RENDICONTI DEL SEMINARIO MATEMATICO

Radial solutions for p-Laplace equation 77t ≤ 0. Consider a trajectory x τ (Q u (τ), t) of the non-autonomous system (8) such thatx τ (Q u (τ), 0) = Q u (τ) ∈ L + . It can be proved thatce nt < |x τ b (t)|p∗ ≤ |x τ (Q u (τ), t)| p∗ ≤ |x τ a (t)|p∗ < Ce ntfor any t ≤ 0, see [18]. Denote by ¯W u (τ) and ¯W s (τ) respectively the subset of W u (τ)and W s (τ) contained in {x | 0 < x < L x }. It can be shown easily that for any pointQ ∈ ¯W u (τ) we have c(Q)e nt ≤ |x τ (Q u (τ), t)| p∗ ≤ C(Q)e nt , where C(Q)/C =K(Q) = c(Q)/c > 0.Now assume that hypothesis ¯M + 2 is satisfied; then there is T 0 > 0 such that˙φ(t) > 0 for any t > T 0 . Hence for any Q ∈ ¯W u (τ) we have(21)H p ∗(Q,τ) =≥ e−nτ K(Q)σ∫ 0−∞˙φ(τ + t) |xτ (Q; t)| p∗dt ≥p ∗[C(φ(T 0 ) − b)e nT 0+ c∫ τT 0]˙φ(ζ)e nζ dζSince ˙φ(ζ)e nζ ̸∈ L 1[ [0,∞) ] , we can find N + > T 0 such that H p ∗(Q,τ) > 0 for anyQ ∈ ¯W u (τ) and τ > N + .We denote by τ,t (Q) the diffeomorphism defined by the flow of (8), precisely τ,t (Q) = x τ (Q; t). Note that for any Q ∈ ¯W u (τ), where τ > N + , and any t ≥ 0, wehave H( τ,t (Q), t + τ) > H(Q,τ) > 0 since ˙φ(s) > 0 for s > τ > N + .Observe that there is a unique t = T u (Q) > 0 such that x τ (Q; t) ∈ E + for anyt < T u (Q) and x τ (Q; T u (Q)) ∈ ξ u (T u (Q) + τ). We choose T + ω = min{T u (Q) +N + | Q ∈ ¯W u (N + )}; it follows that N + ,t[ ¯W u (N + )] ⊃ ˜W u (N + + t), for any t ≥T + ω − N+ . Hence H(Q,τ) > 0 for any Q ∈ ˜W u (τ) for any τ > T + ω .Moreover, for any P ∈ ˜W s (τ) we have∫ +∞H p ∗(P,τ) = − ˙φ(t + τ) |xτ (P, t)| p∗τp ∗ dt < 0,since ˙φ(t) > 0 for t + τ > T 0 . Therefore H p ∗(P,τ) < 0 < H p ∗(Q,τ) for anyP ∈ ˜W s (τ) and any Q ∈ ˜W u (τ). Analogously if ¯M − 1 is satisfied, we can find T − α < 0such that H p ∗(Q,τ) < 0 < H p ∗(P,τ) for any point P ∈ ˜W s (τ) and Q ∈ ˜W u (τ), forany τ < T − α . It follows that there is τ 0 ∈ (T − α , T + ω ) such that ξ s (τ 0 )∩ξ u (τ 0 ) ̸= ∅. So ifQ 0 ∈ ξ s (τ 0 ) ∩ ξ u (τ 0 ) we have that the solution u(r) of (3) corresponding to x τ 0(Q 0 , t)is a G.S. with fast decay.Then repeating the argument of the perturbative case we conclude the proof ofthe Theorem.This way we have proved structure results for positive solutions also in the casep > 2 and corrected the corresponding results in [15]. However we cannot correct theproof of the results concerning the existence of multiple G.S. with fast decay, publishedin [15].