RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 59OXOSXSWsWuPPẋ=0ẋ=0YWsYWuFigure 2: A sketch of the phase portrait for the autonomous system (8) when φ ≡k > 0, 1 < p ≤ 2 and q ≥ 2. The figures show the stable manifold W s (dottedline) and the unstable manifold W u (dashed line). The solid curve S indicates the set{(x q , y q ) | x q ≥ 0 H q (x q , y q ) = 0}. Figure 2A refers to the case q ≥ p ∗ while 2B tothe case p ∗ < q < p ∗ .B All the regular solutions u(d, r) are crossing solutions, and there are uncountablymany S.G.S. with fast decay v(r). There is at least one S.G.S. with slow decay.Namely, if q > p ∗ positive solutions have structure A, while if p ∗ < q < p ∗they have structure B. In both the cases the S.G.S. with slow decay is unique and canbe explicitly computed. If q = p ∗ we are in the border situation, so all the regularsolutions are G.S. with fast decay, see (11), there are uncountably many S.G.S. withslow decay, and uncountably many oscillatory solutions, see [12]. When q ≤ p ∗ , it iseasy to show that all the regular solutions u(r) of (3) are crossing solutions.We conclude this section with some basic results concerning the existence ofregular solutions and positive fast decay solutions for (3) and a wide class of functionsf (u, r). First of all we recall that, if f (u, r) is continuous and locally Lipschitz continuousin the u variable, the existence of regular solution is ensured, and if f (d, 0) > 0we also have local uniqueness of u(d, r). The proof of this standard result can be foundin [19] for the spatially independent case, but the argument can be easily adapted to thegeneral case, see [16], [17]. We give now a result concerning the asymptotic behaviourof positive solutions. The proof of this result can be found in [19], [13], [17].PROPOSITION 1. Consider a solution u(r) of (3) such that u ′ (r) ≤ 0 ≤ u(r)for any r > R for a certain R > 0, and lim r→∞ u(r) = 0.A Assume that there are U > 0 and g(u) ∈ Lloc 1 such that | f (u, r)| < g(u) for r ≥ 0and 0 ≤ u ≤ U, and denote by G(u) = ∫ u0g(s)ds. Moreover assume that∫0 |G(s)|−1/p ds < ∞. Then the support of u(r) is bounded.B Assume that there are C > 0, U > 0 and q 1 ≥ p such that | f (u, r)| < Cu q 1−1 for