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 +
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 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.
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