RENDICONTI DEL SEMINARIO MATEMATICO

76 M. Franca¯M + 1k(r) is increasing for r small and k ′ (r)r −n/(p−1) ̸∈ L 1 (0, 1].¯M − 1k(r) is decreasing for r small and k ′ (r)r −n/(p−1) ̸∈ L 1 (0, 1].¯M + 2¯M − 2k(r) is increasing for r large and k ′ (r)r n ̸∈ L 1 [1,∞).k(r) is decreasing for r large and k ′ (r)r n ̸∈ L 1 [1,∞).Now we can state the following theorem, see [18], [15].THEOREM 12. Consider (3) where q = p ∗ and k(r) ∈ [a, b] for any r ≥ 0,for some b > a > 0. Assume that either hypotheses ¯M + 1 and ¯M − 2 , or ¯M − 1 and ¯M + 2 aresatisfied. Then there is a G.S. with fast decay. Moreover1. If ¯M − 2 is satisfied there are uncountably many G.S. with slow decay and uncountablymany crossing solutions.2. If ¯M + 1is satisfied, there are uncountably many S.G.S. with fast decay and uncountablymany solutions of Dirichlet problem in the exterior of a ball.3. If ¯M + 1 and ¯M − 2 are satisfied positive solutions have structure C.Proof. Consider the autonomous system (8) where q = p ∗ and φ ≡ a, or φ ≡ b respectively.Denote by x a (t) and x b (t) the trajectories of the former and the latter systemsuch that ẋ a (0) = 0 = ẋ b (0). Denote by A + = {x a (t)|t ≤ 0}, A − = {x a (t)|t ≥ 0},B + = {x b (t)|t ≤ 0}, B − = {x b (t)|t ≥ 0}, by A = (A x , A y ) = x a (0) and byB = (B x , B y ) = x b (0). Let us denote by E + (respectively E − ) the bounded subsetsenclosed by A + , B + (resp. A − , B − ) and the isocline ẋ = 0.Note that the flow of the non autonomous system (8) on A + ∪ B + points towardsthe interior of E + while on A − ∪ B − points towards the exterior of E − . So, usingWazewski’s principle, we can construct compact connected sets as follows, see [15].W u (τ) :={Q ∈ E + |W s (τ) :={Q ∈ E − |limt→−∞ xτ (Q, t) = O and x τ (Q, t) ∈ E + for t ≤ 0},limt→+∞ xτ (Q, t) = O and x τ (Q, t) ∈ E − for t ≥ 0}.We denote by ξ u (τ) and ξ s (τ) the intersection of the isocline ẋ = 0 respectivelywith W u (τ) and W s (τ). In analogy to what we have done in the perturbative case wewant to measure the distance with sign of the compact non-empty sets ξ u (τ) and ξ s (τ)evaluating the energy function H on these sets.We wish to stress that we have committed a mistake in [15] in such evaluation,but we can correct it as follows, see [18]. Let us denote by L the line x = B x , andby C + the intersection of L with A + ; finally let L + be the segment of L betweenC + and B. Denote by xa τ (t), the trajectory of the autonomous system where φ ≡ asuch that xa τ(0) = C+ , and by x τ b(t), the trajectory of the autonomous system whereφ ≡ b such that x τ b (0) = B. Recall that we have explicit formulas for xτ a (t) and xτ b (t)√pand that we can find C > c such that ∗ n−pce pt < x τ b (t) < xτ a (t) < p∗ √Cen−ppt for