EXISTENCE AND REGULARITY FOR CRITICAL ANISOTROPIC
EQUATIONS WITH CRITICAL DIRECTIONS
JÉRÔME VÉTOIS
Abstract. We establish existence and regularity results for doubly critical anisotropic equations in domains of the Euclidean space. In particular, we answer a question posed by Fragalà–Gazzola–Kawohl [24] when the maximum of the anisotropic configuration coincides with
the critical Sobolev exponent.
1. Introduction
In this paper, we investigate existence and regularity for doubly critical anisotropic equa−
tions. In dimension n ≥ 2, we provide ourselves with an anisotropic configuration →
p =
→
1,−
p
(p1 , . . . , pn ) with pi > 1 for all i = 1, . . . , n. We let D (Ω) be the anisotropic Sobolev space
defined as the completion of the vector space
all smooth functions with compact support
Pof
n
−
in Ω with respect to the norm kukD1,→
=
p (Ω)
i=1 k∂u/∂xi kLpi (Ω) . We are concerned with the
following anisotropic problem of critical growth
(
p∗ −2
→
− ∆−
u in Ω ,
p u = λ |u|
(1.1)
−
→
u ∈ D1, p (Ω) ,
on domains Ω in the Euclidean space Rn , where λ is a positive real number, p∗ is the critical
→
Sobolev exponent (see (1.3) below), and ∆−
p is the anisotropic Laplace operator defined by
n
X
∂ pi
→
∆−
∇ u,
pu =
∂xi xi
i=1
(1.2)
→
where ∇pxii u = |∂u/∂xi |pi −2 ∂u/∂xi for all i = 1, . . . , n. As one can check, ∆−
p involves directional derivatives with distinct weights. Anisotropic operators appear in several places in
the literature. Recent references can be found in physics [3, 7], in biology [11], and in image
processing [46].
We consider in this paper the doubly critical situation p+ = p∗ , where p+ = max (p1 , . . . , pn )
is the maximum value of the anisotropic configuration and p∗ is the critical Sobolev exponent
−
→
for the embeddings of the anisotropic Sobolev space D1, p (Ω) into Lebesgue spaces. In this
setting, not only the nonlinearity has critical growth, but the operator itself has critical growth
in particular directions of the Euclidean space. As a remark, the notion of critical direction
is a pure anisotropic notion which does not exist when dealing with the Laplace operator or
the p-Laplace operator. Given i = 1, . . . , n, the i-th direction is said to be critical if pi = p∗ ,
resp. subcritical if pi < p∗ . Critical directions induce a failure in the rescaling invariance rule
associated with (1.1).
Date: July 16, 2010.
Published in Advances in Differential Equations 16 (2011), no. 1/2, 61–83.
1
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
2
P
Pn
−
Given an anisotropic configuration →
p satisfying ni=1 1/pi > 1 and pj ≤ n/
i=1
for all j = 1, . . . , n, the critical Sobolev exponent is equal to
n
p ∗ = Pn 1
.
i=1 pi − 1
1
pi
−1
(1.3)
In this paper, we consider weak solutions of problem (1.1). We say that a function u in
−
→
D1, p (Ω) is a weak solution of problem (1.1) if there holds
Z
n Z X
∂u pi −2 ∂u ∂ϕ
∗
dx =
|u|p −2 uϕdx
∂xi ∂xi ∂xi
Ω
i=1 Ω
for all smooth functions ϕ with compact support in Ω.
In this paper, we prove an existence result and a regularity result for problem (1.1). The
regularity result, stated in Theorem 1.2 below, is established on arbitrary domains (bounded or
not), and is motivated in particular by a question posed by Fragalà–Gazzola–Kawohl [24, Section 8.3, Problem 1]. The existence result, stated in Theorem 1.1 below, is established on
cylindric domains. Problem (1.1) on cylindric domains is involved in the description of the asymptotic behavior of Palais–Smale sequences for critical anisotropic problems (see Vétois [44]).
The rescaling phenomenon is described in Section 3. Our existence result states as follows.
P
−
Theorem 1.1. Let n ≥ 3, 1 ≤ n < n, and →
p = (p , . . . , p ), and assume that n 1/p > 1,
+
1
n
i
i=1
p+ = p∗ , pn−n+ +1 = · · · = pn = p+ , and pi < p+ for all i ≤ n − n+ . Let V be a nonempty,
bounded, open subset of Rn+ , and assume that Ω = Rn−n+ × V . Then there exists a positive
real number λ such that problem (1.1) admits at least one nonnegative, nontrivial solution.
Theorem 1.1 is concerned with cylindric domains. Theorem 1.2 below holds true for arbitrary domains Ω, including Ω bounded. This result, which answers the question of the
regularity associated to (1.1), is stated as follows.
P
−
Theorem 1.2. Let n ≥ 3 and →
p = (p , . . . , p ), and assume that n 1/p > 1 and p = p∗ .
1
n
i=1
i
+
Let Ω be a nonempty, open subset of Rn , and λ be a positive real number. Then any solution
of problem (1.1) belongs to L∞ (Ω).
Theorem 1.2 is established on arbitrary domains. In case of bounded domains Ω, Theorem 1.2 answers a question posed by Fragalà–Gazzola–Kawohl [24, Section 8.3, Problem 1].
The boundedness of nonnegative weak solutions of problem (1.1) was established in case
p+ < p∗ by Fragalà–Gazzola–Kawohl [24]. It was suggested in [24] that the result should
remain true in case p+ ≥ p∗ for solutions of the problem
(
p+ −1
→
− ∆−
in Ω ,
p u = λu
(1.4)
−
→
u ∈ D1, p (Ω) ∩ Lp+ (Ω) .
Theorem 1.2 answers positively to this question in case p+ = p∗ . On the other hand, we point
toward a negative answer when p+ > p∗ . More precisely, we prove (by using Proposition 2.1,
−
see Section 2) that for particular anisotropic configurations →
p satisfying p+ > p∗ , for instance
when p1 = · · · = pn− = 2 and pn− +1 = · · · = pn = p+ with p+ > 2∗ , 2∗ = 2n− / (n− − 2), and
2 < n− < n, if we assume the existence of nonnegative, unbounded solutions of the isotropic,
supercritical problem
(
− ∆u = up+ −1 in Ω 0 ,
u ∈ D1,2 (Ω 0 ) ∩ Lp+ (Ω 0 ) ,
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
3
for some domain Ω 0 in Rn− , where ∆ = div (∇u) is the classical Laplace operator, then the
anisotropic problem (1.4) with Ω = Ω 0 × Ω 00 admits nonnegative, unbounded solutions for all
domains Ω 00 in Rn−n− , including Ω 00 bounded. As is well-known, problems with supercritical
growth may admit unbounded solutions (see, for instance, Benguria–Dolbeault–Esteban [8],
Farina [22], and also Fragalà–Gazzola–Kawohl [24]).
In case p+ < p∗ , namely when all directions are subcritical, anisotropic equations with critical nonlinearities have been investigated by Alves–El Hamidi [2], El Hamidi–Rakotoson [19,
20], El Hamidi–Vétois [21], Fragalà–Gazzola–Kawohl [24], Fragalà–Gazzola–Lieberman [25],
and Vétois [43]. Other recent references on anisotropic problems like (1.1) are Antontsev–
Shmarev [4, 5], Bendahmane–Karlsen [9, 10], Bendahmane–Langlais–Saad [11], Cianchi [13],
D’Ambrosio [14], Di Castro [17], Di Castro–Montefusco [18], Garcı́a-Melián–Rossi–Sabina de Lis
[27], Li [30], Lieberman [31, 32], Mihăilescu–Pucci–Rădulescu [35], Mihăilescu–Rădulescu–
Tersian [36], Namlyeyeva–Shishkov–Skrypnik [37], Skrypnik [39], Tersenov–Tersenov [40], and
Vétois [42, 44, 45].
In the isotropic configuration where pi = p for all i = 1, . . . , n, there holds p < p∗ and all
directions are subcritical. In this particular situation, the operator (1.2)
is comparable, though
p−2
slightly different, to the p-Laplace operator ∆p = div |∇u| ∇u . Possible references on
critical p-Laplace equations are Alves–Ding [1], Arioli–Gazzola [6], Demengel–Hebey [15, 16],
Filippucci–Pucci–Robert [23], Gazzola [28], and Guedda–Veron [29]. Needless to say, the above
list does not pretend to exhaustivity.
We illustrate our results with examples in Section 2, we prove Theorem 1.1 in Section 3,
and we prove Theorem 1.2 in Section 4.
2. Examples of solutions
In this section, we are concerned with the situation where the anisotropic configuration
→
−
p consists in two distinct exponents p− and p+ . In other words, we assume that there
exist two indices n− ≥ 2 and n+ ≥ 1 such that n = n− + n+ , p1 = · · · = pn− = p− , and
pn− +1 = · · · = pn = p+ . Proposition 2.1 below is the basic tool in our construction. It relies
on a direct computation.
−
Proposition 2.1. Let n− ≥ 2, n+ ≥ 1, n = n− + n+ , and →
p = (p1 , . . . , pn ), and assume that
p1 = · · · = pn− = p− and pn− +1 = · · · = pn = p+ . Let λ be a positive real number. Let Ω1 be a
nonempty open subset of Rn− and Ω2 be a nonempty open subset of Rn+ . Let v be a solution
of the problem
!

n−
X
∂v p− −2 ∂v
∂

−
= |v|p+ −2 v in Ω1 ,
∂x
∂x
∂x
(2.1)
i
i
i
i=1


v ∈ D1,p− (Ω1 ) ∩ Lp+ (Ω1 ) ,
and let w be a solution of the problem
!

n+
X
∂w p+ −2 ∂w
∂

−
= |w|p+ −2 w − |w|p− −2 w in Ω2 ,
∂x
∂x
∂x
i
i
i
i=1


1,p+
p−
w∈D
(Ω2 ) ∩ L (Ω2 ) .
(2.2)
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
4
−1
Then the function u defined on Ω1 × λ p+ Ω2 by
u (x1 , . . . , xn ) = λ
−1
p+ −p−
1
p1
p+
+
v x1 , . . . , xn− w λ xn− +1 , . . . , λ xn
(2.3)
is a solution of the problem

−1
p+ −2
 − ∆−
→
u in Ω1 × λ p+ Ω2 ,
p u = λ |u|
−1
−1
→
 u ∈ D1,−
p
Ω1 × λ p+ Ω2 ∩ Lp+ Ω1 × λ p+ Ω2 ,
(2.4)
→
where ∆−
p is as in (1.2).
Proof. A direct computation provides the result.
If p+ = p∗ , then a solution of equation (2.1) is given by
! n−p−p−
−
1
Vn− ,p− x1 , . . . , xn− = Cn− ,p−
1+
Pn−
i=1
|xi |
,
p−
p− −1
(2.5)
where
p− −1
Cn− ,p− =
n− (n− − p− )
(p− − 1)p− −1
−
! n− −p
2
p−
.
On the other hand, we search for solutions of equation (2.2) of the form
w xn− +1 , . . . , xn = W (r)
n
X
with r =
|xi |
p+
p+ −1
! p+p −1
+
.
i=n− +1
As one can check, equation (2.2) then rewrites as
0
p −2
−r1−n+ rn+ −1 |W 0 | + W 0 = |W|p+ −2 W − |W|p− −2 W
in R+ .
(2.6)
In case n+ = 1, the unique nonnegative, nontrivial C 1 -solution of (2.6) is given by
(
F −1 (F (W0 ) − r) if r < F (W0 ) ,
W (r) =
0
if r ≥ F (W0 ) ,
where
W0 =
p+
p−
1
+ −p−
p
and F (t) =
p+ − 1
p+
p1 Z t +
0
sp −
sp +
−
p−
p+
− p1
+
ds .
In particular, there hold W (0) = W0 , W 0 (0) = 0, W > 0 and W 0 < 0 in (0, F (W0 )), and
W = 0 in [F (W0 ) , +∞). In case n+ ≥ 2, by Franchi–Lanconelli–Serrin [26], we get that
equation (2.6) admits at least one nonnegative C 1 -solution which satisfies W 0 (0) = 0, W > 0
and W 0 < 0 in (0, R), and W = 0 in [R, +∞) for some positive real number R. Summarizing,
we can state the following corollary of Proposition 2.1.
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
5
−
Corollary 2.1. Let n− ≥ 2, n+ ≥ 1, n = n− + n+ , and →
p = (p1 , . . . , pn ), and assume that
p1 = · · · = pn− = p− , pn− +1 = · · · = pn = p+ , and p+ = p∗ . For any point a = (a1 , . . . , an )
in Rn and for any positive real numbers µ and λ, there exists a nonnegative solution Ua,µ,λ in
−
→
D1, p (Rn ) ∩ C 1 (Rn ) of equation (2.4) of the form
p− −p+
p− −p+
−1
−1 p+ −p−
Ua,µ,λ (x1 , . . . , xn ) = µ λ
U µ p− (x1 − a1 ) , . . . , µ p− xn− − an− ,
1
1
λ p+ xn− +1 − an− +1 , . . . , λ p+ (xn − an ) ,
where

U (x1 , . . . , xn ) = Vn− ,p− x1 , . . . , xn−
W
n
X
|xi |
p+
p+ −1
! p+p −1 
+
,
i=n− +1
where Vn− ,p− is as in (2.5) and where W is such that W > 0 and W 0 < 0 in (0, R), and W = 0
in [R, +∞) for some positive real number R.
Since the function W has compact support, Corollary 2.1 provides a class of solutions of
problem (1.1) on cylindric domains Ω = Rn− × V for all nonempty, open subsets V of Rn+ .
These solutions illustrate the general existence result stated in Theorem 1.1 in the particular
−
p consists in two distinct exponents p− and p+ .
case where the anisotropic configuration →
In the supercritical case p+ > p∗ , suppose there exists a nonnegative, unbounded solution
of problem (2.1) for some domain Ω1 in Rn− . Then we easily get with Proposition 2.1 that
problem (1.4) with Ω = Ω1 × Ω2 admits nonnegative, unbounded solutions for all domains
Ω2 in Rn+ , including Ω2 bounded. Indeed, since the above function W has compact support,
by rescaling W, we get a nonnegative solution of the problem (2.2) on the domain Ω2 . Then
Proposition 2.1 provides the existence of a nonnegative, unbounded solution of the form (2.3)
of the problem (1.4) with Ω = Ω1 × Ω2 .
3. The existence result
−
This section
of Theorem 1.1. We let n ≥ 3 and →
p = (p1 , . . . , pn ). We
Pisn devoted to the proof
∗
assume that i=1 1/pi > 1, p+ = p , and that there exists an index n+ such that pn−n+ +1 =
· · · = pn = p+ , and pi < p+ for all i ≤ n − n+ . Moreover, we assume that Ω = Rn−n+ × V ,
where V is a nonempty, bounded, open subset of Rn+ . Without loss of generality, we may
assume that the point 0 belongs to V .
The proof of Theorem 1.1 is based on concentration-compactness arguments. Let us first
−
→
set some notations. For any function u in D1, p (Rn ) and any subset D of Rn , we let the energy
E (u, D) of u on D be defined by
Z
E (u, D) =
up+ dx .
(3.1)
D
For any positive real number µ and any point a = (a1 , . . . , an ) in Rn , we define the affine
−
→
p
transformation τµ,a
: Rn → Rn by
p1 −p+
pn −p+
−
→
p
p1
p
n
τµ,a (x1 , . . . , xn ) = µ
(x1 − a1 ) , . . . , µ
(xn − an ) .
(3.2)
As is easily checked, (3.2) provides a general rescaling invariance rule associated with equation
−
→
−
→
p −1
p
(1.1). Moreover for any subset D of Rn , we get E (u, D) = E µu ◦ τµ,a
, τµ,a
(D) , where
p+ −p1
p+ −pn
−
→
p −1
p1
p
n
τµ,a
(x1 , . . . , xn ) = µ
x 1 + a1 , . . . , µ
x n + an .
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
6
Of importance in our critical setting is that the set D is only rescaled with respect to noncritical
directions. Therefore, we observe a concentration phenomenon on affine subspaces of Rn
spanned by critical directions. Figure 1 below illustrates the rescaling in case D is a threedimensional ball, the first two directions being noncritical, the third one being critical. In case
of the p-Laplace operator, the ball would have been rescaled to the whole euclidean space.
Figure 1. Rescaling of a ball (n = 3, p1 = p2 = 1.5, p3 = 6). The first line
describes the scale in the rescaling. The second line describes the deformation
of the domain.
−
→
We begin the proof of Theorem 1.1. We let (uα )α be a sequence of functions in D1, p (Ω)
such that
Z
Z Z n
n
X
X
∂uα pi
∂u pi
1
1
p+
|uα | dx = 1 and
lim
∂xi dx = u∈Dinf
∂xi dx . (3.3)
−
1,→
p (Ω)
α→+∞
p
p
i
i
Ω
Ω
Ω
R
i=1
i=1
Ω |u|
p+
dx=1
Taking the absolute value, we may assume that for any α, the function uα is nonnegative.
−
→
Clearly, the sequence (uα )α is bounded in D1, p (Ω).
Step 3.1 below is the first step in the proof of Theorem 1.1. We say that a sequence (vα )α
−
→
in D1, p (Ω) is Palais–Smale for the functional Iλ defined in (3.4) if there hold |Iλ (vα )| ≤ C
−
for some positive constant C independent of α, and kDIλ (vα )kD1,→
p (Ω)0 → 0 as α → +∞.
Step 3.1. Up to a subsequence, (uα )α is a Palais–Smale sequence for the functional
Z Z
n
X
∂u pi
1
λ
Iλ (u) =
dx −
|u|p+ dx ,
p
∂xi
p+ Ω
i=1 i Ω
where
n Z X
∂uα pi
λ = lim
∂xi dx .
α→+∞
i=1 Ω
(3.4)
(3.5)
Proof. It easily follows from (3.3) that there holds |Iλ (uα )| ≤ C for some positive contant C
−
→
independent of α. We then prove that for any bounded sequence (ϕα )α in D1, p (Ω), there
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
7
holds DIλ (uα ) .ϕα → 0 as α → +∞. By (3.3), we get that there exists a sequence (εα )α of
positive real numbers converging to 0 such that for any real number t, there holds
 p i

p i
Z Z n
n
X
X
1
uα + tϕα
1
∂uα dx − εα ≤
∂ 

p1 dx
R
∂xi ∂xi
p
+
p
p
i
i
Ω
Ω
|u + tϕ | dx + i=1
i=1
Ω
=
n
X
i=1
1
pi
α
α
p
− ppi Z ∂uα
+
∂ϕα i
p+
|uα + tϕα | dx
+t
dx .
∂xi Ω
Ω ∂xi
Z
(3.6)
As is easily checked, there exists a positive real number C such that for any i = 1, . . . , n and
for any real numbers x and y, there holds
( pi
|y|
if pi ≤ 2
p
p
p
−2
|x + y| i − |x| i − pi |x| i xy ≤ C
(3.7)
|y|2 |x|pi −2 + |y|pi −2 if pi > 2.
−
→
Since (uα )α and (ϕα )α are bounded in D1, p (Ω), by (3.7) and Hölder’s inequality, we get
Z p
Z Z ∂u
∂uα pi
∂uα pi −2 ∂uα ∂ϕα ∂ϕα i
α
+t
dx
dx −
∂xi dx − pi t
∂xi Ω ∂xi
∂xi ∂xi ∂xi Ω
Ω
p
 Z ∂ϕα i

p
i

if pi ≤ 2

 t Ω ∂xi dx
≤C
pi 2 Z pi pi −2
Z Z 
pi
pi
∂ϕα pi
∂u
∂ϕ

α
α
2
pi

t
+t
∂xi dx
∂xi dx
∂xi dx if pi > 2.
Ω
Ω
( pi Ω
t
if
p
≤
2
i
≤ C0
(3.8)
2
pi −2
t 1+t
if pi > 2.
for all i = 1, . . . , n, and
Z
Z
Z
p
p
−1
p
+
+
+
|uα + tϕα | dx −
uα ϕα dx
uα dx − p+ t
Ω
Ω
Ω
Z



tp+
|ϕα |p+ dx
if p+ ≤ 2


Ω
≤C
p2 Z
Z
p+p −2
Z

+
+

p+
p+
2
p+

t
|uα | dx
|ϕα | dx
+t
|ϕα |p+ dx if p+ > 2.
Ω
Ω
( p+ Ω
t
if p+ ≤ 2
≤ C0
(3.9)
2
p+ −2
t 1+t
if p+ > 2.
for some positive constants C and C 0 independent of α and t. By (3.6), (3.8), (3.9), we get
!Z
!
n Z n Z X
X
∂uα pi −2 ∂uα ∂ϕα
∂uα pi
−εα ≤ t
dx −
upα+ −1 ϕα dx + o (t)
∂xi dx
∂xi ∂x
∂x
i
i
Ω
i=1 Ω
i=1 Ω
!
p i ! Z
Z
n
X
∂uα ≤ t DIλ (uα ) .ϕα + λ −
upα+ −1 ϕα dx + o (t)
∂xi dx
Ω
i=1 Ω
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
8
as t → 0 uniformly with respect to α, where λ is as in (3.5). Passing to the limit as α → +∞,
we get
0 ≤ lim sup (tDIλ (uα ) .ϕα ) + o (t)
α→+∞
as t → 0. Since the real number t takes either positive or negative values, it follows that
DIλ (uα ) .ϕα → 0 as α → +∞. Since this holds true for all bounded sequences (ϕα )α in
−
→
−
D1, p (Ω), we get kDIλ (uα )kD1,→
p (Ω)0 → 0 as α → +∞. This ends the proof of Step 3.1.
Now, for any α, we define the concentration function Qα : R+ → R+ by
−
→
Qα (s) = max E uα , Pyp (s) ,
y∈Ω
where the energy functional E is as in (3.1) and
n
p+ −pi
−
→
Pyp (s) = (x1 , . . . , xn ) ∈ Ω; |xi − yi | < s pi
∀i ∈ {1, . . . , n − n+ }
o
(3.10)
for all positive real number s and for all point y = (y1 , . . . , yn ) in Ω. By the continuity of the
functions Qα and by (3.3), given a real number δ0 in (0, 1), we get the existence of a sequence
(µα )α of positive real numbers such that there holds Qα (µα ) = δ0 for all α. We let xα be a
point in Ω for which Qα (µα ) is reached, so that there holds
−
→
−
→
max E uα , Pyp (µα ) = E uα , Pxpα (µα ) = δ0
(3.11)
y∈Ω
−
→
for all α. By definition of Pxpα (µα ), see (3.10), we may assume that the n+ last coordinates of
the point xα are equal to 0. For any α, we then define the function u
eα by
−1
−
→
,
u
eα = µα uα ◦ τµpα ,xα
−
→
where τµpα ,xα is as in (3.2). Since Ω = Rn−n+ × V , pn−n+ +1 = · · · = pn = p+ , and p+ = p∗ , we
−
→
get τµpα ,xα (Ω) = Ω for all α. As well as (uα )α , we get that (e
uα )α is a Palais–Smale sequence
−
−
for the functional Iλ defined in (3.4). Moreover, there holds ke
uα kD1,→
p (Ω) = kuα kD 1,→
p (Ω) for
→
1,−
p
all α. In particular, the sequence (e
uα )α is bounded in D (Ω). Passing if necessary to a
subsequence, we may assume that (e
uα )α converges weakly to a nonnegative function u∞ in
→
1,−
p
D (Ω) and that (e
uα )α converges to u∞ almost everywhere in Ω. The second step in the
proof of Theorem 1.1 is as follows.
Step 3.2. If the constant δ0 is small enough, then (e
uα )α converges, up to a subsequence, to
p+
(Rn ).
u∞ in Lloc
Proof. We fix a positive real number R, and we let B0 (R) be the (n − n+ )-dimensional ball
of center 0 and radius R. We show that the sequence (e
uα )α converges to u∞ in Lp+ (B0 (R)).
For any α, we let vα = u
eα − u∞ . By Banach–Alaoglu theorem, since the sequence (vα )α is
→
1,−
p
bounded in D (Ω) and since Ω = Rn−n+ × V , where V is bounded, passing if necessary to
a subsequence, we may assume that there exist nonnegative, finite measures µ and ν1 , . . . , νn
on B0 (2R) × Rn+ such that |vα |p+ * µ and |∂vα /∂xi |pi * νi as α → +∞ in the sense of
measures on B0 (2R) × Rn+ , for all i = 1, . . . , n. Moreover, the supports of the measures µ
and ν1 , . . . , νn are included in B0 (2R) × V . Now, we borrow some ideas in Lions [33, 34] with
the tricky difference here that the concentration holds on n+ -dimensional affine subspaces of
Rn . Since p+ = p∗ , by the anisotropic Sobolev inequality in Troisi [41], there exists a positive
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
9
−
constant Λ = Λ (→
p ) such that for any α and any smooth function ϕ with compact support in
n+
B0 (2R) × R , there holds
Z
p+
|vα ϕ|
Ω
p+
np
n Z Y
∂ (vα ϕ) pi
i
dx ≤ Λ
∂xi dx
Ω
i=1
p1 !
∂vα pi
i
dx
ϕ
Ω ∂xi
Z p1 Z
n
Y
∂ϕ pi
i
vα
dx
+
≤Λ
∂xi Ω
i=1
p+
n
.
(3.12)
For i = 1, . . . , n−n+ , by the compact embeddings in Rákosnı́k [38], we get that (vα )α converges
to 0 in Lpi (Supp ϕ). Passing to the limit as α → +∞ into (3.12) gives
Z
n−n+
|ϕ|p+ dµ ≤ Λ
B0 (R)×V
Y Z
|ϕ|pi dνi
i
B0 (R)×V
i=1
n
Y
×
p+
np
Z
B0 (R)×V
i=n−n+ +1
p1
p1 !
Z
∂ϕ p+
+
+
p+
dµ
dν
+
|ϕ|
i
∂xi B0 (R)×V
p+
n
.
By an easy density argument, it follows that for any bounded measurable function ϕ on
B0 (R) × V which does not depend on the variables xn−n+ +1 , . . . , xn , there holds
Z
p+
|ϕ|
dµ ≤ Λ
B0 (R)×V
n Z
Y
i=1
p+
np
pi
|ϕ| dνi
i
.
(3.13)
B0 (R)×V
In particular, for any Borelian set A in B0 (R), taking ϕ = 1A×V , we get
µ A×V ≤Λ
n
Y
νi A × V
p+
np
i
.
(3.14)
i=1
Letting ν =
Pn
i=1
νi , since
Pn
1
i=1 pi
=
n+p+
,
p+
it follows that
n+p+
µ A × V ≤ Λν A × V n .
(3.15)
e (A) = µ A × V and νei (A) =
We let µ
e and νe1 , . . . , νen be the measures defined on B0 (R) by µ
P
νi A × V for all i = 1, . . . , n. We let νe = ni=1 νei . By the Lebesgue decomposition of νe
with respect to µ
e, there exist a nonnegative function f in L1 B0 (R), de
µ and a nonnegative
bounded measure σ on B0 (R) such that there holds νe = f µ
e + σ and such that σ is singular
with respect to µ
e. We may assume in additionthat the function f isidentically zero on the
support of the measure σ. By (3.15), we get µ
e x ∈ B0 (R); f (x) = 0 = 0. For any natural
number β, any real number q ≥ 1, and any Borelian set A in B0 (R), by (3.13) with ϕ = f q 1Aβ ,
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
where Aβ = {x ∈ A ;
Z
10
f (x) ≤ β}, we get
µ≤Λ
f qp+ de
Aβ
f qpi dνei
p+
! np
Z
n
Y
i
f qpi +1 de
µ
Aβ
i=1
n−n+
≤Λ
i
Aβ
i=1
≤Λ
p+
! np
Z
n
Y
p+
! np
i
Z
Y
f qpi +1 de
µ
µ
f qp+ de
β
Aβ
i=1
! nn+
Z
.
Aβ
Choosing q large enough so that q > 1/ (p+ − pi ) for all i = 1, . . . , n − n+ , by Hölder’s
inequality, it follows that
Z
f qp+ de
µ≤β
n+
n
Λν B0 (R) × V
+
p+nqq−1 Pn−n
i=1
n−n
1
− n+
pi
Aβ
Z
n−n+
! nq1 Pi=1
f qp+ de
µ
1
+1
pi
.
Aβ
R
R
µ > Cβ , for some positive constant Cβ
µ = 0 or Aβ f qp+ de
We then get that either Aβ f qp+ de
R
independent of A. It follows that for any β, the measure A → Aβ f qp+ de
µ is a finite linear
combination of Dirac masses. Since µ
e x ∈ B0 (R); f (x) = 0 = 0, it follows that for any
β, the measure A → µ
e (Aβ ) is a finite linear combination of Dirac masses. Passing to the
limit as β → +∞, we
get
that there exists an at most countable index set J of distinct points
j
j
yj = y1 , . . . , yn−n+ in B0 (R), j ∈ J, such that Supp µ
e = {yj ; j ∈ J}. It follows that
[
V yj ,
(3.16)
Supp µ ∩ B0 (R) × V ⊂
j∈J
where
V yj =
j
y1j , . . . , yn−n
+
×V .
(3.17)
We end the proof of Theorem 1.1 by using Palais–Smale properties of the sequence (e
uα )α . For
any smooth function φ with compact support in Ω, we get
pi −2
Z
n Z X
∂e
u
∂e
uα ∂φ
α
dx = λ
u
epα+ −1 φdx + o (1)
(3.18)
∂xi ∂xi ∂xi
Ω
i=1 Ω
as α → +∞. The functions u
epα+ −1 keep bounded in Lp+ /(p+ −1) (Ω) and converge, up to a
subsequence, almost everywhere to up∞+ −1 in Ω as α → +∞. By standard integration theory,
it follows that the functions u
epα+ −1 converge weakly to up∞+ −1 in Lp+ /(p+ −1) (Ω). On the other
hand, for any i = 1, . . . , n, the functions |∂e
uα /∂xi |pi −2 ∂e
uα /∂xi keep bounded in Lpi /(pi −1) (Ω),
and thus converge, up to a subsequence, weakly to a function ψi in Lpi /(pi −1) (Ω) as α → +∞.
Passing to the limit into (3.18) as α → +∞, we get
Z
n Z
X
∂φ
dx = λ
up∞+ −1 φdx .
(3.19)
ψi
∂x
i
Ω
Ω
i=1
−
→
By an easy density argument, (3.19) holds true for all functions φ in D1, p (Ω). Now, we let ϕ
be a nonnegative, smooth function with support in B0 (2R) × Rn+ . Since the sequence (e
uα )α
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
11
is Palais–Smale for the functional Iλ , we get
!
p i
pi −2
Z
Z Z n
X
∂e
u
∂e
u
∂e
u
∂ϕ
α
α
α
uα ) . (e
uα ϕ)
u
eα
dx = λ
u
epα+ ϕdx + DIλ (e
∂xi ϕdx +
∂xi ∂x
∂x
i
i
Ω
Ω
Ω
i=1
Z
(3.20)
u
epα+ ϕdx + o (1)
≤λ
Ω
as α → +∞. For any i = 1, . . . , n − n+ , by the compact embeddings in Rákosnı́k [38], we get
that the sequence (e
uα )α converges to u∞ in Lpi (Supp ϕ), and thus that
pi −2
Z Z
∂e
∂e
uα ∂ϕ
∂ϕ
uα u
eα
dx −→
ψi u∞
dx
(3.21)
∂xi ∂xi ∂xi
∂xi
Ω
Ω
as α → +∞. For any α and any i = n − n+ + 1, . . . , n, we get
Z p+ −2
p+ −1
∂e
∂ϕ ∂e
uα ∂ϕ ∂e
u
uα α
u
eα
dx ≤ ke
uα kLp+ (Ω) ∂xi ∞ n .
Ω ∂xi ∂xi ∂xi ∂xi Lp+ (Ω)
L (R )
(3.22)
Since the sequence (e
uα )α is bounded in Lp+ (Ω) and converges to u∞ almost everywhere in Ω,
by Brezis–Lieb [12], we get
Z
Z
Z
p+
p+
u
eα ϕdx −→
u∞ ϕdx +
ϕdµ .
(3.23)
Ω
Ω
pi
B0 (2R)×V
pi
Since there holds |∂e
uα /∂xi | ≥ |∂vα /∂xi | − |∂u∞ /∂xi |pi , where vα = u
eα − u∞ for all α and
i = 1, . . . , n, we get
p
Z Z Z
∂u∞ pi
∂e
uα i
ϕdνi −
lim inf
ϕdx ≥
(3.24)
∂xi ϕdx
α→+∞ Ω ∂xi Ω
B0 (2R)×V
as α → +∞. By (3.21), (3.22), (3.23), and (3.24), passing to the limit into (3.20) as α → +∞,
we get
n−n+ Z
n Z n Z
X
X
X
∂u∞ pi
∂ϕ
ψi u∞
ϕdνi −
ϕdx
+
dx
∂xi ∂xi
Ω
i=1 Ω
i=1
i=1 B0 (2R)×V
Z
Z
n
X
∂ϕ p+
u∞ ϕdx +
≤λ
(3.25)
ϕdµ + C
∂xi ∞ n
Ω
B0 (2R)×V
L
(R
)
i=n−n +1
+
for some positive constant C independent of ϕ. Increasing if necessary the constant C, it
follows from (3.19) and (3.25) that
n Z
n Z n Z
X
X
X
∂u∞ pi
∂u∞
ϕdx −
ϕdνi −
ψ
ϕdx
i
∂xi ∂x
i
B
(2R)×V
Ω
Ω
0
i=1
i=1
i=1
Z
n
X
∂ϕ ≤λ
ϕdµ + C
(3.26)
∂xi ∞ n .
B0 (2R)×V
L
(R
)
i=n−n +1
+
n−n+
We let η be a smooth cutoff function on R
such that η = 1 in B0 (1), 0 ≤ η ≤ 1 in
B0 (2) \B0 (1), and η = 0 in Rn−n+ \B0 (2). For any point y = y1 , . . . , yn−n+ in B0 (R) and
for any positive real number ε, we let ϕε,y be the function defined on Rn by
1
1
ϕε,y (x1 , . . . , xn ) = η
(x1 − y1 ) , . . . , xn−n+ − yn−n+ .
ε
ε
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
12
Plugging ϕ = ϕε,y into (3.26), and passing to the limit as ε → 0, we get
n
X
νi V y ≤ λµ V y ,
(3.27)
i=1
where V y is as in (3.17). By (3.14) and (3.27), we get that there holds either
−n
p+
µ V y = 0 or λµ V y n+p+ ≥ Λ n+p+
(3.28)
for all points y in Rn−n+ . On the other hand, by (3.11) and by an easy change of variable, for
any α, we get
−
→
E u
eα , Pyp (1) ≤ δ0 ,
(3.29)
−
→
where the energy functional E is as in (3.1) and Pyp (1) is as in (3.10). By (3.23) and since
−
→
there holds V y ⊂ Pyp (1), passing to the limit into (3.29) as α → +∞, it follows that
−
→
(3.30)
E u∞ , Pyp (1) ≤ δ0 .
−
n
−
n+p+
Choosing δ0 small enough so that δ0 < Λ p+ λ p+ , it follows from (3.28) and (3.30) that
there holds µ V y = 0 for all points y in B0 (R). By (3.16), we then get that the measure µ
is identically zero on B0 (R). It follows that |vα |p+ * 0 as α → +∞, where vα = u
eα − u∞ ,
p+
and thus that the sequence (e
uα )α converges to u∞ in Lloc
(B0 (R)). This ends the proof of
Step 3.2.
The next step in the proof of Theorem 1.1 is as follows.
Step 3.3. If the constant δ0 is small enough, then ∇e
uα converges, up to a subsequence, to
∇u∞ almost everywhere in Ω.
Proof. We let ϕ be a smooth function with compact support in Rn . Since the sequence (e
uα )α
is Palais–Smale for the functional Iλ , there holds DIλ (e
uα ) . ((e
uα − u∞ ) ϕ) → 0 as α → +∞,
and thus
pi −2
pi −2
n Z n Z X
X
∂e
∂e
u
∂e
u
∂e
u
∂u
u
∂e
uα
∂ϕ
α
α
α
∞
α
−
ϕdx +
(e
uα − u∞ )
dx
∂xi ∂x
∂x
∂x
∂x
∂x
∂x
i
i
i
i
i
i
Ω
Ω
i=1
i=1
Z
−λ
u
epα+ −1 (e
uα − u∞ ) ϕdx −→ 0 (3.31)
Ω
as α → +∞. By Hölder’s inequality and by Step 3.2, we get
Z
p −1
p
−1
+
u
(e
uα − u∞ ) ϕdx ≤ kϕkL∞ (Ω) ke
uα kL+p+ (Ω) ke
uα − u∞ kLp+ (Supp ϕ) −→ 0
eα
(3.32)
Ω
and
Z pi −2
∂e
∂e
uα
∂ϕ uα (e
uα − u∞ )
dx
Ω ∂xi ∂xi
∂xi pi −1
p+ −pi ∂e
∂ϕ u
α
p+ pi ≤ |Supp ϕ|
ke
uα − u∞ kLp+ (Supp ϕ) −→ 0
∂xi ∞ ∂xi p
L (Ω)
L i (Ω)
as α → +∞ for all i = 1, . . . , n. By (3.31), (3.32), and (3.33), we get
pi −2
n Z X
∂e
∂e
uα ∂e
uα ∂u∞
uα −
ϕdx −→ 0
∂xi ∂xi ∂xi
∂xi
i=1 Ω
(3.33)
(3.34)
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
13
as α → +∞. On the other hand, since the sequence (e
uα )α converges weakly to the function
→
1,−
p
u∞ in D (Ω), we get
Z Z ∂u∞ pi −2 ∂u∞ ∂e
∂u∞ pi
u
α
ϕdx
ϕdx −→
(3.35)
∂xi ∂xi
Ω ∂xi
Ω ∂xi
as α → +∞ for all i = 1, . . . , n. By (3.34) and (3.35), we get
!
pi −2
n Z
X
∂u∞ pi −2 ∂u∞
∂e
u
∂e
u
∂e
u
∂u
α
α
α
∞
− −
ϕdx −→ 0
∂xi ∂x
∂x
∂xi
∂xi
∂xi
i
i
Ω
i=1
(3.36)
as α → +∞. Since (3.36) holds true for all smooth functions ϕ with compact support in Rn ,
we then get that for any i = 1, . . . , n and any bounded domain Ω 0 of Rn , there holds
!
pi −2
Z
∂u∞ pi −2 ∂u∞
∂e
u
∂e
u
∂e
uα ∂u∞
α
α
−
−
dx −→ 0
∂xi ∂xi ∂xi ∂xi
∂xi
∂xi
Ω0
as α → +∞. In particular, up to a subsequence, there holds
!
pi −2
pi −2
∂e
u
∂e
u
∂u
∂u
∂e
u
∂u
α
α
∞
∞
α
∞
−
−
−→ 0 a.e. in Ω
∂xi ∂xi ∂xi ∂xi
∂xi
∂xi
as α → +∞. It easily follows that the functions ∂e
uα /∂xi converge, up to a subsequence,
almost everywhere to ∂u∞ /∂xi in Ω as α → +∞. This ends the proof of Step 3.3.
The final step in the proof of Theorem 1.1 is as follows.
Step 3.4. The function u∞ is a nontrivial, nonnegative solution of the problem (1.1).
Proof. We let ϕ be a smooth function with compact support in Ω. Since the sequence (e
uα )α
is Palais–Smale for the functional Iλ , we get
pi −2
Z
n Z X
∂e
u
∂e
uα ∂ϕ
α
u
epα+ −1 ϕdx −→ 0
dx − λ
(3.37)
∂xi ∂xi ∂xi
i=1
Ω
Ω
as α → +∞. By Step 3.3 (resp. Step 3.2), the functions ∂e
uα /∂xi (resp. u
eα ) converge almost
everywhere to ∂u∞ /∂xi (resp. u∞ ) in Ω as α → +∞. Moreover, |∂e
uα /∂xi |pi −2 ∂e
uα /∂xi (resp.
p+ −1
pi /(pi −1)
p+ /(p+ −1)
u
eα ) keep bounded in L
(Ω) (resp. L
(Ω)). By standard integration theory,
it follows that
Z
Z
u
epα+ −1 ϕdx −→
up∞+ −1 ϕdx
(3.38)
Ω
Ω
and
pi −2
Z Z ∂u∞ pi −2 ∂u∞ ∂ϕ
∂e
∂e
uα ∂ϕ
uα dx −→
dx
(3.39)
∂xi ∂x
∂x
∂x
∂xi ∂xi
i
i
i
Ω
Ω
as α → +∞ for all i = 1, . . . , n. By (3.37), (3.38), and (3.39), we get that u∞ is a solution of
problem (1.1). Moreover, u∞ is nonnegative since the functions u
eα are nonnegative. We finally
claim that u∞ is not identically zero. Indeed, by (3.11) and by an easy change of variable, for
any α, we get
−
→
E u
eα , P0p (1) = δ0 ,
(3.40)
−
→
where the energy functional E is as in (3.1) and P0p (1) is as in (3.10). By Step 3.2, passing
to the limit into (3.40) as α → +∞, we get
−
→
E u∞ , P0p (1) = δ0 .
In particular, u∞ is not identically zero. This ends the proof of Step 3.4.
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
14
Step 3.4 ends the proof of Theorem 1.1
4. The regularity result
In this section, we prove Theorem 1.2.
Proof of Theorem 1.2. Without loss of generality, we may assume that there exists an index
n+ such that pn−n+ +1 = · · · = pn = p+ and pi < p+ for all i ≤ n − n+ . We let u be a solution
of problem (1.1). We begin with proving that u belongs to Lq (Ω) for all real numbers q > p+ .
We let
q−p+ ϕα = min |u| p+ , α
p
for all positive real numbers α. For any j = 1, . . . , n, multiplying equation (1.1) by uϕαj and
integrating by parts on Ω, since u = 0 on ∂Ω, we get
Z
n Z X
∂u pi p
j
|u|p+ ϕpαj dx .
(4.1)
∂xi ϕα dx ≤ λ
i=1
Ω
Ω
Moreover, for any positive real number β, we get
Z
Z
Z
(q−p+ )pj
p+ pj
p+
p+
|u| ϕα dx ≤ β
|u| dx +
Ω
Ω
|u|p+ ϕpαj dx ,
(4.2)
Wβ
where
Wβ = {x ∈ Ω ;
|u (x)| > β} .
(4.3)
By Hölder’s inequality, we get
Z
p+
|u|
ϕpαj dx ≤
Z
p+
|u|
! p+p−pj Z
+
|u|
dx
ϕpα+ dx
ppj
+
.
(4.4)
Ω
Wβ
Wβ
p+
Since p+ = p∗ , by the anisotropic Sobolev inequality in Troisi [41], we get
p+
p i
np
Z
n Z Y
i
∂u
pi
|u|p+ ϕpα+ dx ≤ Λ
∂xi ϕα dx
Ω
Ω
i=1
(4.5)
for some positive constant Λ independent of α and u. By Young’s inequality, it follows that
for any ε > 0, there holds
p i
pp+
−n n−n
Z
+ Z
X
+
i
∂u
Λ
ϕpαi dx
|u|p+ ϕpα+ dx ≤
ε n−n+
n
Ω
Ω ∂xi
i=1
Z n
X
∂u p+ p
+
+ε
(4.6)
∂xi ϕα dx .
i=n−n +1 Ω
+
By (4.1)–(4.6), we get
! p+p−pj
ppj Z
Z Z
+
(q−p+ )pj
∂u pj p
Λ +
p+
p+
ϕαj dx ≤ λβ p+
|u|
dx
+
λ
|u|
dx
n
Ω ∂xj
Ω
Wβ


 ppj 
pj
p i
p+
+
Z
Z
n−n+ −n
n
X
pi
+
∂u p
∂u X
p
n−n+
+
i
 
×
ε
+ ε
 . (4.7)
∂xi ϕα dx
∂xi ϕα dx
i=1
Ω
i=n−n+ +1
Ω
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
15
Choosing ε small enough so that ε < n/ (λΛ), it follows that
n
X
i=n−n+
Z q−p+ Z
∂u p+ p
ϕα+ dx ≤ nλβ
|u|p+ dx
∂xi n
−
λΛε
Ω
+1 Ω
pp+
Z n−n+ −n
X
+
∂u pi p
i
λΛ
ϕαi dx
ε n−n+
+
.
n − λΛε i=1
Ω ∂xi
(4.8)
It follows from (4.7) with ε = 1 and (4.8) with ε < n/ (λΛ) that
p1
p1
n−n+ Z
q−p+ X
X Z ∂u pi
i
i
p
p
+
p
i
≤C β +
|u| dx
∂xi ϕα dx
Ω
Ω
i=1
i=1
!
p
−p
Z
p1
p+ p i
n−n+ Z
n−n+ Z
q−p+
X
X
+ i
+
p+
p+
p+
+
β
+
|u| dx
|u| dx
n−n+
i=1
Wβ
Ω
i=1
(4.9)
p1 ! !
∂u pi p
i
i
∂xi ϕα dx
Ω
for some positive constant C independent of α, β, and u. Since the function u belongs to
Lp+ (Ω), increasing if necessary the constant C, it follows from (4.8) and (4.9) that for β
large, there holds
p1
n Z q−p+
X
∂u pi p
i
p+
ϕαi dx
≤
Cβ
,
(4.10)
∂xi Ω
i=1
where C is independent of α, β, and u. Passing to the limit into (4.10) as α → +∞, we get
p1
n Z X
∂ q p i
i
p+
dx
|u|
< +∞ .
Ω ∂xi
i=1
−
→
q
By the continuity of the embedding of D1, p (Ω) into Lp+ (Ω), it follows that |u| p+ belongs to
Lp+ (Ω), and thus that u belongs to Lq (Ω) for all real numbers q > p+ . Now, we prove that
u belongs to L∞ (Ω). For any positive real number t, we define the function ϕt : R → R by


 s + t if s ≤ −t ,
if − t < s < t ,
ϕt (s) = 0

 s − t if s ≥ t .
Multiplying equation (1.1) by ϕt (u) and integrating by parts on Ω, since u = 0 on ∂Ω, we get
n Z
X
i=1
Z
∂u pi
dx = λ
|u|p+ −2 uϕt (u) dx ,
∂xi Wt
Wt
where Wt is as in (4.3). For any real number q > p+ , by Hölder’s inequality, it follows that
p1
Z
p+q−1 Z
(p+ −1)(q−p+ )
∂u pi
+
q
p+
p+ q
dx ≤ λ |Wt |
|u| dx
|ϕt (u)| dx
.
∂xi Wt
Wt
Wt
n Z
X
i=1
(4.11)
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
Since p+ = p∗ , by the anisotropic Sobolev inequality in
we get
n
Z
n+p
n Z
Y
+
p+
|ϕt (u)| dx
≤Λ
Troisi [41], and by Young’s inequality,
p+
∂u pi
(n+p+ )pi
∂xi dx
Wt
i=1
Z n
∂u pi
p+ Λ X 1
dx
≤
n + p+ i=1 pi Wt ∂xi Wt
16
(4.12)
for some positive constant Λ independent of t and u. Since the function u belongs to Lq (Ω),
it follows from (4.11) and (4.12) that
Z
p1
(n+p+ )(p+ −1)(q−p+ )
+
p+
|ϕt (u)| dx
≤ C |Wt | (np+ −n−p+ )p+ q
Wt
for some positive constant C independent of t and u. By Fubini’s theorem and Hölder’s
inequality, we then get
Z +∞ Z
Z
Z +∞
(p+ −1)(nq−n−p+ )
|Ws | ds =
1Ws dxds =
|ϕt (u)| dx ≤ C |Wt | (np+ −n−p+ )q .
t
t
Wt
Wt
Choosing q large enough so that
(p+ − 1) (nq − n − p+ )
> 1,
(np+ − n − p+ ) q
it easily follows that there holds |Wt | = 0 for t large, and thus that u belongs to L∞ (Ω).
Acknowledgments: The author was partially supported by the ANR grant ANR-08-BLAN0335-01. The author is very grateful to Emmanuel Hebey for many helpful remarks and
suggestions during the preparation of the manuscript. The author is also very grateful to
Alberto Farina for interesting discussions about equations with supercritical growth.
References
[1]
[2]
[3]
[4]
[5]
[6]
[7]
[8]
[9]
[10]
C. O. Alves and Y. H. Ding, Multiplicity of positive solutions to a p-Laplacian equation involving critical
nonlinearity, J. Math. Anal. Appl. 279 (2003), no. 2, 508–521.
C. O. Alves and A. El Hamidi, Existence of solution for a anisotropic equation with critical exponent,
Differential Integral Equations 21 (2008), no. 1, 25–40.
S. Antontsev, J. I. Dı́az, and S. Shmarev, Energy methods for free boundary problems: Applications to
nonlinear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications,
vol. 48, Birkhäuser, Boston, 2002.
S. Antontsev and S. Shmarev, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal. 65 (2006), no. 4, 728–761.
, Elliptic equations with anisotropic nonlinearity and nonstandard growth conditions, Handbook of
Differential Equations: Stationary Partial Differential Equations, vol. 3, Elsevier, Amsterdam, 2006.
G. Arioli and F. Gazzola, Some results on p-Laplace equations with a critical growth term, Differential
Integral Equations 11 (1998), no. 2, 311–326.
J. Bear, Dynamics of Fluids in Porous Media, American Elsevier, New York, 1972.
R. D. Benguria, J. Dolbeault, and M. J. Esteban, Classification of the solutions of semilinear elliptic
problems in a ball, J. Differential Equations 167 (2000), no. 2, 438–466.
M. Bendahmane and K. H. Karlsen, Renormalized solutions of an anisotropic reaction-diffusion-advection
system with L1 data, Commun. Pure Appl. Anal. 5 (2006), no. 4, 733–762.
, Nonlinear anisotropic elliptic and parabolic equations in RN with advection and lower order terms
and locally integrable data, Potential Anal. 22 (2005), no. 3, 207–227.
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
17
[11] M. Bendahmane, M. Langlais, and M. Saad, On some anisotropic reaction-diffusion systems with L1 -data
modeling the propagation of an epidemic disease, Nonlinear Anal. 54 (2003), no. 4, 617–636.
[12] H. Brézis and E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983), no. 3, 486–490.
[13] A. Cianchi, Symmetrization in anisotropic elliptic problems, Comm. Partial Differential Equations 32
(2007), no. 4-6, 693–717.
[14] L. D’Ambrosio, Liouville theorems for anisotropic quasilinear inequalities, Nonlinear Anal. 70 (2009),
no. 8, 2855–2869.
[15] F. Demengel and E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev
growth, Adv. Differential Equations 3 (1998), no. 4, 533–574.
, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth and pertur[16]
bation terms, Appl. Anal. 72 (1999), no. 1–2, 75–109.
[17] A. Di Castro, Existence and regularity results for anisotropic elliptic problems, Adv. Nonlin. Stud. 9
(2009), 367–393.
[18] A. Di Castro and E. Montefusco, Nonlinear eigenvalues for anisotropic quasilinear degenerate elliptic
equations, Nonlinear Anal. 70 (2009), no. 11, 4093–4105.
[19] A. El Hamidi and J.-M. Rakotoson, On a perturbed anisotropic equation with a critical exponent, Ricerche
Mat. 55 (2006), no. 1, 55–69.
, Extremal functions for the anisotropic Sobolev inequalities, Ann. Inst. H. Poincaré Anal. Non
[20]
Linéaire 24 (2007), no. 5, 741–756.
[21] A. El Hamidi and J. Vétois, Sharp Sobolev asymptotics for critical anisotropic equations, Arch. Ration.
Mech. Anal. 192 (2009), no. 1, 1–36.
[22] A. Farina, On the classification of solutions of the Lane-Emden equation on unbounded domains of Rn , J.
Math. Pures Appl. (9) 87 (2007), no. 5, 537–561.
[23] R. Filippucci, P. Pucci, and F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J.
Math. Pures Appl. (9) 91 (2009), no. 2, 156–177.
[24] I. Fragalà, F. Gazzola, and B. Kawohl, Existence and nonexistence results for anisotropic quasilinear
elliptic equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 21 (2004), no. 5, 715–734.
[25] I. Fragalà, F. Gazzola, and G. Lieberman, Regularity and nonexistence results for anisotropic quasilinear
elliptic equations in convex domains, Discrete Contin. Dyn. Syst. suppl. (2005), 280–286.
[26] B. Franchi, E. Lanconelli, and J. Serrin, Existence and uniqueness of nonnegative solutions of quasilinear
equations in Rn , Adv. Math. 118 (1996), no. 2, 177–243.
[27] J. Garcı́a-Melián, J. D. Rossi, and J. C. Sabina de Lis, Large solutions to an anisotropic quasilinear
elliptic problem, Ann. Mat. Pura Appl. (4) 189 (2010), no. 4, 689–712.
[28] F. Gazzola, Critical growth quasilinear elliptic problems with shifting subcritical perturbation, Differential
Integral Equations 14 (2001), no. 5, 513–528.
[29] M. Guedda and L. Véron, Quasilinear elliptic equations involving critical Sobolev exponents, Nonlinear
Anal. 13 (1989), no. 8, 879–902.
[30] F. Q. Li, Anisotropic elliptic equations in Lm , J. Convex Anal. 8 (2001), no. 2, 417–422.
[31] G. M. Lieberman, Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann.
Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994), no. 4, 497–522.
[32]
, Gradient estimates for anisotropic elliptic equations, Adv. Differential Equations 10 (2005), no. 7,
767–812.
[33] P.-L. Lions, The concentration-compactness principle in the calculus of variations. The limit case. I, Rev.
Mat. Iberoamericana 1 (1985), no. 1, 145–201.
[34]
, The concentration-compactness principle in the calculus of variations. The limit case. II, Rev.
Mat. Iberoamericana 1 (1985), no. 2, 45–121.
[35] M. Mihăilescu, P. Pucci, and V. Rădulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), no. 1, 687–698.
[36] M. Mihăilescu, V. Rădulescu, and S. Tersian, Eigenvalue problems for anisotropic discrete boundary value
problem, Journal of Difference Equations and Applications 15 (2009), no. 6, 557–567.
[37] Y. V. Namlyeyeva, A. E. Shishkov, and I. I. Skrypnik, Isolated singularities of solutions of quasilinear
anisotropic elliptic equations, Adv. Nonlinear Stud. 6 (2006), no. 4, 617–641.
[38] J. Rákosnı́k, Some remarks to anisotropic Sobolev spaces. I, Beiträge Anal. 13 (1979), 55–68.
[39] I. I. Skrypnik, Removability of an isolated singularity for anisotropic elliptic equations with absorption,
Math. Sb. 199 (2008), no. 7, 1033-1050.
CRITICAL ANISOTROPIC EQUATIONS WITH CRITICAL DIRECTIONS
18
[40] A. S. Tersenov and A. S. Tersenov, The problem of Dirichlet for anisotropic quasilinear degenerate elliptic
equations, J. Differential Equations 235 (2007), no. 2, 376–396.
[41] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat. 18 (1969), 3–24 (Italian).
[42] J. Vétois, A priori estimates for solutions of anisotropic elliptic equations, Nonlinear Anal. 71 (2009),
no. 9, 3881–3905.
, Asymptotic stability, convexity, and Lipschitz regularity of domains in the anisotropic regime,
[43]
Commun. Contemp. Math. 12 (2010), no. 1, 35–53.
, The blow-up of critical anistropic equations with critical directions, Nonlinear Differ. Equ. Appl.
[44]
18 (2011), no. 2, 173–197.
, Strong maximum principles for anisotropic elliptic and parabolic equations (2010). Preprint.
[45]
[46] J. Weickert, Anisotropic diffusion in image processing, European Consortium for Mathematics in Industry,
B. G. Teubner, Stuttgart, 1998.
Jérôme Vétois, Université de Nice - Sophia Antipolis, Laboratoire J.-A. Dieudonné, UMR
CNRS-UNS 6621, Parc Valrose, 06108 Nice Cedex 2, France
E-mail address: [email protected]