4 J. MAWHIN

In 1990, Pucci and Serrin have used similar identities to extend to the semilinear

Dirichlet problem on a ball for polyharmonic operators (with n 2K and s =

(n + 2K)/(n − 2K))

(−Δ)K

u = λu +

|u|s−1u

in B ⊂

Rn,

u = Du = . . . =

DK−1u

= 0 on ∂B

a famous result of Brezis-Nirenberg when K = 1, recovering existence of non-trivial

solutions by addition of a linear term. Pucci and Serrin have shown that, when

K ≥ 2, the lower bound of λ for which a positive solution exists is positive for some

critical dimensions, for example, if K = 2, when n = 5, 6, 7.

Even if their main contributions deal with partial differential equations, Pucci

and Serrin have devoted between 1989 and 1995 a dozen of papers to the study

of continuation, asymptotic behavior and stability of solutions of some nonlinear

ordinary differential equations and systems of the second order of the form

(A(u )u ) + δ(r)A(u )u + f(r, u) = 0.

Special emphasis is made upon systems having a variational structure or coming

from radial solutions of some quasilinear partial differential equations. Special cases

are the Lane-Emden, Emden-fowler, and Haraux-Weissler equations. The obtained

conditions are related to, and substantially generalize in several directions, earlier

work of Arstein-Infante, Levin-Nohel, Salvadori, Thurston-Wong, Ballieu-Peiffer

and others. The underlying technique is based upon the construction Lyapunov

functions based upon the use of Pucci-Serrin’s variational identity. The results pro-

vide in many cases sharp suﬃcient conditions, especially on the damping term, for

obtaining various types of asymptotic behavior. In particular, asymptotic stability

can hold when the damping is intermittently suppressed. Studies in this direction

have been pursued by Giovanni Leoni, a student of Serrin, in a series of papers

which include a joint one with Maria Manfredini and Pucci.

It is not too surprising that this work has also inspired to our authors, between

1996 and 1998, the obtention of various asymptotic stability and non continuation

conditions for some classes of evolution equations in Banach spaces, like dissipative

wave systems

utt − Δu + Q(t, x, u, ut) + f(x, u) = 0,

nonlinear parabolic systems

A(t)|ut|m−2ut

= Δu − f(x, u),

and abstract evolution equations

[P (u (t))] + A(u(t)) + Q(t, u (t)) + F (u(t)) = 0.

Another interesting question, considered in two papers of 1998 by Pucci and

Serrin, is the study of some properties of ground states for quasilinear elliptic op-

erators. From uniqueness conditions for the positive solutions u = u(r) of ordinary

differential problems of the form

−[rn−1g(|u

|)u ] = f

n−1f(u)

in R+, u (0) = 0 = u(∞),

linked to the radial solutions of the elliptic equation

−div (g(|Du|)Du) = f(u) in

Rn,

they have obained uniqueness conditions for the ground state solutions under con-

ditions which, in the important cases where g(t) =

tm−2

(m 1), f(t) =

tq

−

tp