RENDICONTI DEL SEMINARIO MATEMATICO

62 M. Francaspeaking, when f is positive for u small, we have seen that solutions with fast andslow decay may coexist, while when it is negative we can have either solutions withfast decay or oscillatory solutions. Analogously when f (u, r) is positive and supercriticalwith respect to p ∗ , for u large and r small, we can have regular solutions u(d, r)such that u(d, 0) = d and u ′ (d, 0) = 0, and singular solutions v(r) that are such thatlim r→0 v(r) = +∞. More preciselyPROPOSITION 4. Assume that there are s > p ∗ , ρ > 0 and positive functionsb(r) ≥ a(r) such that, for any 0 ≤ r ≤ ρ we have0 < a(r) ≤ lim infu→+∞f (u, r)u s−1≤ lim supu→+∞f (u, r)u s−1 ≤ b(r) < ∞.If Q ∈ W u (τ) then the solution u(r) corresponding to x τ s (Q, t) is a regular solution.Moreover any singular solution, if it exists, is such that u(r)r p/(s−p) is bounded for rsmall and, if s ̸= p ∗ , u(r)r p/(s−p) is uniformly positive, too.Assume further that s ̸= p ∗ f (u,0)and that the limit lim u→+∞ = k(∞) > 0u s−1exists and is finite. Then lim r→∞ u(r)r p/(s−p) = P x > 0 where P= (P x , P y ) is thecritical point of system (5) where l = q and g ≡ k(∞)x|x| s−2 .Assume that there are q > p ∗ , R > 0 and positive functions B(r) ≥ A(r) suchthat, for any r > R we have0 < A(r) ≤ lim infu→0f (u, r)u q−1≤ lim supu→0f (u, r)u q−1 ≤ B(r) < ∞.Then, if Q ∈ ˜W s (τ), the solution u(r) corresponding to xq τ (Q, t) has fast decay, thatis the limit lim r→∞ u(r)r (n−p)/(p−1) > 0 exists and is finite. A slow decay solution(if it exists), is such that u(r)r p/(q−p) is bounded for r large; moreover if q ̸= p ∗ ,u(r)r p/(q−p) is uniformly positive, too.limAssume further that the limit lim r→∞ f (u,r)u→0 = k(∞) > 0 exists and isu q−1finite. Then lim r→∞ u(r)r p/(q−p) = P x > 0 where P= (P x , P y ) is the critical point ofsystem (5) where l = q and g ≡ k(∞)x|x| q−2 .These results are proved in [12], [13], [17] using dynamical arguments.3. When the Pohozaev function does not change sign3.1. The case f (u, r) = k(r)u|u| q−2In this subsection we discuss positive solutions of equation (3) in the case f (u, r) =k(r)u|u| q−2 and q > p ∗ . This problem has been subject to rather deep investigations inthe ’90s also for the relevance it has in different applied areas. First of all, when p = 2eq. (1) can be regarded as a nonlinear Schroedinger equation. Moreover, when q = p ∗and again p = 2, this equation is known with the name of scalar curvature equation. Infact the existence of a G.S. u(x) amounts to the existence of a metric g conformal to a