RENDICONTI DEL SEMINARIO MATEMATICO

70 M. Francar > 0. Now we can state the following result proved in [43], using the Pohozaevidentity and the Kelvin transformation.THEOREM 6. Consider (3) where p = 2 and assume ρ + > 0 and ρ − < ∞.If Z(r 1 ) > 0 for some r 1 ∈ (0,ρ + ] and Z(r 2 ) > 0 for some r 2 ∈ [ρ − ,∞), there isD > 0 such that u(D, r) is a G.S. with fast decay.Let us set λ := (n−p)(q−p∗ )p; following [43], we get the following more explicit result.COROLLARY 4. Consider (3) where p = 2 and suppose q ̸= p ∗ and that k(r)is nonnegative and satisfies:k(r) = Ar σ + o(r σ ) at r = 0 k(r) = Br l + o(r l ) at r = ∞,where A, B > 0 and l < λ < σ , then there is a G.S. with fast decay.Now assume q = p ∗ and that k(r) satisfiesk(r) = A 0 + A 1 r σ + o(r σ ) at r = 0 k(r) = B 0 + B 1 r l + o(r l ) at r = ∞,where A 1 , B 1 > 0, A 0 , B 0 > 0, −n < l < 0 < σ < n. Then there is a G.S. with fastdecay.Note that the case q = p ∗ is more delicate; we stress that the restriction |l|,|σ|smaller than n is needed even if it was not required in [43]. However when A 0 = 0 wedo not need the restriction on |l| and when B 0 = 0 we do not need the restriction on|σ|.Using a similar argument Kabeya, Yanagida and Yotsutani, in [32] found ananalogous result for the case p ̸= 2.THEOREM 7. Consider (3) where k(r) ≥ 0 for any r. Assume that eitherrklim inf ′ (r) r→0 k(r)> λ or k(r) = Ar σ + o(r σ ) at r = 0 for some A > 0 and σ > λ.rkMoreover assume that either lim sup ′ (r)r→∞ k(r)< λ or k(r) = Br l +o(r l ) at r = 0 forsome B > 0, l < λ. Then there is a strictly increasing sequence d j > 0, j = 0,...,∞,such that u(d j , r) has exactly j zeroes and has fast decay. So in particular u(d 0 , r) isa G.S. with fast decay.Note that the conditions of the previous Theorem at r = 0 (and at r = ∞) aresimilar, but they do not imply each other. In fact the former is useful when k(r) hasa logarithmic term, e. g. k(r) = | ln(r)|r σ , and the latter when k(r) behaves like apower at r = 0 (and at r = ∞). However, in both the cases, when q = p ∗ we havek(0) = k(∞) = 0.We wish to mention that Bianchi and Egnell in [5], [7] have some other sufficientconditions for the existence of G.S. with fast decay. Each condition, as the onesof Corollary 4 and of Theorem 7, in some sense, requires a change in the sign of thefunction J + (r) which is “sufficiently large to be detected”. We stress that the situation