Tiểu Luận Quasilinear and Hessian equations of Lane-Emden type

Quasilinear and Hessian equations
of Lane-Emden type
By Nguyen Cong Phuc and Igor E. Verbitsky*
Abstract
The existence problem is solved, and global pointwise estimates of solu-tions are obtained for quasilinear and Hessian equations of Lane-Emden type,
including the following two model problems:

p
u = u
q
+ ; F
k
[ u] = u
q
+ ; u  0;
on R
n
, or on a bounded domain
 R
n
. Here

p
is the p-Laplacian dened
by

p
u = div (rujruj
p 2
), and F
k
is the k-Hessian dened as the sum of
k  k principal minors of the Hessian matrix D
2
u (k = 1; 2; : : : ; n);  is a
nonnegative measurable function (or measure) on
.
The solvability of these classes of equations in the renormalized (entropy)
or viscosity sense has been an open problem even for good data  2 L
s
(
),
s > 1. Such results are deduced from our existence criteria with the sharp
exponents s =
n(q p+1)
pq
for the rst equation, and s =
n(q k)
2kq
for the second
one. Furthermore, a complete characterization of removable singularities is
given.
Our methods are based on systematic use of Wol's potentials, dyadic
models, and nonlinear trace inequalities. We make use of recent advances in
potential theory and PDE due to Kilpelainen and Maly, Trudinger and Wang,
and Labutin. This enables us to treat singular solutions, nonlocal operators,
and distributed singularities, and develop the theory simultaneously for quasi-linear equations and equations of Monge-Ampere type.
1. Introduction
We study a class of quasilinear and fully nonlinear equations and in-equalities with nonlinear source terms, which appear in such diverse areas
as quasi-regular mappings, non-Newtonian
uids, reaction-diusion problems,
and stochastic control. In particular, the following two model equations are of
*N. P. was supported in part by NSF Grants DMS-0070623 and DMS-0244515. I. V. was
supported in part by NSF Grant DMS-0070623.
860 NGUYEN CONG PHUC AND IGOR E. VERBITSKY
substantial interest:
(1.1)
p
u = f (x; u); F
k
[ u] = f (x; u);
on R
n
, or on a bounded domain
 R
n
, where f (x; u) is a nonnegative func-tion, convex and nondecreasing in u for u  0. Here

p
u = div (ru jruj
p 2
)
is the p-Laplacian (p > 1), and F
k
is the k-Hessian (k = 1; 2; : : : ; n) dened
by
(1.2) F
k
=
X
1i1<


<ikn

i1




ik
;
where 
1
; : : : ; 
n
are the eigenvalues of the Hessian matrix D
2
u. In other
words, F
k
is the sum of the k  k principal minors of D
2
u, which coincides
with the Laplacian F1
=
u if k = 1, and the Monge{Ampere operator
Fn
= det (D
2
u) if k = n.
The form in which we write the second equation in (1.1) is chosen only
for the sake of convenience, in order to emphasize the profound analogy be-tween the quasilinear and Hessian equations. Obviously, it may be stated as
( 1)
k
F
k
= f (x; u), u  0, or F
k
= f (x; u), u  0.
The existence and regularity theory, local and global estimates of sub-and super-solutions, the Wiener criterion, and Harnack inequalities associated
with the p-Laplacian, as well as more general quasilinear operators, can be
found in [HKM], [IM], [KM2], [M1], [MZ], [S1], [S2], [SZ], [TW4] where many
fundamental results, and relations to other areas of analysis and geometry are
presented.
The theory of fully nonlinear equations of Monge-Ampere type which
involve the k-Hessian operator F
k
was originally developed by Caarelli,
Nirenberg and Spruck, Ivochkina, and Krylov in the classical setting. We re-fer to [CNS], [GT], [Gu], [Iv], [Kr], [Tru2], [TW1], [Ur] for these and further
results. Recent developments concerning the notion of the k-Hessian measure,
weak continuity, and pointwise potential estimates due to Trudinger and Wang
[TW2]{[TW4], and Labutin [L] are used extensively in this paper.
We are specically interested in quasilinear and fully nonlinear equations
of Lane-Emden type:
(1.3)
p
u = u
q
; and F
k
[ u] = u
q
; u  0 in
;
where p > 1, q > 0, k = 1; 2; : : : ; n, and the corresponding nonlinear inequali-ties:
(1.4)
p
u  u
q
; and F
k
[ u]  u
q
; u  0 in
:
The latter can be stated in the form of the inhomogeneous equations with
measure data,
(1.5)
p
u = u
q
+ ; F
k
[ u] = u
q
+ ; u  0 in
;
where  is a nonnegative Borel measure on
.