RENDICONTI DEL SEMINARIO MATEMATICO

58 M. FrancaY-POXSPẋ=0Figure 1: A sketch of the phase portrait for the autonomous system (8) when φ ≡ k >0, and q = p ∗ . The lines also represents the level curves for the function H q for tfixed, when g q (x q , t) = φ(t)x q |x q | q−2 .gets into R 2 + , which correspond to the positive solutions u(r) of (3) we are interestedin. The existence of trajectories converging to the origin either in the past or in thefuture can be inferred from invariant manifold theory, whenever 1 < p ≤ 2 and q ≥ 2.In such a case we directly prove the existence of a stable and an unstable manifold,denoted respectively by W s and by W u , see [12].When these regularity hypotheses are not satisfied the proofs become more difficult,due to the lack of local uniqueness of the trajectories crossing the coordinateaxes. But using Wazewski’s principle and the fact that the trajectories we are interestedin do not cross the coordinate axes, it is possible to obtain a similar result. However,with this different proof, a priori W u and W s are just compact and connected sets. Butin the autonomous case k ≡ const > 0, we can exploit the invariance of the systemwith respect to t, to conclude that W s and W u are in fact graph of a trajectory havingthe origin respectively as ω-limit set and α-limit set. Therefore, even in this case, theyare 1 dimensional manifolds, see [15], [17]. We think it is worth mentioning the factthat, when the system is not Lipschitz, a priori the trajectories could reach the origin atsome t = T finite, either in the past or in the future. However it is easy to show thatthis possibility cannot take place when q ≥ p, see [17] for a detailed proof.Note that, if k > 0 is a constant, we also have that H q (x q (t), y q (t), t) is increasingalong the trajectories if and only if p ∗ < q < p ∗ , and it is decreasing if and only ifq > p ∗ . Moreover for any trajectory converging to the origin as t → ±∞, we havelim t→±∞ H p ∗(x p ∗(t), y p ∗(t), t) = 0. Putting together all these results, we can drawfig. 2, and classify positive solutions in one of the following structures.A All the regular solutions are monotone decreasing G.S. with slow decay. There areuncountably many solutions of the Dirichlet problem in the exterior of the ball.More precisely, for any R > 0 there is a solution v(r) such that v(R) = 0, v(r)is positive for any r > R and it has fast decay. There is at least one S.G.S. withslow decay.