A TRIBUTE TO PATRIZIA PUCCI 5

(under restrictions upon 0 p q depending upon m and n), generalize a large

number of existing results. They also have found conditions which can be applied

to the case where f(t) =

tp−1[exp(tp)

− 1 − (α +

1)tp].

With Filippo Gazzola and

Bert Peletier, they have analyzed in 2003 the asymptotic behavior of the radially

symmetric ground state of the perturbed p-Laplacian equation

−Δpu =

−δup−1

+

uq−1

in

Rn

when n p 1 and q, subcritical, tends to the critical growth np/(n − p). With

Garc´ ıa-Huidobro and Man´ asevich, Pucci and Serrin have initiated in 2006 the prob-

lem of uniqueness and qualitative properties of the non-negative radial solutions of

singular quasilinear elliptic equations with weights

div

{g(|x|)|Du|p−2Du}

+ h(|x|)f(u) = 0, x ∈

Rn

\ {0},

with applications to the case where f(t) =

−tq

+

ts.

The maximum principle for nonlinear elliptic equations has been a source of

inspiration for Pucci and Serrin between 1999 (in joint work with Henghui Zou)

and 2011. They have extended V´ azquez’ strong maximum principle for semilinear

inequalities involving a p-Laplacian to more general classes of quasilinear differential

inequalities of the form

div (A(|Du|)Du) − f(u) ≤ 0, u ≥ 0, (4)

in a domain Ω ⊂ Rn. Similar results are proved for fully quasilinear inequalities.

A compact support principle is proved as well when Ω contains the exterior of some

ball and u is a solution of the reverse inequality tending to zero at infinity : u

must vanish identically or have a compact support. New original contributions,

simplified proofs and a description of the state of the art for those questions have

been the object of a substantial survey paper in 2004. The main results of this line

of papers can be summarized as follows. If one defines for t 0

Φ(t) := tA(t), H(t) :=

Φ(t)

0

Φ−1(s)

ds, F (u) :=

u

0

f(s) ds,

then

(1) In order that the strong maximum principle holds for (4), it is necessary

and suﬃcient that either f(s) ≡ 0 in some neighborhood of 0 or that

f(s) 0 for s ∈ (0,δ) and

δ

0

ds

H−1(F (s))

= ∞.

(2) If f(u) 0 for u 0, in order that the compact support principle holds

for

div (A(|Du|)Du) − f(u) ≥ 0, u ≥ 0,

in an exterior domain, it is necessary and suﬃcient that

δ

0

ds

H−1(F (s))

∞.

Existence results are also given for dead cores, i.e. solutions u for which an open

set Ω1 ⊂ Ω1 ⊂ Ω exists such that u ≡ 0 in Ω1 and is positive in Ω \ Ω1. Those

results have been integrated in 2007, together with a superb presentation of classi-

cal results in a historical perspective, in the beautiful monograph ‘The Maximum

Principle’, published by Birkh¨ auser. The last joint paper of Pucci and Serrin, from

2011, provides simple proofs of properties of entire solutions of quasilinear elliptic

equations based upon a comparison principle contained in the monograph.