Electronic Journal of Differential Equations, Vol. 1999(1999), No. 14, pp. 1–12.
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
ftp ejde.math.swt.edu ejde.math.unt.edu (login: ftp)
EXISTENCE OF MULTIPLE SOLUTIONS FOR QUASILINEAR
DIAGONAL ELLIPTIC SYSTEMS
MARCO SQUASSINA
Abstract. Nonsmooth-critical-point theory is applied in proving multiplicity
results for the quasilinear symmetric elliptic system
n
X
−
Dj (akij (x, u)Di uk ) +
i,j=1
1
2
N
n
X
X
Dsk ah
ij (x, u)Di uh Dj uh = gk (x, u) ,
i,j=1 h=1
for k = 1, .., N .
1. Introduction
Many papers have been published on the study of multiplicity of solutions for
quasilinear elliptic equations via nonsmooth-critical-point theory; see e.g. [2, 3, 4, 7,
8, 9, 10, 18, 20]. However, for the vectorial case only a few multiplicity results have
been proven: [20, Theorem 3.2] and recently [4, Theorem 3.2], where systems with
multiple identity coefficients are treated. In this paper, we consider the following
diagonal quasilinear elliptic system, in an open bounded set Ω ⊂ Rn with n ≥ 3,
−
n
X
Dj (akij (x, u)Di uk ) +
i,j=1
N
n
1 X X
Dsk ahij (x, u)Di uh Dj uh =
2 i,j=1
h=1
= Dsk G(x, u)
(1)
in Ω ,
for k = 1, .., N , where u : Ω → RN and u = 0 on ∂Ω. To prove the existence of
weak solutions, we look for critical points of the functional f : H01 (Ω, RN ) → R,
f (u) =
1
2
Z
N
n X
X
Ω i,j=1 h=1
ahij (x, u)Di uh Dj uh dx −
Z
G(x, u) dx .
Ω
1991 Mathematics Subject Classification. 35D05, 35J20, 35J60.
Key words and phrases. Quasilinear elliptic differential systems,
Nonsmooth critical point theory.
c
1999
Southwest Texas State University and University of North Texas.
Submitted January 4, 1999. Published May 10, 1999.
1
(2)
2
MARCO SQUASSINA
EJDE–1999/14
This functional is not locally Lipschitz if the coefficients ahij depend on u; however,
as pointed out in [2, 7], it is possible to evaluate f 0 ,
Z X
N
n X
ahij (x, u)Di uh Dj vh dx +
f 0 (u)(v) =
Ω i,j=1 h=1
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, u) · vDi uh Dj uh dx −
Z
Ω
Ds G(x, u) · v dx
for all v ∈ H01 (Ω, RN ) ∩ L∞ (Ω, RN ). We shall apply the nonsmooth-critical-point
theory developed in [11, 13, 15, 16]. For notation and related results, the reader is
referred to [9]. To prove our main result and to provide some regularity of solutions,
we consider the following assumptions.
(A1) The matrix ahij (·, s) is measurable in x for every s ∈ RN , and of class C 1
in s for a.e. x ∈ Ω with
ahij = ahji .
Furthermore, we assume that there exist ν > 0 and C > 0 such that for a.e. x ∈ Ω,
all s ∈ RN and ξ ∈ RnN
N
n X
X
ahij (x, s)ξih ξjh ≥ ν|ξ|2 , ahij (x, s) ≤ C, Ds ahij (x, s) ≤ C
(3)
i,j=1 h=1
and
N
n X
X
s · Ds ahij (x, s)ξih ξjh ≥ 0.
(4)
i,j=1 h=1
(A2) There exists a bounded Lipschitz function ψ : R → R, such that for a.e.
x ∈ Ω, for all ξ ∈ RnN , σ ∈ {−1, 1}N and r, s ∈ RN
N n X
X
1
Ds ahij (x, s) · expσ (r, s) + ahij (x, s)Dsh (expσ (r, s))h ξih ξjh ≤ 0,
2
i,j=1 h=1
(5)
where (expσ (r, s))i := σi exp[σi (ψ(ri ) − ψ(si ))] for each i = 1, .., N .
(G1) The function G(x, s) is measurable in x for all s ∈ RN and of class C 1 in
s for a.e. x ∈ Ω, with G(x, 0) = 0. Moreover for a.e. x ∈ Ω we will denote with
g(x, ·) the gradient of G with respect to s.
(G2) For every ε > 0 there exists aε ∈ L2n/(n+2) (Ω) such that
|g(x, s)| ≤ aε (x) + ε|s|(n+2)/(n−2)
(6)
for a.e. x ∈ Ω and all s ∈ RN and that there exist q > 2, R > 0 such that for all
s ∈ RN and for a.e. x ∈ Ω
|s| ≥ R =⇒ 0 < qG(x, s) ≤ s · g(x, s).
(7)
(AG) There exists γ ∈ (0, q − 2) such that for all ξ ∈ RnN , s ∈ RN and a.e. in Ω
N
n X
X
i,j=1 h=1
s · Ds ahij (x, s)ξih ξjh ≤ γ
N
n X
X
i,j=1 h=1
ahij (x, s)ξih ξjh .
(8)
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QUASILINEAR DIAGONAL ELLIPTIC SYSTEMS
3
Under these assumptions we will prove the following.
Theorem 1. Assume that for a.e. x ∈ Ω and for each s ∈ RN
ahij (x, −s) = ahij (x, s),
g(x, −s) = −g(x, s) .
Then there exists a sequence (um ) ⊆ H01 (Ω, RN ) of weak solutions to (1) such that
lim f (um ) = +∞ .
m
The above result is well known for the semilinear scalar problem
−
n
X
Dj (aij (x)Di u) = g(x, u) in Ω
i,j=1
u=0
on ∂Ω .
A. Ambrosetti and P. H. Rabinowitz in [1, 19] studied this problem using techniques
of classical critical point theory. The quasilinear scalar problem
−
n
X
Dj (aij (x, u)Di u) +
i,j=1
n
1 X
Ds aij (x, u)Di uDj u = g(x, u) in Ω
2 i,j=1
u=0
on ∂Ω ,
was studied in [7, 8, 9] and in [18] in a more general setting. In this case the
functional
Z X
Z
n
1
f (u) =
aij (x, u)Di uDj u dx −
G(x, u) dx
2 Ω i,j=1
Ω
is continuous under appropriate conditions, but it is not locally Lipschitz. Consequently, techniques of nonsmooth-critical-point theory have to be applied. In
the vectorial case, to my knowledge, problem (1) has only been considered in
[20, Theorem 3.2] and recently in [4, Theorem 3.2] for coefficients of the type
hk
ahk
ij (x, s) = δ αij (x, s). In [4] a new technical condition is introduced to be
compared with our (5). They assume that there exist K > 0 and an increasing bounded Lipschitz function ψ : [0, +∞[→ [0, +∞[, with ψ(0) = 0, ψ 0 nonincreasing, ψ(t) → K as t → +∞ and such that for all ξ ∈ Rn , for a.e. x ∈ Ω and
for all r, s ∈ RN
n X
N
X
i,j=1 k=1
|Dsk aij (x, s)ξi ξj | ≤ 2e−4K ψ 0 (|s|)
n
X
aij (x, s)ξi ξj .
(9)
i,j=1
The proof itself of [4, Lemma 6.1] shows that condition (9) implies our assumption
(A2). On the other hand, if N ≥ 2, the two conditions look quite similar. However,
condition (A2) seems to be preferable, because when N = 1 it reduces to the
inequality
n
n
X
X
0
≤
2ψ
D
a
(x,
s)ξ
ξ
(s)
aij (x, s)ξi ξj ,
s ij
i j
i,j=1
i,j=1
which is not so restrictive in view of (3), while (9) is in this case much stronger.
4
MARCO SQUASSINA
EJDE–1999/14
2. Boundedness of concrete Palais-Smale sequences
Definition 2. Let c ∈ R. A sequence (um ) ⊆ H01 (Ω; RN ) is said to be a concrete
Palais-Smale sequence at level c ((CP S)c −sequence, in short) for f , if f (um ) → c,
N
n X
X
m
−1
Dsk ahij (x, um )Di um
(Ω; RN )
h Dj u h ∈ H
i,j=1 h=1
eventually as m → ∞, and
−
n
X
Dj (akij (x, um )Di um
k )+
i,j=1
N
n
1 X X
m
m
Dsk ahij (x, um )Di um
h Dj uh − gk (x, u )
2 i,j=1
h=1
−1
converges to zero strongly in H (Ω; RN ). We say that f satisfies the concrete
Palais-Smale condition at level c ((CP S)c in short), if every (CP S)c −sequence for
f admits a strongly convergent subsequence in H01 (Ω; RN ).
Next we state and prove a vectorial version of Brezis-Browder’s Theorem [6].
Lemma 3. Let T ∈ L1loc (Ω, RN ) ∩ H −1 (Ω, RN ), v ∈ H01 (Ω, RN ) and η ∈ L1 (Ω)
with T · v ≥ η. Then T · v ∈ L1 (Ω) and
Z
T · v dx
hT, vi =
Ω
Cc∞ (Ω, RN )
Proof. Let (vh ) ⊆
support in Ω by setting
with vh → v. Define Θh (v) ∈ H01 ∩L∞ with compact
v
Θh (v) = min{|v|, |vh |} q
|v|2 +
Since
and
T ·v
min{|v|, |vh |} q
|v|2 +
*
+
v
T, min{|v|, |vh |} q
|v|2 +
1
h
1
h
1
h
.
≥ −η − ∈ L1 (Ω),
Z
=
T ·v
min{|v|, |vh |} q
Ω
|v|2 +
1
h
dx,
R
a variant of Fatou’s Lemma implies Ω T · v dx ≤ hT, vi, so that T · v ∈ L1 (Ω).
Finally, since
T
·
v
≤ |T · v|,
min{|v|, |vh |} q
1
2
|v| + h
Lebesgue’s Theorem yields
Z
hT, vi =
Ω
T · v dx,
and the proof is complete.
The first step for the (CP S)c to hold is the boundedness of (CP S)c sequences.
Lemma 4. Assume (A1), (G1), (G2) and (AG). Then for all c ∈ R each
(CP S)c sequence of f is bounded in H01 (Ω, RN ).
EJDE–1999/14
QUASILINEAR DIAGONAL ELLIPTIC SYSTEMS
5
Proof. Let a0 ∈ L1 (Ω) be such that for a.e. x ∈ Ω and all s ∈ RN
qG(x, s) ≤ s · g(x, s) + a0 (x).
Now let (um ) be a (CP S)c sequence for f and let wm → 0 in H −1 (Ω, RN ) such
that for all v ∈ Cc∞ (Ω, RN ),
Z
m
hw , vi =
N
n X
X
Ω i,j=1 h=1
1
+
2
Z
ahij (x, um )Di um
h Dj vh dx +
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, um )
·
m
vDi um
h Dj uh dx
Z
−
Ω
g(x, um ) · v .
Taking into account the previous Lemma, for every m ∈ N we obtain
−kwm kH −1 (Ω,RN ) kum kH01 (Ω,RN ) ≤
Z X
N
n X
m
ahij (x, um )Di um
≤
h Dj uh dx +
Ω i,j=1 h=1
+
1
2
Z
≤
Z
N
n X
X
Ω i,j=1 h=1
N
n X
X
Ω i,j=1 h=1
+
1
2
Z
Z
−q
m
Ds ahij (x, um ) · um Di um
h Dj uh dx −
Z
Ω
g(x, um ) · um dx ≤
m
ahij (x, um )Di um
h Dj uh dx +
N
n X
X
Ω i,j=1 h=1
m
Ds ahij (x, um ) · um Di um
h Dj uh dx +
G(x, um ) dx +
Ω
Z
Ω
a0 dx .
Taking into account the expression of f and assumption (AG), we have that for
each m ∈ N,
−kwm kH −1 (Ω,RN ) kum kH01 (Ω,RN ) ≤
N
n X
Z X
q
m
−1
ahij (x, um )Di um
≤ −
h Dj uh dx +
2
Ω i,j=1
1
+
2
Z
h=1
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, um )
·u
m
m
Di u m
h Dj uh dx
Ω
Z
+ qf (u ) +
Z X
N
n X
γ
m
−1−
ahij (x, um )Di um
≤ −
h Dj uh dx +
2
2 Ω i,j=1
h=1
Z
a0 dx .
+qf (um ) +
q
m
Ω
a0 dx ≤
6
MARCO SQUASSINA
EJDE–1999/14
Because of (A1), for each m ∈ N,
ν(q − 2 − γ)kDum k22
Z
N
n X
X
m
ahij (x, um )Di um
h Dj uh dx
≤
(q − 2 − γ)
≤
2kwm kH −1 (Ω,RN ) kum kH01 (Ω,RN ) + 2qf (um ) + 2
Ω i,j=1 h=1
Z
Ω
a0 dx .
Since (wm ) converges to 0 in H −1 (Ω, RN ), we conclude that (um ) is a bounded
sequence in H01 (Ω, RN ).
Lemma 5. If condition (6) holds, then the map
H01 (Ω, RN )
u
−→
7−→
L2n/(n+2) (Ω, RN )
g(x, u)
is completely continuous.
Proof. This is a direct consequence of [9, Theorem 2.2.7].
3. Compactness of concrete Palais-Smale sequences
The next result is crucial for the (CP S)c condition to hold for our elliptic system.
Lemma 6. Assume (A1) and (A2), let (um ) be a bounded sequence in H01 (Ω, RN ),
and set
Z X
N
n X
hwm , vi =
ahij (x, um )Di um
h Dj vh dx +
Ω i,j=1 h=1
+
Z
1
2
N
n X
X
Ω i,j=1 h=1
m
Ds ahij (x, um ) · vDi um
h Dj uh dx
for all v ∈ Cc∞ (Ω, RN ). If (wm ) is strongly convergent to some w in H −1 (Ω, RN ),
then (um ) admits a strongly convergent subsequence in H01 (Ω, RN ).
Proof. Since (um ) is bounded, we have um * u for some u up to a subsequence.
m
Each component um
k satisfies (2.5) in [5], so we may suppose that Di uk → Di uk
a.e. in Ω for all k = 1, . . . , N (see also [12]). We first prove that
Z
N
n X
X
Ω i,j=1 h=1
ahij (x, u)Di uh Dj uh dx +
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, u) · uDi uh Dj uh dx = hw, ui.
(10)
Let ψ be as in assumption (A2) and consider the following test functions
m
v m = ϕ(σ1 exp[σ1 (ψ(u1 ) − ψ(um
1 ))], . . . , σN exp[σN (ψ(uN ) − ψ(uN ))]),
where ϕ ∈ Cc∞ (Ω), ϕ ≥ 0 and σl = ±1 for all l. Therefore, since we have
m
m
Dj vkm = (σk Dj ϕ + (ψ 0 (uk )Dj uk − ψ 0 (um
k )Dj uk )ϕ) exp[σk (ψ(uk ) − ψ(uk ))],
EJDE–1999/14
QUASILINEAR DIAGONAL ELLIPTIC SYSTEMS
7
we deduce that for all m ∈ N,
Z
N
n X
X
0
m
ahij (x, um )Di um
h (σh Dj ϕ+ψ (uh )Dj uh ϕ) exp[σh (ψ(uh )−ψ(uh ))] dx+
Ω i,j=1 h=1
Z
N
n
X
X
σl
m
m
Dsl ahij (x, um ) exp[σl (ψ(ul ) − ψ(um
+
l ))]Di uh Dj uh ϕ dx +
2
Ω i,j=1
h,l=1
Z
−
N
n X
X
Ω i,j=1 h=1
m 0 m
m
ahij (x, um )Di um
h Dj uh ψ (uh ) exp[σh (ψ(uh ) − ψ(uh ))]ϕ dx =
= hwm , v m i .
Let us study the behavior of each term of the previous equality as m → ∞. First
of all, if v = (σ1 ϕ, . . . , σN ϕ), we have that v m * v implies
limhwm , v m i = hw, vi.
(11)
m
Since um * u, by Lebesgue’s Theorem we obtain
Z
lim
m
N
n X
X
Ω i,j=1 h=1
Z
=
ahij (x, um )Di um
h (Dj (σh ϕ)+
+ϕψ 0 (uh )Dj uh ) exp[σh (ψ(uh ) − ψ(um
h ))] dx =
n
X
N
X
Ω i,j=1 h=1
(12)
ahij (x, u)Di uh (Dj vh + ϕψ 0 (uh )Dj uh ) dx .
Finally, note that by assumption (A2) we have
N X
n X
N
X
σl
i,j=1 h=1
l=1
2
Dsl ahij (x, um ) exp[σl (ψ(ul ) − ψ(um
l ))]+
−ahij (x, um )ψ 0 (um
h ) exp[σh (ψ(uh )
m
m
− ψ(um
h ))] Di uh Dj uh
≤ 0.
Hence, we can apply Fatou’s Lemma to obtain
lim sup
n1 Z
m
Z
−
2
N
n
X
X
Ω i,j=1 h,l=1
N
n X
X
Ω i,j=1 h=1
≤
1
2
Z
−
o
m 0 m
m
ahij (x, um )Di um
D
u
ψ
(u
)
exp[σ
(ψ(u
)
−
ψ(u
))]ϕ
dx
≤
j h
h
h
h
h
h
N
n
X
X
Ω i,j=1 h,l=1
Z
m
m
Dsl ahij (x, um ) exp[σl (ψ(ul ) − ψ(um
l ))]Di uh Dj uh (σl ϕ) dx+
N
n X
X
Ω i,j=1 h=1
Dsl ahij (x, u)Di uh Dj uh (σl ϕ) dx +
ahij (x, u)Di uh Dj uh ψ 0 (uh )ϕ dx ,
8
MARCO SQUASSINA
EJDE–1999/14
which, together with (11) and (12), yields
Z X
N
n X
ahij (x, u)Di uh Dj vh dx +
Ω i,j=1 h=1
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, u) · vDi uh Dj uh dx ≥ hw, vi
for all test functions v = (σ1 ϕ, . . . , σN ϕ) with ϕ ∈ Cc∞ (Ω, RN ), ϕ ≥ 0. Since we
may exchange v with −v we get
Z X
N
n X
ahij (x, u)Di uh Dj vh dx +
Ω i,j=1 h=1
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Ds ahij (x, u) · vDi uh Dj uh dx = hw, vi
for all test functions v = (σ1 ϕ, . . . , σN ϕ), and since every function v ∈ Cc∞ (Ω, RN )
can be written as a linear combination of such functions, taking into account Lemma
3, we infer (10). Now, let us prove that
Z X
Z X
N
N
n X
n X
m
lim sup
ahij (x, um )Di um
D
u
dx
≤
ahij (x, u)Di uh Dj uh dx.
j
h
h
m
Ω i,j=1 h=1
Ω i,j=1 h=1
(13)
Because of (4), Fatou’s Lemma implies that
Z X
N
n X
u · Ds ahij (x, u)Di uh Dj uh dx ≤
Ω i,j=1 h=1
≤
lim inf
m
Z
N
n X
X
Ω i,j=1 h=1
m
um · Ds ahij (x, um )Di um
h Dj uh dx .
Combining this fact with (10), we deduce that
Z X
N
n X
m
ahij (x, um )Di um
lim sup
h Dj uh dx =
m
=
≤
Ω i,j=1 h=1
N
n X
h 1Z X
i
m
m
m
um · Ds ahij (x, um )Di um
D
u
dx
+
hw
,
u
i
≤
lim sup −
j h
h
2 Ω i,j=1
m
−
Z
=
1
2
Z
h=1
n
X
N
X
Ω i,j=1 h=1
n
X
N
X
Ω i,j=1 h=1
u · Ds ahij (x, u)Di uh Dj uh dx + hw, ui =
ahij (x, u)Di uh Dj uh dx ,
so that (13) is proved. Finally, by (3) we have
νkDum − Duk22 ≤
Z X
N
n X
m
m
≤
ahij (x, um ) (Di um
h Dj uh − 2Di uh Dj uh + Di uh Dj uh ) dx.
Ω i,j=1 h=1
EJDE–1999/14
QUASILINEAR DIAGONAL ELLIPTIC SYSTEMS
9
Hence, by (13) we obtain
lim sup kDum − Duk2 ≤ 0
m
which proves that u
m
→ u in H01 (Ω, RN ).
We now come to one of the main tools of this paper, the (CP S)c condition for
system (1).
Theorem 7. Assume (A1), (A2), (G1), (G2), (AG). Then f satisfies (CP S)c
condition for each c ∈ R.
Proof. Let (um ) be a (CP S)c sequence for f . Since (um ) is bounded in H01 (Ω, RN ),
from Lemma 5 we deduce that, up to a subsequence, (g(x, um )) is strongly convergent in H −1 (Ω, RN ). Applying Lemma 6, we conclude the present proof.
4. Existence of multiple solutions for elliptic systems
We now prove the main result, which is an extension of theorems of [7, 9] and a
generalization of [4, Theorem 3.2] to systems in diagonal form.
Proof of Theorem 1. We want to apply [9, Theorem 2.1.6]. First of all, because of
Theorem 7, f satisfies (CP S)c for all c ∈ R. Whence, (c) of [9, Theorem 2.1.6] is
satisfied. Moreover we have
Z
Z
ν
|Du|2 dx −
G(x, u) dx ≤ f (u) ≤
2 Ω
Ω
Z
Z
1
nN C
|Du|2 dx −
G(x, u) dx.
≤
2
Ω
Ω
We want to prove that assumptions (a) and (b) of [9, Theorem 2.1.6] are also
satisfied. Let us observe that, instead of (b) of [9, Theorem 2.1.6], it is enough
to find a sequence (Wn ) of finite dimensional subspaces with dim(Wn ) → +∞
satisfying the inequality of (b) (see also [17, Theorem 1.2]). Let W be a finite
dimensional subspace of H01 (Ω; RN ) ∩ L∞ (Ω, RN ). From (7) we deduce that for all
s ∈ RN with |s| ≥ R
s
G x, R |s|
|s|q ≥ b0 (x)|s|q ,
G(x, s) ≥
Rq
where
b0 (x) = R−q inf{G(x, s) : |s| = R} > 0
a.e. x ∈ Ω. Therefore there exists a0 ∈ L1 (Ω) such that
G(x, s) ≥ b0 (x)|s|q − a0 (x)
N
1
(14)
a.e. x ∈ Ω and for all s ∈ R . Since b0 ∈ L (Ω), we may define a norm k · kG on
W by
Z
1/q
q
b0 |u| dx
.
kukG =
Ω
Since W is finite dimensional and q > 2, from (14) it follows
lim
kukG →+∞
u∈W
f (u) = −∞
10
MARCO SQUASSINA
EJDE–1999/14
and condition (b) of [9, Theorem 2.1.6] is clearly fulfilled too for a sufficiently large
R > 0. Let now (λh , uh ) be the sequence of eigenvalues and eigenvectors for the
problem
∆u = −λu in Ω
u=0
on ∂Ω .
Let us prove that there exist h0 , α > 0 such that
where V +
we find
∀u ∈ V + : kDuk2 = 1 =⇒ f (u) ≥ α,
= span uh ∈ H01 (Ω, RN ) : h ≥ h0 . In fact, given u ∈ V + and ε > 0,
2n
∞
(2)
n+2 (Ω),
a(1)
ε ∈ Cc (Ω), aε ∈ L
(2)
2n ≤ ε and
such that kaε k n+2
n+2
(2)
n−2 .
|g(x, s)| ≤ a(1)
ε (x) + aε (x) + ε|s|
If u ∈ V + , it follows that
Z
ν
kDuk22 −
G(x, u) dx
f (u) ≥
2
Ω
Z 2n
ν
n−2
2
(1)
(2)
kDuk2 −
ε|u| n−2 dx
aε + aε |u| +
≥
2
2n
Ω
2n
ν
n−2
(2)
2n kDuk2 − εc2 kDuk
kDuk22 − ka(1)
≥
ε k2 kuk2 − c1 kaε k n+2
2
2
2n
ν
n−2
kDuk22 − ka(1)
≥
.
ε k2 kuk2 − c1 εkDuk2 − εc2 kDuk2
2
Then if h0 is sufficiently large, from the fact that (λh ) diverges, for all u ∈ V + ,
kDuk2 = 1 implies
ν
.
ka(1)
ε k2 kuk2 ≤
6
Hence, for ε > 0 small enough, kDuk2 = 1 implies that f (u) ≥ ν/6.
Finally, set V − = span uh ∈ H01 (Ω, RN ) : h < h0 , we have the decomposition
H01 (Ω; RN ) = V + ⊕ V − .
Therefore, since the hypotheses for [9, Theorem 2.1.6] are fulfilled, we can find a
sequence (um ) of weak solution of system (1) such that
lim f (um ) = +∞,
m
and the theorem is now proven.
5. Regularity of weak solutions for elliptic systems
Assume conditions (A1) and (G1), and consider the nonlinear elliptic system
Z
Z X
N
n
X
ahk
(x,
u)D
u
D
v
dx
=
b(x, u, Du) · v dx
(15)
i
h
j
k
ij
Ω i,j=1 h,k=1
Ω
for all v ∈ H01 (Ω; RN ). For l = 1, .., N , we choose


N
n
 X

X
Dsl ahk
(x,
u)D
u
D
u
+
g
(x,
u)
.
bl (x, u, Du) = −
i
h
j
k
l
ij


i,j=1 h,k=1
EJDE–1999/14
QUASILINEAR DIAGONAL ELLIPTIC SYSTEMS
Assume that there exist c > 0 and q <
n+2
n−2
11
such that for all s ∈ RN and a.e. in Ω
|g(x, s)| ≤ c (1 + |s|q ) .
(16)
Then it follows that for every M > 0, there exists C(M ) > 0 such that for a.e.
x ∈ Ω, for all ξ ∈ RnN and s ∈ RN with |s| ≤ M
|b(x, s, ξ)| ≤ c(M ) 1 + |ξ|2 .
(17)
A nontrivial regularity theory for quasilinear systems (see, [14, Chapter VI]) yields
the following :
Theorem 8. For every weak solution u ∈ H 1 (Ω, RN ) ∩ L∞ (Ω, RN ) of the system
(1) there exist an open subset Ω0 ⊆ Ω and s > 0 such that
n
∀p ∈ (n, +∞) : u ∈ C 0,1− p (Ω0 ; RN ),
Hn−s (Ω\Ω0 ) = 0 .
Proof. For the proof, see [14, Chapter VI].
hk
We now consider the particular case when ahk
ij (x, s) = αij (x, s)δ , and provide
an almost everywhere regularity result.
Lemma 9. Assume condition (17). Then the weak solutions u ∈ H01 (Ω, RN ) of the
system
Z X
N
n X
aij (x, u)Di uh Dj vh dx +
Ω i,j=1 h=1
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Z
Ds aij (x, u) · vDi uh Dj uh dx =
Ω
g(x, u) · v dx
(18)
for all v ∈ Cc∞ (Ω, RN ), belong to L∞ (Ω, RN ).
Proof. By [20, Lemma 3.3], for each (CP S)c sequence (um ) there exist u ∈ H01 ∩L∞
and a subsequence (umk ) with umk * u. Then, given a weak solution u, consider
the sequence (um ) such that each element is equal to u and the assertion follows.
We can finally state a partial regularity result for our system.
Theorem 10. Assume condition (17) and let u ∈ H01 (Ω, RN ) be a weak solution
of the system
Z X
N
n X
aij (x, u)Di uh Dj vh dx +
Ω i,j=1 h=1
+
1
2
Z
N
n X
X
Ω i,j=1 h=1
Z
Ds aij (x, u) · vDi uh Dj uh dx =
Ω
g(x, u) · v dx
(19)
for all v ∈ Cc∞ (Ω, RN ). Then there exist an open subset Ω0 ⊆ Ω and s > 0 such
that
n
∀p ∈ (n, +∞) : u ∈ C 0,1− p (Ω0 ; RN ),
Hn−s (Ω\Ω0 ) = 0.
Proof. It suffices to combine the previous Lemma with Theorem 8.
12
MARCO SQUASSINA
EJDE–1999/14
Acknowledgment. The author wishes to thank Professor Marco Degiovanni for
providing helpful discussions.
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Marco Squassina
Dipartimento di Matematica, Milan University, Via Saldini 50, 20133 Milano, Italy
E-mail address: squassin@ares.mat.unimi.it