RENDICONTI DEL SEMINARIO MATEMATICO

78 M. FrancaNote that with this approach it is possible to prove the existence of G.S. withfast decay also when ¯M − 1 and ¯M + 2are satisfied, while the approach of [43], [32] failsin that case. However the latter article is able to deal also with the case q ̸= p ∗ . Wewish to stress that the condition on the integrability of k ′ (r)r n and k ′ (r)r −n/(p−1) is insome sense optimal, in view of Theorem 8. Moreover observe that we can combine theexistence results for G.S. with fast decay given in Theorem 6, 11, 12 with the structureresult of Theorem 5 to obtain uniqueness. Furthermore we have the following, see [18],[15].REMARK 7. Assume that hypotheses ¯M + 1 and ¯M − 2are satisfied. Then there areB ≥ A > 0 such that u(d, r) is a crossing solution for any d > B and it is a G.S. withslow decay for 0 < d < A.Assume that hypotheses ¯M − 1 and ¯M + 2are satisfied. Then there are B ≥ A > 0such that u(d, r) is a crossing solution for any d > B and any 0 < d < A. Moreoverthere are R ≥ ρ > 0 such that the Dirichlet problem in the ball of radius r admits 2solutions for r > R and 0 solutions for 0 < r < ρ.Roughly speaking, if k(r) ∈ C 1 is uniformly positive and bounded, admits justone critical point which is a maximum and it is not too flat for r small and r large,regular solutions have structure 3 of Theorem 5 (and positive solutions have structureC). But if the critical point is a minimum, the situation is more complicated. We knowfrom Theorem 12 a sufficient condition to have a G.S. with fast decay. However weconjecture, that, in such a case, we may have multiple G.S. with fast decay, perhapseven infinitely many.Theorem 12 also helps to understand what happens in the perturbative case.When we have a regular perturbation, the stripes E + and E − are very narrow. So,when we approximate the trajectory of the perturbed system with a trajectory of theunperturbed one, we commit a small mistake. In the singular perturbation case we havethat φ varies slowly, so ˙φ has constant sign in long intervals. Since the trajectory ofthe stable and unstable sets have an exponential decay, the sign of the energy functionH mainly depends on the sign of φ(ǫt + τ)x p∗ (t) evaluated when x(t) is far from theorigin. Choose Q either in ξ s (τ) or in ξ u (τ). The idea hidden in Theorem 11 is that,playing with the values of the parameters τ and ǫ, we can make the sign of H p∗ (Q,τ)depend just on the sign of ˙φ evaluated at t = τ.5. f subcritical for u small and supercritical for u largeIn this section we collect few results about an equation for which even some basicquestions are still unsolved. We consider Eq. (1) where f (u) = u|u| q 1−2 + u|u| q 2−2 ,and p ∗ < q 1 < p ∗ < q 2 . In fact as far as we are aware there are only two articles,[11] and [1], concerning the argument and they deal with the case p = 2. Recall that2 ∗ = 2n/(n − 2) and 2 ∗ = 2(n − 1)/(n − 2).Zhou in [44] established that G.S. for (2), in this case have to be radial. So wecan in fact consider directly an equation of the form (3) (with p = 2).