for any x l > 0. Then each regular solution u(r) of (3) corresponds to atrajectory x τ l (Qu , t) such that Q u ∈ ˜W u (τ), and viceversa. Moreover any trajectoryx τ (Q s , t) where Q s ∈ ˜W s (τ), corresponds to a solution u(r) of (3) with fast decay.REMARK 2. Take f (u, r) = −k 1 (r)u|u| q 1−2 + k 2 (r)u|u| q 1−2 , where q 1 < q 2and the functions k i (r) are continuous, uniformly positive and bounded as r → ∞.Then if q 1 < p we are in the Hypotheses of claim A of Proposition 1, while if q 1 ≥ pHyp. B is satisfied. Moreover if q 1 > p ∗ , then we are in Hyp. 2 of Corollary 1, whileif p < q 1 ≤ p ∗ Hyp. 3 of Corollary 1 holds. To satisfy Hyp. 4 we need to assume thatp < q 1 ≤ p ∗ and lim r→∞ k 1 (r) = k(∞) > 0.We recall that, roughly speaking, positive solutions of (3) can have two asymptoticbehaviours, both as r → 0 and as r → ∞. Obviously the asymptotic behavioras r → 0 is influenced by the behaviour of f for u large and r small, while their behavioras r → ∞ depends on the behaviour of f for u small and r large. Generally