RENDICONTI DEL SEMINARIO MATEMATICO

84 M. Franca0. Furthermore assume that we are in the Hypotheses of Proposition 1 B, then we alsohave that lim d→D R 1 (d) = ∞. Therefore the Dirichlet problem in the ball of radiusR > 0 for equation (3) admits at least one solution for any R > 0.THEOREM 21. Assume that Hyp. F1, F2 and F4 are satisfied, then (3) admitsuncountably many S.G.S.REMARK 13. Assume that f is as in (23), that the functions k i (r) are uniformlypositive and bounded for any r ≥ 0 and p < q M < p ∗ . Then hypotheses F1 and F2are satisfied. Moreover if −k i (r) and k j (r) are decreasing for any r > 0, 1 ≤ i ≤ Nand N < j ≤ M, hypothesis F3 is satisfied; finally if q M > p ∗ F4 holds.The proof of Theorem 20 can be found in [17] and follows, with some minorchanges, the scheme introduced in [25] and then used in [19], [22] and [16].When Hyp. F1 and F2 are satisfied the initial value problem (3), with u(0) =d > 0, u ′ (0) = 0 admits at least a solution. Moreover such a solution, denoted byu(d, r) is unique for any d ≥ A and u ′ (r) ≤ 0 for r small. All these solutions can becontinued in J(d) = (0, R d ) = {r > 0|u ′ (r) < 0 < u(r)}, where R d can also beinfinite. This was proved for the spatial independent problem in [19] and then adaptedwith some trivial changes to the spatial dependent problem in [17]. Since u(d, r) ispositive and decreasing for r < R d , the limit lim r→Rd u(d, r) exists and is nonnegative,so we can define the following set:I := {d ≥ A | limr→R du ′ (d, r) < 0}Using an energy analysis we can prove that A ̸∈ I , when F1, F2 and F3 are satisfied.Moreover using a continuity argument on the auxiliary system (5), we prove that I isopen in [A,∞) whenever F1 and F2 hold, see [17]. The difficult part of the proof is toshow that I ̸= ∅. In fact we show that, if F1 and F2 are satisfied there is D ̸∈ I suchthat (D,∞) ⊂ I .For this purpose we introduce a dynamical system of the form (5), using (4)with l = q, where q is the parameter defined in [12]. Then we show that for r smalland u large we can approximate our system with an autonomous subcritical system oftype (8). Through a careful analysis of the phase portrait we are able to construct abarrier set E τ ⊂ {(x, y) ∈ R 2 | y ≤ 0 ≤ x}. Then, using Wazewski’s principle, weshow that there are M > 0 and δ > 0 such that, for any τ < −M, there is an unstableset ˜W u (τ) ⊂ E τ , which intersects the y negative semi-axis in a compact connectedset, say ζ(τ). It follows that the trajectories x τ q (Qu (τ), t) ∈ E τ for any t < 0 and thatlim t→−∞ x τ q (Qu (τ), t) = O. So they correspond to regular solutions u(d(τ), r), thatare positive and decreasing for r ≤ exp(τ) and they become null with nonzero slope atr = exp(τ), so they are crossing solutions.It follows that if F1, F2 and F3 hold there is D > A, such that (D,∞) ∈ I , butD ̸∈ I . Then we show that u(D, r) is a monotone decreasing G.S. using a continuityargument on (5), and Theorem 20 and Corollary 5 follow.To prove the existence of S.G.S. we have to consider system (7) and to construct