Electronic Journal of Differential Equations, Vol. 2008(2008), No. 115, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
ON THE EXISTENCE OF WEAK SOLUTIONS FOR
p, q-LAPLACIAN SYSTEMS WITH WEIGHTS
OLIMPIO H. MIYAGAKI, RODRIGO S. RODRIGUES
Abstract. This paper studies degenerate quasilinear elliptic systems involving p, q-superlinear and critical nonlinearities with singularities. Existence
results are obtained by using properties of the best Hardy-Sobolev constant
together with an approach developed by Brezis and Nirenberg.
1. Introduction
In a well-known paper, Brezis and Nirenberg [11] proved that, under certain
conditions, the elliptic problem with Dirichlet boundary condition
∗
−∆u = λuq + u2 −1
u > 0 in Ω,
in Ω,
(1.1)
u = 0 on ∂Ω
possesses at least a solution, for all λ > 0, where 1 < q < 2∗ = 2N /(N − 2), N ≥ 3,
2∗ is said to be the critical Sobolev exponent, and Ω ⊂ RN (N ≥ 3) is a bounded
smooth domain. In general, the main difficulty in this type of problem is the lack
∗
of compactness of the injection H01 (Ω) ,→ L2 (Ω).
We recall that the perturbation λuq is essential in this kind of the problem. By
Pohozaev identity [30], problem (1.1) does not possess any solution when λ ≤ 0.
Garcı́a and Peral in [19] studied the existence of nontrivial solution for a class
of problems involving the p-laplacian operator, namely,
∗
−∆p u ≡ − div(|∇u|p−2 ∇u) = λ|u|q−2 u + µ|u|p
u ≥ 0 in Ω,
−2
u
in Ω,
u = 0 on ∂Ω,
in a bounded smooth domain Ω ⊂ RN (N > p), with 1 < p ≤ q < p∗ = N p/(N − p).
When p < q < p∗ , we say that the above problem is p-superlinear. These type of
problems, which are related to the Brezis and Nirenberg problem [11] (problem
(1.1) with p = 2), have been widely treated by several authors and we would like
2000 Mathematics Subject Classification. 35B25, 35B33, 35D05, 35J55, 35J70.
Key words and phrases. Degenerate quasilinear equations; elliptic system;
critical exponent; singular perturbation.
c
2008
Texas State University - San Marcos.
Submitted May 19, 2008. Published August 20, 2008.
The first author was supported by the CNPq-Brazil and AGIMB–Millenium Institute
MCT/Brazil. The second author was supported by Capes-Brazil.
1
2
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
to mention some of them, e.g., [14, 20, 21] for 1 < p < N and [26, 28, 29] for p = 2,
see also references cited there.
Caffarelli, Kohn and Nirenberg in [12] proved that if 1 < p < N , −∞ < a <
(N − p)/p, a ≤ c1 ≤ a + 1, d1 = 1 + a − c1 , and p∗ = p∗ (a, c1 , p) := N p/(N − d1 p),
there exists Ca,p > 0 such that the following Hardy-Sobolev type inequality with
weights is satisfied
Z
p/p∗
Z
∗
∗
|x|−c1 p |u|p dx
≤ Ca,p
|x|−ap |∇u|p dx , ∀u ∈ C0∞ (RN ).
RN
RN
Note that several papers have been appeared on this subject, mainly, the works
about the existence of solution for a class of quasilinear elliptic problems of the
type
∗
−Luap = g(x, u) + |x|−e1 p |u|q−2 u
in Ω,
where Luap = div(|x|−ap |∇u|p−2 ∇u), under certain suppositions on the exponents
1 < p < N , −∞ < a < (N − p)/p, a ≤ e1 < a + 1, d = 1 + a − e1 , and
p∗ = N p/(N − dp), and on the function g : Ω × R → R. See, for instance,
[4, 7, 13, 16, 34, 35] and references therein. The lack of compactness is overcame
proving that all the Palais Smale sequence at the level c, ( (P S)c -sequence, in short),
∗ N/dp
∗ N/dp
)
is so called the
)
, is relatively compact. (d/N )(Ca,p
with c < (d/N )(Ca,p
∗
critical level and Ca,p is the best Hardy-Sobolev constant and it is characterized by
R
o
n
|x|−ap |∇u|p dx
∗
∗
Ω
.
Ca,p = Ca,p (Ω) :=
inf
R
∗
p/p
u∈W01,p (Ω,|x|−ap )\{0}
|x|−e1 p∗ |u|p∗ dx
Ω
Besides the great number of the applications known for the scalar case, for
instance, in fluid mechanics, in newtonian fluids, in flow through porous media,
reaction-diffusion problems, nonlinear elasticity, petroleum extraction, astronomy,
glaciology, etc, see [15], the above systems can involve another phenomena, such as
competition model in population dynamics, see [18] and reference therein. For the
systems case we would like to mention the papers [2, 32] and a survey paper [17]
as well as in the references therein.
In our work, we will use a version of the well-known mountain pass theorem [6] to
establish conditions for the existence of a nontrivial solution for a quasilinear elliptic
system involving the above operator and a p, q-superlinear nonlinear perturbation
−Luap = λθ|x|−β1 |u|θ−2 |v|δ u + µα|x|−β2 |u|α−2 |v|γ u
−β1
−Lvbq = λδ|x|
θ
δ−2
|u| |v|
−β2
v + µγ|x|
α
γ−2
|u| |v|
v
in Ω,
in Ω,
(1.2)
u = v = 0 on ∂Ω,
where
Ω is a bounded smooth domain of RN with 0 ∈ Ω,
(1.3)
the parameters λ, µ are positive real numbers and the exponents satisfy
1 < p,
q < N,
a ≤ c1 < a + 1,
−∞ < a < (N − p)/p,
b ≤ c2 < b + 1,
∗
p = N p/(N − d1 p),
α, γ, θ, δ > 1,
−∞ < b < (N − q)/q,
d1 = 1 + a − c1 ,
∗
d2 = 1 + b − c2 ,
q = N q/(N − d2 q),
β1 , β2 ∈ R,
(1.4)
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
3
with one of the following two sets of conditions satisfied:
θ δ α γ
+ , + > 1 (p, q-superlinear)
p q p
q
(1.5)
δ α
γ
θ
+ ∗, ∗ + ∗ < 1
p∗
q p
q
(p, q-subcritical),
or
δ
θ δ
γ
θ
α
+ ∗ < 1 < + and ∗ + ∗ = 1 p, q-superlinear/critical case)
(1.6)
∗
p
q
p q
p
q
However, the variational systems behave, in a certain sense, like in the scalar case,
there exist some additional difficulties mainly coming from the mutual actions of
the variables u and v, see e. g. [23, 33]. Another difficulty, even in the regular case,
are the systems involving p-laplacian and q-laplacian operators and their respective
critical exponents. In this situation, it is hard to find a well appropriated critical
level, mainly, when p 6= q. This open question was pointed out in Adriouch and
Hamidi [1]. But, recently Silva and Xavier in [31] were able to prove, in a certain
context and in the regular case, the existence of weak solution for a system involving
p-laplacian and q-laplacian operators with p 6= q. Still in the regular case and p = q,
we would like to mention the papers [2, 5, 27, 32, 36], also a survey paper [17]. In
particular, Morais and Souto in [27] defined the following critical level number
SH /p, where
n R |∇u|p + |∇v|p dx o
Ω
SH = inf
p/p∗ ,
R
W \{0}
H(u, v)dx
Ω
W = W01,p (Ω) × W01,q (Ω) and H is homogeneous nonlinearity of degree p∗ . In
this work, we will improve the critical level by proving that all the Palais Smale
sequences at the level c are relatively compact provided that
−p
p∗
1
1
1
1
c < ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p + λ( − )M,
p p
p p1
∗
and M = M (un , vn ) ≥ 0 depends of Palais Smale sequence.
where S̃ depends of Ca,p
Our first result deals with p, q-superlinear and subcritical nonlinear perturbation.
Theorem 1.1. In addition to (1.3), (1.4), and (1.5), assume that pi ∈ (p, p∗ ),
qi ∈ (q, q ∗ ), i = 1, 2, with θ/p1 + δ/q1 = α/p2 + γ/q2 = 1 and
pi qi βi < min (a + 1)pi + N 1 −
, (b + 1)qi + N 1 −
, i = 1, 2.
(1.7)
p
q
Then system (1.2) possesses a weak solution, where each component is nontrivial
and nonnegative, for each λ ≥ 0 and µ > 0.
The next result treats the p, q-superlinear and critical case.
Theorem 1.2. Assume (1.3), (1.4) and (1.6), with p = q and a = b ≥ 0. Suppose
also p1 = q1 ∈ (p, p∗ ), with θ/p1 + δ/q1 = 1, p∗ = q ∗ , β2 = c1 p∗ , and β1 =
(a + 1)p1 − c with
−N [1 − (p1 /p)] < c <
(p1 − p + 1)N − (a + 1)p1
(N − p − ap)(p1 − p)
−
·
p−1
p(p − 1)
Then, system (1.2) possesses a weak solutions, where each component is nontrivial
and nonnegative, for each λ, µ > 0.
4
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
The p, q-superlinear and critical case with p 6= q is studied in the following result.
Theorem 1.3. In addition to (1.3), (1.4), and (1.6), assume that p1 ∈ (p, p∗ ),
q1 ∈ (q, q ∗ ), with θ/p1 + δ/q1 = 1, β2 = c1 p∗ = c2 q ∗ , and β1 as in (1.7). Then there
exists µ0 sufficiently small such that system (1.2) posesses a weak solution, where
each component is nontrivial and nonnegative, for each λ > 0 and 0 < µ < µ0 .
2. Preliminaries
We will set some spaces and their norms. If α ∈ R and l ≥ 1, we define Ll (Ω, |x|α )
as being the subspace of Ll (Ω) of the Lebesgue measurable functions u : Ω → R
satisfying
Z
1/l
kukLl (Ω,|x|α ) :=
|x|α |u|l dx
< ∞.
Ω
If 1 < p < N and −∞ < a < (N − p)/p, we define W01,p (Ω, |x|−ap ) as being the
completion of C0∞ (Ω) with respect to the norm k · k defined by
1/p
Z
.
|x|−ap |∇u|p dx
kuk :=
Ω
First of all, from the Caffarelli, Kohn and Nirenberg inequality (see [12]) and by
the boundedness of Ω, it is easy to see that there exists C > 0 such that
p/r
Z
Z
|x|−ap |∇u|p dx , ∀u ∈ W01,p (Ω, |x|−ap ),
|x|−δ |u|r dx
≤C
Ω
Ω
where 1 ≤ r ≤ N p/(N − p) and δ ≤ (a + 1)r + N [1 − (r/p)].
Lemma 2.1. Suppose that Ω is a bounded smooth domain of RN with 0 ∈ Ω, 1 <
p < N , −∞ < a < (N − p)/p, a ≤ e1 < a + 1, d1 = 1 + a − e1 , p∗ = N p/(N − d1 p),
and α + γ = p∗ , then
n R |x|−ap (|∇u|p + |∇v|p )dx o
Ω
S̃ := inf
∗ ,
R
−e1 p∗ |u|α |v|γ dx p/p
(u,v)∈W̃
|x|
Ω
where
2
W̃ = (u, v) ∈ W01,p (Ω, |x|−ap ) : |ukv| 6≡ 0 ,
satisfies
∗
∗
∗
S̃ = (α/γ)γ/p + (α/γ)−α/p Ca,p
.
The proof of the above lemma is similar to the proof of [5, Theorem 5] (see also
[27, Lemma 3] for p 6= 2).
Let us consider Ω a smooth domain of RN (not necessarily bounded), 0 ∈ Ω, 1 <
p < N , 0 ≤ a < (N − p)/p, a ≤ c1 < a + 1, d1 = 1 + a − c1 , and p∗ = N p/(N − d1 p).
We define the space
∗
∗
1,p
Wa,c
(Ω) = u ∈ Lp (Ω, |x|−c1 p ) : |∇u| ∈ Lp (Ω, |x|−ap ) ,
1
equipped with the norm
1,p
kukWa,c
1
(Ω)
= kukLp∗ (Ω,|x|−c1 p∗ ) + k∇ukLp (Ω,|x|−ap ) .
We consider the best Hardy-Sobolev constant given by
R
n
o
|x|−ap |∇u|p dx
RN
.
S̃a,p = 1,p inf
R
∗
p/p
Wa,c1 (RN )\{0}
|x|−c1 p∗ |u|p∗ dx
RN
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
5
Also, we define
1,p
1,p
Ra,c
(Ω) = u ∈ Wa,c
(Ω) : u(x) = u(|x|) ,
1
1
endowed with the norm
1,p
kukRa,c
1
(Ω)
1,p
= kukWa,c
1
(Ω) .
Actually, Horiuchi in [24] proved that
R
o
n
|x|−ap |∇u|p dx
RN
= S̃a,p
S̃a,p,R = 1,p inf
R
∗
p/p
−c1 p∗ |u|p∗ dx
Ra,c1 (RN )\{0}
|x|
N
R
and it is achieved by functions of the form
y (x) := ka,p ()Ua,p, (x),
∀ > 0,
where
2
ka,p () = c0 (N −d1 p)/d1 p
Moreover, y satisfies
Z
−ap
|x|
N −d p
d1 p(N −p−ap) −( d p1 )
1
and Ua,p, (x) = + |x| (p−1)(N −d1 p)
.
p
Z
|∇y | dx =
RN
∗
∗
|x|−c1 p |y |p dx.
(2.1)
RN
See also Clément, Figueiredo and Mitidieri [16, Proposition 1.4].
The next lemma can be proved arguing as in [11] (see also [35, Lemma 5.1]). For
the sake of the completeness we will give the proof in the appendix.
Lemma 2.2. In addition to (1.3) and (1.4), assume that p1 = q1 ∈ (p, p∗ ), θ/p1 +
δ/q1 = 1, β2 = c1 p∗ = c2 q ∗ , and β1 = (a + 1)p1 − c with
−N [1 − (p1 /p)] < c.
Let R0 ∈ (0, 1) be such that B(0, 2R0 ) ⊂ Ω and ψ ∈ C0∞ (B(0, 2R0 )) with ψ ≥ 0 in
B(0, 2R0 ) and ψ ≡ 1 in B(0, R0 ), then the function
u (x) =
ψ(x)Ua,p, (x)
kψUa,p, kLp∗ (Ω,|x|−c1 p∗ )
satisfies
∗
ku kpLp∗ (Ω,|x|−c1 p∗ ) = 1,
k∇u kpLp (Ω,|x|−ap ) ≤ S̃a,p,R + O((N −d1 p)/d1 p ),
and
ku kpL1p1 (Ω,|x|−β1 )

2
−(a+1)p1
O((N −d1 p)p1 /d1 p ) if c > (p1 −p+1)N
,

p−1



2
(p
−p+1)N
−(a+1)p

1
O((N −d1 p)p1 /d1 p |ln()|) if c = 1
,
p−1
(N −d1 p)(p−1)(N −p1 −ap1 +c) (N −d1 p)(p−1)p1 ≥
−

d1 p(N −p−ap)
d1 p2

O 



−(a+1)p1
if c < (p1 −p+1)N
.
p−1
(2.2)
The following result, which will be useful in the proof of our results, was proved
by Kavian in [25, Lemma 4.8].
Lemma 2.3. Let Ω be an open subset of RN , {fn } ∈ Lr (Ω), for some 1 < r < ∞,
a bounded sequence such that fn (x) → f (x), for a.e. x ∈ Ω, as n → ∞. Then,
f ∈ Lr (Ω) and fn * f weakly in Lr (Ω) as n → ∞.
6
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
Definition. Let us consider {(un , vn )} in W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ). We
say that the sequence {(un , vn )} is a Palais Smale sequence for operator I at the
level c (or simply, (P S)c -sequence) if
I(un , vn ) → c and I 0 (un , vn ) → 0,
as n → ∞.
Our approach will be to use variational techniques; that is, we have to find the
critical points of the Euler-Lagrange functional
I : W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) → R
given by
1
I(u, v) =
p
Z
−ap
1
|∇u| dx +
q
Z
|x|
|x|−bq |∇v|q dx
Ω
Ω
Z
Z
γ
−β1 θ δ
− λ |x|
u+ v+ dx − µ |x|−β2 uα
+ v+ dx,
p
Ω
Ω
1
which is well defined and is of class C , with the Gâteaux derivative
Z
Z
0
−ap
p−2
hI (u, v), (w, z)i = |x|
|∇u| ∇u∇w dx + |x|−bq |∇v|q−2 ∇v∇z dx
Ω
Ω
Z
Z
δ−1
−β1 θ−1 δ
z dx
− λθ |x|
u+ v+ w dx − λδ |x|−β1 uθ+ v+
Ω
Ω
Z
Z
γ−1
γ
w
dx
−
µγ
|x|−β2 uα
− µα |x|−β2 uα−1
v
+ v+ z dx,
+
+
Ω
Ω
W01,p (Ω, |x|−ap )
where u± = max{0, ±u} which is in
(Similarly v± = max{0, ±v}
1,q
−bq
which is in W0 (Ω, |x| ); see [3]).
First of all, we are going to show the geometric conditions of the mountain pass
theorem.
Lemma 2.4. In addition to (1.3) and (1.4), assume that one of the following
conditions hold:
(i) the case (1.5), pi ∈ (p, p∗ ), qi ∈ (q, q ∗ ), with θ/p1 + δ/q1 = α/p2 + γ/q2 = 1,
and βi as in (1.7), for i = 1, 2.
(ii) the case (1.6), p1 ∈ (p, p∗ ), q1 ∈ (q, q ∗ ), with θ/p1 + δ/q1 = 1, β1 as in
(1.7), p2 = p∗ , q2 = q ∗ , and β2 = c1 p∗ = c2 q ∗ .
Then the Euler-Lagrange functional I satisfies:
(a) There exist σ, ρ > 0 such that
I(u, v) ≥ σ if k(u, v)k = ρ.
(b) There exists e ∈
W01,p (Ω, |x|−ap )
I(e) ≤ 0,
×
kek ≥ R
W01,q (Ω, |x|−bq )
(2.3)
such that
for some R > ρ.
Proof. Part (a). For (u, v) ∈ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) with k(u, v)k ≤ 1,
we have
1
θC p1 /p
αC p2 /p
kukp − λ
kukp1 − µ
kukp2
I(u, v) ≥
p
p1
p2
1
q1 /q
δC
γC q2 /q
+
kvkq − λ
kvkq1 − µ
kvkq2
q
q1
q2
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
≥
7
θC p1 /p
1
αC p2 /p kukmin{p1 ,p2 }
kukp − λ
+µ
p
p1
p2
δC q1 /q
1
γC q2 /q kvkmin{q1 ,q2 } .
+ kvkq − λ
+µ
q
q1
q2
Hence, as p < min{p1 , p2 } and q < min{q1 , q2 }, we can choose ρ ∈ (0, 1) such that
I(u, v) ≥ σ
if k(u, v)k = ρ.
Part (b). The proof follows by taking (u0 , v0 ) ∈ W01,p (Ω, |x|−ap )×W01,q (Ω, |x|−bq )
1
with u0+ .v0+ 6≡ 0. Then, defining (ut , vt ) = (t1/p u0 , t q v0 ), for t > 0, we obtain
Z
1
α
1
γ
+ γq
p
q
p
(2.4)
I(ut , vt ) ≤
ku0 k + kv0 k t − µt
|x|−β2 uα
0+ v0+ dx → −∞,
p
q
Ω
as t → ∞.
From the mountain pass theorem [6] we get a (P S)c -sequence {(un , vn )} in
W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ), where
0 < σ ≤ c = inf max I(h(t))
h∈Γt∈[0,1]
(2.5)
and
Γ = h ∈ C([0, 1], W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq )) : h(0) = 0, h(1) = e , (2.6)
with I(e) ≡ I(t0 u0 , t0 v0 ) < 0.
Lemma 2.5. In addition to (1.3) and (1.4), assume that one of the two following
conditions hold:
(i) the case (1.5), pi ∈ (p, p∗ ), qi ∈ (q, q ∗ ), with θ/p1 + δ/q1 = α/p2 + γ/q2 = 1,
and βi as in (1.7), for i = 1, 2.
(ii) the case (1.6), p1 ∈ (p, p∗ ), q1 ∈ (q, q ∗ ), with θ/p1 + δ/q1 = 1, β1 as in
(1.7), p2 = p∗ , q2 = q ∗ , and β2 = c1 p∗ = c2 q ∗ .
Let {(un , vn )} ⊂ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) be a (P S)c -sequence. Then
{(un+ , vn+ )} is a (P S)c -sequence which is bounded uniformly in µ > 0.
Proof. Let θ1 = min{p1 , p2 } and θ2 = min{q1 , q2 }, we have
c + k(un , vn )k + On (1) ≥ I(un , vn ) − hI 0 (un , vn ), (un /θ1 , vn /θ2 )i
1
1
1
1
≥ ( − )kun kp + ( − )kvn kq
p θ1
q
θ2
Z
θ
δ
+ λ( +
− 1) |x|−β1 uθn+ vnδ + dx
θ1
θ2
ZΩ
α
γ
γ
+ µ( +
− 1) |x|−β2 uα
n+ vn+ dx
θ1
θ2
Ω
1
1
1
1
≥ ( − )kun kp + ( − )kvn kq .
p θ1
q
θ2
Therefore, independently of λ ≥ 0 and µ > 0, we conclude that {(un , vn )} is
a bounded sequence in W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ). In particular, we have
that {(un− , vn− )} and {(un+ , vn+ )} are bounded sequences in W01,p (Ω, |x|−ap ) ×
W01,q (Ω, |x|−bq ), then
−kun− kp = I 0 (un , vn ), (un− , 0) → 0 as n → ∞
(2.7)
8
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
and similarly
−kvn− kq = I 0 (un , vn ), (0, vn− ) → 0
as n → ∞.
(2.8)
Moreover, we get
1
1
I(un+ , vn+ ) = I(un , vn ) + kun− kp + kvn− kq = I(un , vn ) + On (1).
p
q
Therefore, from (2.7) and (2.8), we obtain I(un+ , vn+ ) → c as n → ∞. Similarly, if
(w, z) ∈ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ), we prove that
hI 0 (un+ , vn+ ), (w, z)i = hI 0 (un , vn ), (w, z)i + On (1),
hence I 0 (un+ , un+ ) → 0 as n → ∞.
3. Proof of theorem 1.1
Lemma 3.1. Suppose that (1.3) and (1.4) hold. Assume that pi ∈ (p, p∗ ), qi ∈
(q, q ∗ ), i = 1, 2, with θ/p1 + δ/q1 = α/p2 + γ/q2 = 1, and βi , i = 1, 2, as in
(1.7). Then, every (P S)c -sequence {(un , vn )} with un , vn ≥ 0, for a.e. in Ω, is
precompact.
Proof. From lemma 2.5, the sequence {(un , vn )} is bounded in W01,p (Ω, |x|−ap ) ×
W01,q (Ω, |x|−bq ). We can assume, passing to a subsequence if necessary, there exists
(u, v) ∈ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) satisfying un * u and vn * v weakly, as
n → ∞. From the compact embedding theorem [35, Theorem 2.1], we obtain
un → u
vn → v
in Lp1 (Ω, |x|−β1 ) ∩ Lp2 (Ω, |x|−β2 )
−β1
q1
in L (Ω, |x|
−β2
q2
) ∩ L (Ω, |x|
)
as n → ∞,
as n → ∞.
Since there exist f ∈ Lp1 (Ω, |x|−β1 ) and g ∈ Lq1 (Ω, |x|−β1 ) such that |un |(x) ≤ f (x)
and |vn |(x) ≤ g(x), for a.e. x ∈ Ω and all n ∈ N, applying the Lebesgue’s dominated
convergence theorem we infer that
Z
δ
lim
|x|−β1 uθ−1
(3.1)
n vn (un − u)dx = 0 ,
n→∞
Ω
and similarly
Z
lim
n→∞
|x|−β2 uα−1
vnγ (un − u)dx = 0.
n
(3.2)
Ω
Now, taking the upper limit in the equation
Z
|x|−ap |∇un |p−2 ∇un − |∇u|p−2 ∇u ∇(un − u)dx
Ω
Z
= hI 0 (un , vn ), (un − u, 0)i − |x|−ap |∇u|p−2 ∇u∇(un − u)dx
Ω
Z
Z
−β1 θ−1 δ
+ λθ |x|
un vn (un − u)dx + µα |x|−β2 unα−1 vnγ (un − u)dx .
Ω
Ω
Using the definition of (P S)c -sequence, the weak convergence, (3.1), and (3.2), we
obtain
Z
lim sup |x|−ap |∇un |p−2 ∇un − |∇u|p−2 ∇u ∇(un − u)dx = 0.
n→∞
Ω
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
9
Consequently, by a well known lemma (see e.g. [20, lemma 4.1]) we achieve, up to
a subsequence, that un → u strongly in W01,p (Ω, |x|−ap ) as n → ∞. Analogously,
we get vn → v strongly in W01,q (Ω, |x|−bq ) as n → ∞.
Proof of theorem 1.1. By combining lemmata 2.4 and 2.5, there exists a (P S)c sequence {(un , vn )} in W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) with un , vn ≥ 0, for a.e. in
Ω. Moreover, from lemma 3.1 there exist (u, v) ∈ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq )
and a subsequence of {(un , vn )}, that we will denote by {(un , vn )}, such that un → u
strongly in W01,p (Ω, |x|−ap ) and vn → v strongly in W01,q (Ω, |x|−bq ), as n → ∞.
Then, we conclude that
I(u, v) = c > 0
and I 0 (u, v) = 0,
that is, (u, v) is a nonnegative weak solution of system (1.2). Moreover, it is easy
to check that u, v 6≡ 0.
4. Proof of theorem 1.2
First of all, notice that by lemma 2.4 the geometric conditions of the mountain
pass theorem for the functional I are satisfied.
The next three lemmata are crucial in the proof of this theorem.
Lemma 4.1. Let {(un , vn )} ⊂ (W01,p (Ω, |x|−ap ))2 be a bounded (P S)c -sequence
such that un , vn ≥ 0, for a.e. in Ω, and there exists (u, v) ∈ (W01,p (Ω, |x|−ap ))2
satisfying un * u and vn * v weakly in W01,p (Ω, |x|−ap ), as n → ∞. Then, (u, v)
is a weak solution of system (1.2) and u, v ≥ 0 for a.e. in Ω.
Proof. Arguing as in the proof of lemma 3.1, by combining the compact embedding
theorem [35, Theorem 2.1] with the Lebesgue’s dominated convergence theorem,
we obtain that u, v ≥ 0 for a.e. in Ω,
Z
Z
−β1 θ−1 δ
|x|
un vn wdx = |x|−β1 uθ−1 v δ wdx, ∀w ∈ W01,p (Ω, |x|−ap ), (4.1)
lim
n→∞ Ω
Ω
and
Z
lim
n→∞ Ω
|x|−β1 uθn vnδ−1 zdx
Z
=
|x|−β1 uθ v δ−1 zdx,
∀z ∈ W01,p (Ω, |x|−ap ).
(4.2)
Ω
Notice that ∇un (x) → ∇u(x) and ∇vn (x) → ∇v(x), for a.e. x ∈ Ω, as n → ∞.
These facts can be proved arguing as in [9] (see also [8, 20, 22]).
Since {(un , vn )} is bounded in (W01,p (Ω, |x|−ap ))2 , we have {|∇un |p−2 ∇un } and
p
{|∇vn |p−2 ∇vn } are bounded in (L p−1 (Ω, |x|−ap ))N . On the other hand, since α +
γ = p∗ , by the Hölder’s inequality, we infer that {un α−1 vn γ } and {un α vn γ−1 } are
p∗
∗
bounded in L p∗ −1 (Ω, |x|−e1 p ). Therefore, by lemma 2.3 we get
∇un * ∇u
and ∇vn * ∇v
p
weakly in (L p−1 (Ω, |x|−ap ))N
(4.3)
and
γ−1
uα
* uα v γ−1 ,
n vn
uα−1
vnγ * uα−1 v γ
n
p∗
∗
weakly in L p∗ −1 (Ω, |x|−c1 p ),
(4.4)
as n → ∞. Consequently, using (4.1) − (4.4) we obtain
hI 0 (u, v), (w, z)i = lim hI 0 (un , vn ), (w, z)i = 0,
n→∞
that is, (u, v) is a weak solution of system (1.2).
∀(w, z) ∈ (W01,p (Ω, |x|−ap ))2 ,
10
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
Lemma 4.2. In addition to (1.3), (1.4), and (1.6), assume that p = q, 0 ≤ a = b,
p1 = q1 ∈ (p, p∗ ), with θ/p1 + δ/q1 = 1, p∗ = q ∗ , and β2 = c1 p∗ . Then, all the
Palais Smale sequences {(un , vn )} ⊂ (W01,p (Ω, |x|−ap ))2 for the operator I at the
level c, with un , vn ≥ 0 for a.e. in Ω, are precompact provided that
−p
p∗
1
1
c < ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p + K(λ),
p p
(4.5)
where
1
1
K(λ) = λp1 ( − ) lim
p p1 n→∞
Z
|x|−β1 uθn vnδ dx.
Ω
Proof. By Lemma 2.5 the sequence {(un , vn )} is bounded in (W01,p (Ω, |x|−ap ))2 ;
consequently, there exists (u, v) ∈ (W01,p (Ω, |x|−ap ))2 such that un * u and vn * v
weakly in W01,p (Ω, |x|−ap ), as n → ∞. Then, by combining the compact embedding
theorem [35, Theorem 2.1] with the Lebesgue’s dominated convergence theorem,
we infer that un (x) → u(x), vn (x) → v(x), for a.e. in Ω, as n → ∞, and
Z
Z
−β1 θ δ
|x|
lim
un vn dx = |x|−β1 uθ v δ dx.
(4.6)
n→∞ Ω
Ω
Moreover, as in Lemma 4.1 we can suppose that ∇un (x) → ∇u(x) and ∇vn (x) →
∇v(x), for a.e. x ∈ Ω, as n → ∞.
Define ũn = un − u and ṽn = vn − v. By Brezis and Lieb [10, Theorem 1] we
have
(i) kun kp = kũn kp + kukp + On (1), as n → ∞.
(ii) kvn kp = kṽn kp + kvkp + On (1), as n → ∞.
Z
Z
(iii)
∗
∗
|x|−c1 p |un |α |vn |γ dx −
|x|−c1 p |ũn |α |ṽn |γ dx
Ω
Ω
Z
∗
=
|x|−c1 p |u|α |v|γ dx + On (1), as n → ∞.
Ω
We recall that the proof of identity iii. follows arguing as in [27, Lemma 8].
By Lemma 4.1 we have that (u, v) is a weak solution of system (1.2), that is,
hI 0 (u, v), (w, z)i = 0 for all (w, z) ∈ (W01,p (RN , |x|−ap ))2 . By using (4.6) and (i)–
(iii), we get
Z
∗
p
kũn k − µα |x|−c1 p |ũn |α |ṽn |γ dx
Ω
Z
hZ
i
∗
p
p
−c1 p∗ α γ
= kun k − kuk − µα
|x|
un vn dx − |x|−c1 p uα v γ dx + On (1)
Ω
Ω
= hI 0 (un , vn ), (un , 0)i − hI 0 (u, v), (u, 0)i + On (1)
= On (1),
as n → ∞.
Analogously, we obtain
kṽn kp − µγ
Z
∗
|x|−c1 p |ũn |α |ṽn |γ dx = On (1).
Ω
Thus, we can take l ≥ 0 such that
Z
∗
kũn kp
kṽn kp
l = lim
= lim
= µ lim
|x|−c1 p |ũn |α |ṽn |γ dx.
n→∞
n→∞ Ω
n→∞
α
γ
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
11
If l = 0 the result is proved. Suppose by contradiction that l > 0. By the definition
of (P S)c -sequence we get
c + On (1)
1 0
hI (un , vn ), (un , vn )i
p1
Z
∗
1
1
α+γ
γ
= ( − )(kun kp + kvn kp ) + µ(
− 1) |x|−c1 p uα
n vn dx
p p1
p1
Ω
1
1
1
1
p
p
p
= ( − )(kũn k + kṽn k ) + ( − )(kuk + kvkp )
p p1
p p1
Z
hZ
i
∗
p∗
−c1 p∗ α γ
γ
+ µ( − 1)
|x|
ũn ṽn dx + |x|−c1 p uα
n vn dx + On (1)
p1
Ω
Ω
1 ∗
1
1
1
p
= ( − )p l + ( − )(kuk + kvkp )
p p1
p p1
Z
i
∗
1
1 ∗h
γ
+ ( − ∗ )p l + µ |x|−c1 p uα
n vn dx + On (1)
p1
p
Ω
Z
Z
i
∗
1
1 ∗
1
1 h
= ( − ∗ )p l + ( − ) λp1 |x|−β1 uθ v δ dx + µp∗ |x|−c1 p uα v γ dx
p p
p p1
Ω
Ω
Z
1
1 ∗
−c1 p∗ α γ
|x|
un vn dx + On (1)
+ µ( − ∗ )p
p1
p
Ω
Z
1
1 ∗
1
1
= ( − ∗ )p l + λp1 ( − ) |x|−β1 uθ v δ dx
p p
p p1 Ω
Z
∗
1
1
γ
+ µ( − ∗ )p∗ |x|−c1 p uα
n vn dx + On (1)
p p
Ω
Z
1
1 ∗
1
1
≥ ( − ∗ )p l + λp1 ( − ) |x|−β1 uθ v δ dx + On (1).
p p
p p1 Ω
= I(un , vn ) −
(4.7)
Using the definition of S̃ we have
Z
p/p∗
∗
γ
|x|−c1 p uα
S̃ ≤ kun kp + kvn kp , ∀n.
n vn dx
Ω
Hence, taking the limit in the above inequality we get
l p/p∗
S̃ ≤ (α + γ)l = p∗ l
µ
then
−p
−p∗
p∗
l ≥ (µ) p∗ −p (p∗ ) p∗ −p S̃ p∗ −p .
(4.8)
Substituting (4.8) in (4.7) and taking the limit, we obtain
−p
p∗
1
1
c ≥ ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p + K(λ),
p p
which contradicts the inequality (4.5).
Lemma 4.3. We can choose e in (2.6) such that c given by (2.5) satisfies
−p
p∗
1
1
c < ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p .
p p
(4.9)
12
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
−1
−1
γ p∗
γ p∗
Proof. Let us consider s0 = s1 (sα
and t0 = t1 (sα
, where s1 , t1 > 0 and
1 t1 )
1 t1 )
1/p
s1 /t1 = (α/γ) , and u the function defined in lemma 2.2. Then, it is suffices to
prove that there exists > 0 such that
−p
p∗
1
1
supI(t(s0 u ), t(t0 u )) < ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p .
p p
t≥0
Due to the geometric conditions of the mountain pass theorem, for each > 0,
there exists t > 0 such that
0 < σ ≤ supI(t(s0 u ), t(t0 u )) = I(t (s0 u ), t (t0 u )).
t≥0
Moreover, supposing by contradiction that there exists a subsequence {tn } with
tn → 0 as n → ∞, we obtain
0 < σ ≤ I(tn (s0 un ), tn (t0 un ))
tpn tp0
tpn sp0
p
kun k +
kun kp
≤
p
p
tp
≤ n (sp0 + tp0 )(S̃a,p,R + O(n(N −d1 p)/d1 p )) → 0
p
as n → ∞,
which is an absurd. Then, there exists l > 0 with t ≥ l, for all > 0. Consequently,
by using lemma 2.2 and putting c0 = lp1 sθ0 tδ0 , we get
supI(t(s0 u ), t(t0 u )) ≤
t≥0
tp sp1 + tp1
p
− λc0
∗ ku k
p/p
p (sα tγ )
1 1
Z
∗
|x|−β1 up 1 dx − µtp . (4.10)
Ω
Note that
−1
t1 = (µp∗ ) p∗ −p
sp + tp p∗1−p
p
1
1
ku k p∗ −p
γ p/p∗
α
(s1 t1 )
(4.11)
is the unique maximum point of f : (0, ∞) → R, given by
f (t) =
(sp1 + tp1 ) tp
p
p∗
γ p/p∗ ku k − µt .
α
(s1 t1 )
p
Also we know that
(A + B)k ≤ Ak + k(A + B)k−1 B,
(4.12)
for all A, B ≥ 0 and k ≥ 1 [26]. Observe that the following identity holds
i
sp1 + tp1 h
∗
γ/p∗
+ (α/γ)−α/p .
γ p/p∗ = (α/γ)
α
(s1 t1 )
(4.13)
1,p
By the Caffarelli-Kohn-Nirenberg’s inequality, W01,p (Ω, |x|−ap ) ⊂ Wa,c
(RN ). Then
1
∗
S̃a,p ≤ Ca,p
.
(4.14)
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
13
Substituting (4.11) in (4.10), from (4.12), (4.13), (4.14), and using lemma 2.2, we
obtain
n α
−p
∗
∗
1
1
α
supI(t(s0 u ), t(t0 u )) ≤ ( − ∗ )(µp∗ ) p∗ −p [( )γ/p + ( )−α/p ]S̃a,p,R
p p
γ
γ
t≥0
Z
p∗
o
N −d1 p
p∗ −p
− λc0 |x|−β1 up 1 dx
+ O( d1 p )
Ω
p∗
n α γ
o p∗ −p
−p
α −α
1
1
≤ ( − ∗ )(µp∗ ) p∗ −p [( ) p∗ + ( ) p∗ ]S̃a,p
p p
γ
γ
Z
N −d1 p
+ O( d1 p ) − λc0 |x|−β1 up 1 dx
(4.15)
Ω
p∗
n α γ
−p
1
1
α −α ∗ o p∗ −p
≤ ( − ∗ )(µp∗ ) p∗ −p [( ) p∗ + ( ) p∗ ]Ca,p
p p
γ
γ
Z
N −d1 p
+ O( d1 p ) − λc0 |x|−β1 up 1 dx
Ω
Now, from lemma 2.1 and (4.15), we get
N −d1 p
−p
p∗
1
1
supI(t(s0 u ), t(t0 u )) ≤ ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p + O( d1 p )
p p
t≥0
Z
− λc0 |x|−β1 up 1 dx.
(4.16)
Ω
Supposing that c <
(p1 −p+1)N −(a+1)p1
p−1
−
(N −p−ap)(p1 −p)
p(p−1)
we have
(N − p1 + ap1 + c)(p − 1)(N − d1 p) (N − d1 p)(p − 1)p1
N − d1 p
−
,
<
2
d1 p(N − p − ap)
d1 p
d1 p
then, by lemma 2.2 and by (4.16), we can take a > 0 small enough such that
N −d1 p
−p
p∗
1
1
supI(t(s0 u ), t(t0 u )) ≤ ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p + O( d1 p )
p p
t≥0
(N −d1 p)(p−1)(N −p1 −ap1 +c)
d1 p(N −p−ap)
− O(
−p
p∗
1
1
< ( − ∗ )(µp∗ ) p∗ −p S̃ p∗ −p .
p p
−
(N −d1 p)(p−1)p1
d1 p2
)
This completes the proof.
Proof of theorem 1.2. From lemmata 2.4, 2.5, and 4.3, there exists a bounded
(P S)c -sequence {(un , vn )} in (W01,p (Ω, |x|−ap ))2 with c > 0 satisfying (4.9) and
un , vn ≥ 0 for a.e. in Ω. Since that p1 ∈ (p, p∗ ), it follows that c verifies (4.5).
Thus, we have by lemma 4.2 that there exists (u, v) ∈ (W01,p (Ω, |x|−ap ))2 with
un → u and vn → v in W01,p (Ω, |x|−ap ), as n → ∞. Hence, we conclude
I(u, v) = c > 0
and I 0 (u, v) = 0,
that is, (u, v) is a nontrivial and nonnegative weak solution of system (1.2).
14
O. H. MIYAGAKI, R. S. RODRIGUES
EJDE-2008/115
5. Proof of theorem 1.3
The proof follows the steps the proof of theorem 1.2. By lemmata 2.4 and 2.5,
there exists a (P S)c -sequence {(un , vn )} in W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) with
c > 0 given as in (2.5) and un , vn ≥ 0 for a.e. in Ω. Moreover, {(un , vn )} is bounded
uniformly in µ > 0, that is, there exist M > 0 such that k(un , vn )k ≤ M for all
n ∈ N, uniformly in µ > 0. Consequently, we get that c ≤ M uniformly in µ > 0.
Due to the boundedness of {(un , vn )}, there exists a subsequence, that we will
denote by {(un , vn )}, and (u, v) ∈ W01,p (Ω, |x|−ap ) × W01,q (Ω, |x|−bq ) with un * u
weakly in W01,p (Ω, |x|−ap ) and vn * v weakly in W01,q (Ω, |x|−bq ), as n → ∞. Then,
arguing as in lemma 4.1 we obtain that (u, v) is a weak solution of system (1.2)
with u, v ≥ 0 for a.e. in Ω.
Now, we will prove that there exists µ0 > 0 such that u, v is nontrivial, provided
that 0 < µ < µ0 .
Supposing by contradiction that u(x) ≡ 0 for a.e. x ∈ Ω and proceeding as in
the proof of theorem 1.2, we obtain l > 0 such that
Z
∗
kvn kq
kun kp
γ
= lim
= µ lim
|x|−c1 p uα
l = lim
n vn dx
n→∞
n→∞ Ω
n→∞
α
γ
and
α γ
+ − 1)l > 0.
(5.1)
n→∞
p
q
∗
∗
and Cb,q
, we obtain
On the other hand, by Young’s inequality and definitions of Ca,p
c = lim I(un , vn ) = (
∗
∗
l
α(p +p)/p ∗ −p∗ /p p∗ /p γ (q +q)/q ∗ −q∗ /q q∗ /q
≤
(Ca,p )
l
+
(Cb,q )
l
µ
p∗
q∗
∗
∗
α(p +p)/p ∗ −p∗ /p γ (q +q)/q ∗ −q∗ /q τ
≤
(Ca,p )
+
(Cb,q )
l ,
p∗
q∗
where τ = max{p∗ /p, q ∗ /q} if l > 1, and τ = min{p∗ /p, q ∗ /q} if l ≤ 1. Therefore,
∗
−1
h α(p∗ +p)/p
γ (q +q)/q ∗ −q∗ /q i τ −1
∗ −p∗ /p
l≥ µ
(C
)
+
(C
)
.
a,p
b,q
p∗
q∗
Thus substituting the above inequality in (5.1) and taking µ0 > 0 small enough we
conclude
∗
−1
h α(p∗ +p)/p
α γ
γ (q +q)/q ∗ −q∗ /q i τ −1
∗ −p∗ /p
c ≥ ( + − 1) µ
(C
)
+
(C
)
≥ M,
a,p
b,q
p
q
p∗
q∗
for all 0 < µ < µ0 , which is an absurd.
6. Appendix
Proof of lemma 2.2. From equation (2.1) we obtain
∗
k∇y kpLp (RN ,|x|−ap ) = (S̃a,p,R )p
∗
∗
ky kpLp∗ (RN ,|x|−c1 p∗ ) = (S̃a,p,R )p
/(p∗ −p)
= (ka,p ())p k∇Ua,p, kpLp (RN ,|x|−ap ) ,
/(p∗ −p)
= (ka,p ())p kUa,p, kpLp∗ (RN ,|x|−c1 p∗ ) .
∗
∗
We observe that


if |x| < R0
∇Ua,p, (x)
∇(ψ(x)Ua,p, (x)) = Ua,p, (x)∇ψ(x) + ψ(x)∇Ua,p, (x) if R0 ≤ |x| < 2R0


0
if |x| ≥ 2R0
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
15
and
∇Ua,p, (x) = −
Therefore,
Z
−ap
|x|
N − p − ap
|x|[d1 p(N −p−ap)/(p−1)(N −d1 p)]−2 x
·
·
p−1
( + |x|d1 p(N −p−ap)/(p−1)(N −d1 p) )N/d1 p
Z
p
|x|−ap |∇Ua,p, (x)|p dx
|∇(ψUa,p, )(x)| dx = O(1) +
RN
Ω
p∗
= O(1) + (S̃a,p,R ) p∗ −p (ka,p ())−p
and
Z
∗
∗
|x|−c1 p |ψ(x)Ua,p, (x)|p dx = O(1) +
Z
∗
∗
|x|−c1 p |Ua,p, (x)|p dx
RN
Ω
p∗
∗
= O(1) + (S̃a,p,R ) p∗ −p (ka,p ())−p .
Consequently,
Z
|x|−ap |∇u (x)|p dx =
∗
∗
O(1) + (S̃a,p,R )p /(p −p) (ka,p ())−p
[O(1) + (S̃a,p,R )p∗ /(p∗ −p) (ka,p ())−p∗ ]p/p∗
Ω
∗
∗
(ka,p ())p [O(1) + (S̃a,p,R )p /(p −p) (ka,p ())−p ]
[O(ka,p ()p∗ ) + (S̃a,p,R )p∗ /(p∗ −p) ]p/p∗
=
≤ S̃a,p,R + O(ka,p ()p )
= S̃a,p,R + O((N −d1 p)/d1 p ).
Now, we prove that ku kpL1p1 (Ω,|x|−β1 ) is as in (2.2). Considering the changes of
variables by the polar coordinates and s = R0−1 −1/α r with α =
obtain
Z
|x|−(a+1)p1 +c |ψUa,p, |p1 dx
Ω
Z
≥ O(1) +
|x|−(a+1)p1 +c |Ua,p, |p1 dx
d1 p(N −p−ap)
(p−1)(N −d1 p) ,
we
|x|<R0
Z
= O(1) + ωN
0
R0
r−(a+1)p1 +c+N −1
dr
( + rα )(N −d1 p)p1 /d1 p
−(N −d1 p)p1
+
(6.1)
(N −p1 −ap1 +c)(p−1)(N −d1 p)
d1 p
d1 p(N −p−ap)
= O(1) + ωN (R0α )
Z −1/α −(a+1)p1 +c+N −1
s
×
ds.
−α
α (N −d1 p)p1 /d1 p
(R
0
0 +s )
Assuming that c = [(p1 − p + 1)N − (a + 1)p1 ]/(p − 1), we see that
−(N − d1 p)p1
(p − 1)(N − p1 − ap1 + c)(N − d1 p)
+
= 0,
d1 p
d1 p(N − p − ap)
(N − p − ap)p1
−(a + 1)p1 + c + N −
= 0·
(p − 1)
Therefore, by (6.1),
Z
Z
−(a+1)p1 +c
p1
|x|
|ψUa,p, | dx ≥ ωN
Ω
1
−1/α −(a+1)p1 +c+N −1−α
(N −d1 p)p1
d1 p
s
ds
[(R0 s)−α + 1](N −d1 p)p1 /d1 p
16
O. H. MIYAGAKI, R. S. RODRIGUES
≥
R0−α
EJDE-2008/115
ωN
(N −d1 p)p1 /d1 p | ln()| = O(| ln()|).
+1
Hence, we obtain
Z
O(| ln()|)
|x|−(a+1)p1 +c |u |p1 dx ≥ p1 /p∗
p
Ω
O(1) + (Sa,p,R ) ∗ /(p∗ −p) (ka,p ())−p∗
O(| ln()|)
=
p1 /p∗
−p
(ka,p ()) 1 O(ka,p ()p∗ ) + (Sa,p,R )p∗ /(p∗ −p)
2
≥ O((N −d1 p)p1 /d1 p | ln()|).
Assuming that c > [(p1 − p + 1)N − (a + 1)p1 ]/(p − 1), we have
−(N − d1 p)p1
(N − p1 − ap1 + c)(p − 1)(N − d1 p)
+
> 0,
d1 p
d1 p(N − p − ap)
(N − p − ap)p1
−(a + 1)p1 + c + N −
> 0.
(p − 1)
Consequently, by (6.1),
Z
|x|−(a+1)p1 +c |ψUa,p, |p1 dx
Ω
≥ O(1) + ωN (R0α )
×
−(N −d1 p)p1
d1 p
1
(R0−α
+
1)(N −d1 p)p1 /d1 p
Hence, we get
Z
|x|−(a+1)p1 +c |u |p1 dx ≥
Ω
≥
+
(N −p1 −ap1 +c)(p−1)(N −d1 p)
d1 p(N −p−ap)
Z
1
s−(a+1)p1 +c+N −1 ds ≥ O(1).
1/2
O(1)
(ka,p ())−p1 [O(ka,p ()p∗ ) + (Sa,p,R )p∗ /(p∗ −p) ]p1 /p∗
O(ka,p ()p1 )
[O(1) + (Sa,p,R )p∗ /(p∗ −p) ]p1 /p∗
2
≥ O((N −d1 p)p1 /d1 p ).
If c < [(p1 − p + 1)N − (a + 1)p1 ]/(p − 1), we see that
−(N − d1 p)p1
(N − p1 − ap1 + c)(p − 1)(N − d1 p)
+
< 0,
d1 p
d1 p(N − p − ap)
(N − p − ap)p1
−(a + 1)p1 + c + N −
< 0.
(p − 1)
Using (6.1) we obtain
Z
|x|−(a+1)p1 +c |ψU |p1 dx
Ω
≥ O(1)
−(N −d1 p)p1
d1 p
+
(N −p1 −ap1 +c)(p−1)(N −d1 p)
d1 p(N −p−ap)
Z
1
1/2
≥O −(N −d1 p)p1
d1 p
+
(N −p1 −ap1 +c)(p−1)(N −d1 p)
d1 p(N −p−ap)
.
s−(a+1)p1 +c+N −1 ds
EJDE-2008/115
ON THE EXISTENCE OF WEAK SOLUTIONS
17
Hence, we conclude
Z
−(a+1)p1 +c
|x|
Ω
p1
O −(N −d1 p)p1
d1 p
+
(N −p1 −ap1 +c)(p−1)(N −d1 p)
d1 p(N −p−ap)
|u | dx ≥ p1 /p∗
O(1) + (Sa,p,R )p∗ /(p∗ −p) (ka,p ())−p∗
(N −p1 −ap1 +c)(p−1)(N −d1 p) (N −d1 p)(p−1)p1 −
d1 p(N −p−ap)
d1 p2
.
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Olimpio H. Miyagaki
Departamento de Matemática, Universidade Federal de Viçosa, 36571-000, Viçosa, MG,
Brazil
E-mail address: olimpio@ufv.br
Rodrigo S. Rodrigues
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905, São Carlos, SP, Brazil
E-mail address: rodrigosrodrigues@ig.com.br