RENDICONTI DEL SEMINARIO MATEMATICO

86 M. Francah. We denote by f (v, s) = h(s) ¯ f (v, r(s)) and obtain (3) from (26), with r replacedby s.REMARK 14. Note that, if for any fixed v > 0, f ¯(v, r) grows like either apositive or a negative power in r for r small, we can play with the parameter N in orderto have that, for any fixed u > 0, f (u, 0) is positive and bounded. E.g., if g(r) ≡ 1and f ¯(u, r) = r l u|u| q−1 , we can set N = p(n+l)−np+l−1, so that, switching from r to s asindependent variable (26) takes the form(27) [s N−1 v s |v s | p−2 ] s + Cs N−1 v|v| q−1 = 0,where C = ∣ N−p∣ ∣ p n−1p−1N−pp−1 ∣N−1∣p > 0. So we can directly study the spatial independentequation (27), recalling that the natural dimension is N and this changes the values ofthe critical exponents and the asymptotic behaviors of positive solutions as r → 0 andas r → ∞.Observe that N does not need to be an integer and that in literature such an assumptionis not really used to prove the results. Thus all the theorems obtained for (3) can betrivially extended to an equation of the form (25), where g satisfies either H1 or H2.8. AcknowledgementThis survey is based on a seminar delivered by prof. R. Johnson at the Workshops onDynamics held in Torino in September 2005 and organized by prof. G. Zampieri. Theauthor wishes to thank prof. Johnson for having introduced him to the study of thistopic.References[1] BAMON R., FLORES I. AND DEL PINO M., Ground states of semilinear elliptic equations: ageometrical approach, Ann. Inst. Poincare 17 (2000), 551-581.[2] BATTELLI F. AND JOHNSON R., Singular ground states of the scalar curvature equation in R n ,Diff. Int. Eq. 14 (2000), 123–139.[3] BATTELLI F. AND JOHNSON R., On positive solutions of the scalar curvature equation when thecurvature has variable sign, Nonlinear Analysis 47 (2001), 1029–1037.[4] BATTELLI F. AND JOHNSON R., On transversal smoothness of invariant manifold, Comm. Appl.Anal. 5 (2001), 383–401.[5] BIANCHI G., The scalar curvature equation on R n and S n , Advances in Diff. Eq. 1 (5) (1996),857–880.[6] BIANCHI G., Non-existence of positive solutions to semi-linear elliptic equations on R n or R n +through the method of moving planes, Comm. Part. Diff. Eq. 22 (1997), 1671–1690.[7] BIANCHI G. AND EGNELL H., An ODE approach to the equation u + K u n+2n−2 = 0, in R n , Math.Zeit. 210 (1992), 137–166.[8] CITTI G. AND UGUZZONI F., Positive solutions of − p u + u p−2 − q(x)u α = 0, NonlinearDifferential Equations Appl. 9 (2002), 1–14.