RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 65REMARK 3. Assume that there is T such that J + (t) ≥ 0 (resp. J + (t) ≤ 0),but J + (t) ̸≡ 0 for any t ≤ T , and consider a regular solution u(r) which is positiveand decreasing for any 0 < r < R = ln(T). Then H p ∗(x p ∗(t), t) ≥ 0 (resp.H p ∗(x p ∗(t), t) ≤ 0) for any t ≤ T .Analogously assume that J − (t) ≥ 0 (resp. J − (t) ≤ 0) but J − (t) ̸≡ 0 forany t ≥ T , and consider a solution u(r) which is positive and decreasing for anyr > R = ln(T) and has fast decay. Then H p ∗(x p ∗(t), t) ≥ 0 (resp. H p ∗(x p ∗(t), t) ≤ 0)for any t ≥ T .Using Remark 3 and Lemma 1 we obtain the following result.THEOREM 1. Assume that either J + (r) ≥ 0 and J + (r) ̸≡ 0, or J − (r) ≥ 0and J − (r) ̸≡ 0 for any r > 0. Then all the regular solutions are crossing solutionsand there exists uncountably many S.G.S. with fast decay.Assume that either J + (r) ≤ 0 and J + (r) ̸≡ 0 or J − (r) ≤ 0 and J − (r) ̸≡ 0for any r > 0. Then all the regular solutions are G.S. with slow decay. Moreover thereare uncountably many solutions u(r) of the Dirichlet problem in the exterior of a ball.The proof of the result concerning regular solutions can be find in [34], andinvolves just a shooting argument and the use of J + and J − in relation with the Pohozaevidentity. Translating this argument in this dynamical context we easily get aclassification also of singular solutions, see also [12].Proof. Assume that J + (r) ≤ 0 for any r > 0, but J + ̸≡ 0; consider a regular solutionu(r) which is positive and decreasing in the interval [0, R) and the correspondingtrajectory x p ∗(t). Using (16) we easily deduce that H p ∗(x p ∗(t), t) ≤ 0 for anyt < T = ln(R). From our assumption we easily get that there is l < p ∗ such thatg(x l , t) is uniformly positive for t large and lim t→∞ H l (x l (t), t) < 0. It follows thatx l (t) is forced to stay in a compact subset of the open 4 th quadrant for t large, so u(r)is a G.S. with slow decay.Analogously consider a trajectory ¯x l (t) converging to O as t → +∞. Then thecorresponding solution ū(r) has fast decay, is positive and decreasing for any r > Rwhere R > 0 is a constant. Assume for contradiction that R = 0; then from (16)we find that lim inf t→−∞ H p ∗(¯x p ∗(t), t) > lim t→∞ H p ∗(¯x p ∗(t), t) = 0. Hence, fromLemma 1, we deduce that ¯x p ∗(t) has to cross the coordinate axes indefinitely as t →−∞. Thus R > 0 and ū(r) is a solution of the Dirichlet problem in the exterior of aball.The other claims can be proved reasoning in the same way, see again [12].Reasoning similarly we can complete the previous result proving the existenceof S.G.S. with slow decay, to obtain the following Corollary.COROLLARY 2. Assume that J − (r) ≥ 0 for any r but J − (r) ̸≡ 0, and thatthere is p ∗ < m ≤ p ∗ such that the limit lim t→∞ φ(t)e α m(m−q)t= k(∞) exists ispositive and finite. Then positive solutions have a structure of type A.