RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 63standard metric g 0 on R n (g = u n−2 4g 0 ), whose scalar curvature is k(|x|). Furthermore,if the G.S. has fast decay, the metric g gives rise, via the stereographic projection, toa metric on the sphere deprived of a point S n \ {a point} which is equivalent to thestandard metric.Moreover when q > 1 and k(r) takes the form k(r) =rα eq. (2) is also1+rknown as Matukuma equation and it was proposed as a model in astrophysics. β Thisproblem will be investigated in details also in section 4, where we will assume that thePohozaev functions change their sign, so that positive solutions have a richer structure.We begin by some preliminary results concerning forward and backward continuabilityand long time behaviour for positive solutions, in relation with the Pohozaevfunction. These results can be proved using directly the Pohozaev identity, or through adynamical argument exploiting our knowledge of the level sets of the function H(x, t),see [34] and [13].LEMMA 1. Let u(r) be a solution of (3), and x p ∗(t) the corresponding trajectory.Assume that lim inf t→±∞ H(x p ∗(t), t) > 0, then x p ∗(t) has to cross the coordinateaxes indefinitely as t → ±∞, respectively.Assume that lim sup t→±∞ H(x p ∗(t), t) < 0, then x p ∗(t) cannot converge to theorigin or cross the coordinate axes.When p = 2, the standard tool to understand the behaviour of solutions withfast decay is the Kelvin transformation. Let us set(12)s = r −1 ũ(s) = r n−2 u(r) ˜K(s) = r 2λ K(r −1 ) λ = (n + 2)(q − 2∗ )2Then (3) is transformed into(13) [ũ s (s)s n−1 ] s + ˜K(s)ũ|ũ| q−2 (s)s n−1 = 0.Note that a regular solution u(d, r) of (3) is transformed into a fast decay solutionsũ(s) of (13) such that lim r→∞ ũ(r)r n−pp−1 = d, and viceversa. So we can reduce theproblem of discussing fast decay solutions to an analysis of regular solutions for thetransformed problem.However we do not have an analogous result for the case p ̸= 2, so we need thefollowing Lemma, that, when p = 2, is a trivial consequence of the existence of theKelvin inversion.LEMMA 2. Assume that f (u, r) > 0 for any u > 0 and consider a solutionu(r) which is positive and decreasing for any r > R. Then u(r)r n−pp−1 is increasing forany r > R.Proof. Consider system (8) where l = p ∗ and the trajectory x p∗ (t) corresponding tou(r). Note that x p∗ (t) = u(r)r n−pp−1 and that γ l = 0; hence ẏ p∗ (t) < 0 wheneverx p∗ (t) > 0. Assume for contradiction that there is t 1 > T = ln(R) such that ẋ p∗ (t 1 )