RENDICONTI DEL SEMINARIO MATEMATICO

82 M. Franca∑ Mi=N+1A i > 0. Then there is at least one singular solution v(r) of (3).THEOREM 18. Consider (3) where f satisfies F0. Moreover assume that thereare positive constants C i and c j such that k i (r) > C i and k j (r) < C j for r large andi ≤ N and j > N. Then all the regular and singular solutions are defined and positivefor any r ≥ 0 and lim sup r→∞ u(d, r) > 0 for any d > 0.Assume further that −k i (r) and k j (r) are decreasing and bounded for r largeand i ≤ N and j > N, then there is is a computable constant b ∗ , such that all theregular solutions u(r) (and the singular, if they exist) are such that0 < lim infr→∞u(r) ≤ lim inf u(r) < b∗r→∞REMARK 10. The Hypotheses of Theorem 18 are satisfied for example if wetake f as in (22), q 1 ≤ p ∗ ≤ q 2 , q 1 < q 2 , and the functions k i (r) uniformly positive,bounded and −k 1 (r) and k 2 (r) are decreasing.It is possible to give some ad hoc condition for the existence of G.S. even in the caseq 1 ≤ p ∗ ≤ q 2 and q 2 > q 1 . In fact we have to lower the contribution given by thenegative term −k 1 (r)u|u| q 1−2 , taking a strongly decreasing function k 1 (r), see [13].More preciselyTHEOREM 19. Assume F0, and that the limits lim t→∞ φ i (t)e α p ∗(p∗ −q i )t = B i ≥0 exist and are finite for i ≤ M, and that ∑ Mi=N+1 B i > 0.Then all the regular solutions u(r) are G.S. with slow decay. Finally, if there is asingular solution it is a S.G.S. with slow decay.REMARK 11. The Hypotheses of Theorem 19 and Remark 9 are satisfied forexample if we take f as in (22), q 1 = p ∗ < q 2 , k 1 (r) uniformly positive, bounded andincreasing, k 2 (r) = a + br α p ∗(q 2−p ∗) where a, b > 0; or if we take q 1 < p ∗ = q 2 ,k 2 (r) uniformly positive, bounded and increasing, and k 1 (r) = a/(1 + br α p ∗(p∗ −q 1 ) ),where a, b > 0.As we said at the beginning of the section, the situation becomes more interestingwhen f is subcritical both as u → 0 and as u → ∞. A first important stepto understand equation (2) in this setting was made in [25], where the authors provedthe existence of a G.S. in the case p = 2 and assuming that f (u, r) is as in (22),q 1 = 2 < q 2 < 2 ∗ and −k 1 (r) and k 2 (r) non-increasing. These results have beenextended to more general operators, including the p-Laplacian for p > 1, in [19] andto a wider class of nonlinearities f . They just require that there is A > 0 such thatF(u) < 0 for 0 < u < A, F(A) = 0 and f (A) > 0, where F(u) := ∫ u0 f (s)ds.In [19] the non-linearity f is assumed to be spatially independent and subhalflinear,namely either there is b > A such that f (b) = 0, or lim inf u→∞ u p < ∞.F(u)If we consider the prototypical case (22) the assumptions of [19] reduce to 1 < q 1