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Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities
J. Math. Anal. Appl. 396 (2012) 280–293
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Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities Nemat Nyamoradi Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran
article
abstract
info
Article history: Received 5 July 2011 Available online 23 June 2012 Submitted by Manuel del Pino
This paper is concerned with a singular elliptic system, which involves the Caffarelli– Kohn–Nirenberg inequality and critical Sobolev–Hardy exponents. The existence and multiplicity results of positive solutions are obtained by variational methods. © 2012 Elsevier Inc. All rights reserved.
Keywords: Nehari manifold Critical Hardy–Sobolev exponent Elliptic system Multiple positive solutions Concave–convex nonlinearities
1. Introduction The aim of this paper is to establish the existence of nontrivial solutions to the following quasilinear elliptic system
∗ |u|p (t )−2 u ηα |u|α−2 |v|β u |u|q−2 u |u|p−2 u = + +λ , − ∆ u − µ p p t t | x| |x| α+β |x| |x|s ∗ |v|p−2 v |v|p (t )−2 v ηβ |u|α |v|β−2 v |v|q−2 v − ∆ v − µ = + + θ , p |x|p |x|t α+β |x|t |x|s u = v = 0, where ∆p u = div(|∇ u|
(1)
x ∈ Ω, x ∈ ∂Ω
∇ u), 0 ∈ Ω is a bounded domain in R (N ≥ 3) with a smooth boundary ∂ Ω , λ > p p > 0, 0 < η < ∞, 1 < p < N , 0 ≤ µ < µ , N − , 0 ≤ s, t < p, 1 ≤ q < p, p p−2
0, θ
x ∈ Ω,
α + β = p∗ (t ) ,
p(N −t ) N −p
N
is the Hardy–Sobolev critical exponent.
1,p
We denote by D0 (Ω ) the completion of C0∞ (Ω ) with respect to the norm
Ω
|∇ · |p dx
1p
.
Problem (1) is related to the well known Caffarelli–Kohn–Nirenberg inequality in [1]:
pr |u|r dx ≤ Cr ,t ,p |∇ u|p dx, for all u ∈ D10,p (Ω ), t | x | Ω Ω where p ≤ r < p∗ (t ). If t = r = p, the above inequality becomes the well known Hardy inequality [1–3]: |u|p 1 dx ≤ |∇ u|p dx, for all u ∈ D10,p (Ω ). p µ Ω Ω | x|
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.06.026
(2)
(3)
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
281
1,p
In the space D0 (Ω ) we employ the following norm:
∥u∥ = ∥u∥D1,p (Ω ) :=
Ω
0
|∇ u|p − µ
|u|p |x|p
1p
dx
,
µ ∈ [0, µ).
By using the Hardy inequality (3) this norm is equivalent to the usual norm
|·|p−2
L := |∇ · |p−2 ∇ · −µ |x|p
Ω
1p |∇ u|p dx . The elliptic operator
1,p
is positive in D0 (Ω ) if 0 ≤ µ < µ. 1 ,p
1 ,p
Now, we define the space W = D0 (Ω ) × D0 (Ω ) with the norm
∥(u, v)∥p = ∥u∥p + ∥v∥p . Also, by the Hardy inequality and the Hardy–Sobolev inequality, for 0 ≤ µ < µ, 0 ≤ t < p and p ≤ r ≤ p∗ (t ) we can define the best Hardy–Sobolev constant: Aµ,t ,r (Ω ) =
inf
1,p u∈D0 (Ω )\{0}
p |∇ u|p − µ ||ux||p dx Ω . pr r |u| Ω | x| t
dx
In the important case when r = p∗ (t ), we simply denote Aµ,t ,p∗ (t ) as Aµ,t . Note that Aµ,0 is the best constant in the Sobolev inequality, namely,
p |∇ u|p − µ ||ux||p dx Ω . Aµ,0 (Ω ) = inf p 1,p p∗ dx p∗ u∈D0 (Ω )\{0} | u | Ω For any 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ), by (2), (3), 0 ≤ t < p and the Young inequality, the following best constants are well defined S (µ) :=
inf
1,p u∈D0 (Ω )\{0}
|u|p p dx |∇ u | − µ p Ω | x| = Aµ,t p ∗ p∗ (t ) |u| |x|t
Ω
S˜α,β (µ) :=
Ω
inf
(u,v)∈W \{(0,0)}
p (t )
dx
p
|∇ u|p + |∇ u|p − µ |u| |x+|v| |p p ∗ α β |u| |v| Ω |x|t
dx
p
dx (4)
p (t )
Sη,α,β (µ) :=
p p |∇ u|p + |∇v|p − µ |u| |x+|v| dx |p p ∗ . ∗ ∗ p (t ) |u|p (t ) +|v|p (t ) +η|u|α |v|β Ω
inf
(u,v)∈W \{(0,0)}
Ω
(5)
dx
|x|t
Throughout this paper, let R0 be the positive constant such that Ω ⊂ B(0; R0 ), where B(0; R0 ) = {x ∈ RN : |x| < R0 }. By 1,p Holder and Sobolev–Hardy inequalities, for all u ∈ W0 (Ω ), we obtain
Ω
|u|q ≤ |x|s
|x|−s
p∗ (∗s)−q
B(0;R0 ) R0
≤
Ω
r N −s+1 dr
q p∗p(∗s()− s)
|u|p (s) |x|s ∗
p (s)
q p∗ (s)
− pq
Aµ,s ∥u∥q
0
≤ where ωN =
N ωN RN0 −s N −s
p∗ (∗s)−q p (s)
− pq
Aµ,s ∥u∥q ,
(6)
N
2π 2 NΓ ( N ) 2
is the volume of the unit ball in RN .
The existence of nontrivial non-negative solutions for elliptic equations with singular potentials has been recently studied by several authors, but, essentially, only with a solely critical exponent. We refer, e.g., in bounded domains and for p = 2
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N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
to [4,3,5,6], and for general p > 1 to [7–11] and the references therein. For example, Kang in [11] studied the following elliptic equation via the generalized Mountain-Pass Theorem [12]:
|u|p−2 u |u|p (t )−2 u |u|p−2 u = +λ , p t | x| |x| |x|s ∗
−∆p u − µ u = 0,
x ∈ Ω,
(7)
x ∈ ∂Ω
where Ω ⊂ RN is a bounded domain, 1 < p < N , 0 ≤ s, t < p and 0 ≤ µ < µ , Mountain-Pass Theorem of Ambrosetti and Rabinowitz [14], proved that
−∆ p u − µ
up−1
|x|p
∗ −1
= |u|p
+
∗ up (s)−1
,
|x|s
N −p p p
. Also, the authors in [13] via the
in RN
N −p p
admits a positive solution in RN , whenever µ < µ , . p Also, in recent years, several authors have used the Nehari manifold to solve semilinear and quasilinear problems (see [15–22] and references therein). Brown and Zhang [23] have studied a subcritical semi-linear elliptic equation with a sign-changing weight function and a bifurcation real parameter in the case p = 2 and Dirichlet boundary conditions. In [24] the author studied Eq. (7) via the Nehari manifold. Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ (tu) where Jλ is the Euler function associated with the equation), they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter λ crosses the bifurcation value. In this work, we give a variational method which is similar to the fibering method (see [17,23,24]) to prove the existence and multiplicity of nontrivial non-negative solutions of problem (1). Now, we will begin to state the main theorem. First, we define the following constants
p−q
C =
p p∗ (t )−p
(1 + η)(p∗ (t ) − q) p−p q q C0 = C.
p∗ (t ) − p
p−p q
N ωN R0N −s
p∗ (t ) − q
− p(∗p∗ (s)−q)
N −s
p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
,
p
Then, we have the following result. Theorem 1. Assume that 0 ≤ s, t < p, N ≥ 3, 0 ≤ µ < µ and 1 ≤ q < p. Then we have the following results. p
p
(i) If λ, θ > 0 satisfy λ p−q + θ p−q < C , then (1) has at least one positive solution in W . p
p
(ii) If λ, θ > 0 satisfy 0 < λ p−q + θ p−q < C0 , then (1) has at least two positive solutions in W . This paper is divided into three sections, organized as follows. In Section 2, we establish some elementary results. Finally, in Section 3, we prove our main result (Theorem 1). 2. Notations and preliminaries First, we conclude the relationship between S (µ) and Sη,α,β (µ). To continue, we recall a recent result on the extremal functions of Aµ,t [25]. Lemma 1 ([25]). Assume that 1 < p < N , 0 ≤ t < p and 0 ≤ µ < µ. Then the limiting problem
∗ p−1 |u|p (t )−1 −∆ u − µ |u| = , p |x|p |x|t u ∈ D1,p (RN ), u > 0,
in RN \ {0}, in RN \ {0},
has positive radial ground states Vϵ (x) , ϵ
p−N p
Up,µ
x ϵ
=ϵ
p−N p
Up,µ
|x| , ϵ
∀ϵ > 0,
(8)
that satisfy
Ω
|∇ Vϵ (x)|p − µ
|Vϵ (x)|p |x|p
∗ (t )
dx = Ω
|Vϵ (x)|p |x|t
N −t
N −t
dx = (Aµ,t ) p−t = (S (µ)) p−t ,
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
283
where Up,µ (x) = Up,µ (|x|) is the unique radial solution of the limiting problem with
Up,µ (1) =
(N − t )(µ − µ) N −p
1 p∗ (t )−p
.
Furthermore, Up,µ have the following properties: lim r a(µ) Up,µ (r ) = C1 > 0,
r →0
lim r b(µ) Up,µ (r ) = C2 > 0,
r →+∞
lim r a(µ)+1 |Up′ ,µ (r )| = C1 a(µ) ≥ 0,
r →0
lim r b(µ)+1 |Up′ ,µ (r )| = C2 b(µ) > 0,
r →+∞
where Ci (i = 1, 2) are positive constants and a(µ) and b(µ) are zeros of the function f (ζ ) = (p − 1)ζ p − (N − p)ζ p−1 + µ,
ζ ≥ 0, 0 ≤ µ < µ,
that satisfy 0 ≤ a(µ) <
N −p
N −p
< b(µ) ≤
p
p−1
.
Now, for any α, β > 1 and α + β = p∗ (t ), we define g (τ ) :=
1 + τp
,
p
(1 + ητ β + τ α+β ) α+β
τ ≥ 0,
(9)
g (τmin ) := min g (τ ) > 0,
(10)
τ ≥0
where τmin > 0 is a minimal point of g (τ ) and therefore a root of the equation ∗ p∗ (t )τ p (t )−2 ηβτ β−2 − ηατ β − p∗ (t ) = 0,
τ ≥ 0.
(11)
Theorem 2. Suppose N ≥ 3, 0 ≤ t < p, 0 < η < ∞ and 0 ≤ µ < µ. Then: (i) Sη,α,β (µ) = g (τmin )S (µ); (ii) Sη,α,β (µ) has the minimizers (Vϵ (x), τmin Vϵ (x)), ∀ϵ > 0, where Vϵ (x) are the extremal functions of S (µ) defined as in (8). Proof. (i) Note that limτ →0+ g (τ ) = limτ →+∞ g (τ ) = 1. Thus minτ ≥0 g (τ ) must be achieved at finite τmin ≥ 0. Furthermore, direct calculation shows that there exists a positive constant C such that 0 < C ≤ g (τmin ) = min g (τ ) ≤ 1, τ ≥0
0 < τmin < ∞.
From the fact that g ′ (τmin ) = 0 we deduce that τmin is a root of Eq. (11). 1,p To continue, we follow the argument in [26]. Suppose 0 ≤ µ < µ and {wn } ⊂ D0 (Ω ) \ {0} is a minimizing sequence for S (µ). Let e1 , e2 > 0 be chosen later. Taking un = e1 wn and vn = e2 wn in (5) we have p e1
α+β
(e1
+
α β
|wn |p p |∇w | − µ dx n p Ω | x| ≥ Sη,α,β (µ). p ∗ p∗ (t )
p e2 p α+β α+β )
+ η e1 e2 + e2
|wn | Ω |x|t
dx
(12)
p (t )
Note that
g
e2 e1
p
=
p
e1 + e2 α+β
(e1
p α+β α+β )
β
+ ηeα1 e2 + e2
Choose e1 and e2 in (12) such that
e2 e1
.
= τmin . Passing to the limit as n → ∞ we have
g (τmin )S (µ) ≥ Sη,α,β (µ). Let {(un , vn )} be a minimizing sequence of Sη,α,β (µ) and define zn = sn vn , where
sn =
Ω
(13)
1 α+β −1 |vn |α+β |un |α+β dx dx . |x|t |x|t Ω
(14)
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N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
Then
Ω
|zn |α+β dx = |x|t
Ω
|un |α+β dx. |x|t
(15)
From the Young inequality it follows that
Ω
|un |α |zn |β α dx ≤ |x|t α+β
Ω
|un |α+β β dx + |x|t α+β
|zn |α+β dx. |x|t
Ω
By (15) we have
Ω
|un |α |zn |β dx ≤ |x|t
Ω
|un |α+β dx = |x|t
Ω
|zn |α+β dx. |x|t
(16)
Consequently,
p p p p n| n| |∇ un |p + |∇vn |p − µ |un | |x+|v |∇ un |p + |∇vn |p − µ |un | |x+|v dx dx Ω |p |p ≥ p p α+β α+β −β −(α+β) |un |α+β |un |α+β +|vn |α+β +η|un |α |vn |β dx dx ( 1 + η s + s ) n n Ω Ω |x|t | x| t p |un | p |∇ u | − µ dx n p Ω |x| ≥ p α+β −β −(α+β) |un |α+β (1 + ηsn + sn ) Ω |x|t dx
Ω
−p
sn
Ω
p
|∇ zn | −
p µ ||zxn||p
+ −β
−(α+β)
(1 + ηsn + sn
)
dx
|zn |α+β Ω |x|t
p α+β
dx
1 ≥ g ( s− n )S (µ) ≥ g (τmin )S (µ).
As n → ∞ we have Sη,α,β (µ) ≥ g (τmin )S (µ).
(17)
From (13) and (17) it follows that Sη,α,β (µ) = g (τmin )S (µ),
∀µ ∈ [0, µ).
(18)
(ii) From (5), (8) and (18) the desired result follows.
The corresponding energy functional of problem (1) is defined by Jλ,θ (u, v) =
1 p
∥(u, v)∥p −
1 p∗ (t )
B(u, v) −
1 q
Kλ,θ (u, v),
for each (u, v) ∈ W , where
∗ |u|α |v|β + |v|p (t ) dx + η dx, |x|t |x|t Ω Ω |u|q |v|q Kλ,θ (u, v) = λ dx + θ dx. s s Ω |x| Ω |x|
B(u, v) =
∗ (t )
|u|p
Then Jλ,θ ∈ C 1 (W , R). Now, we consider the problem on the Nehari manifold. Define the Nehari manifold (cf. [27]): ′ Nλ,θ = (u, v) ∈ W \ {(0, 0)}|⟨Jλ,θ (u, v), (u, v)⟩ = 0 ,
where ′ ⟨Jλ,θ (u, v), (u, v)⟩ = ∥(u, v)∥p − B(u, v) − Kλ,θ (u, v).
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
285
Note that Nλ,θ contains every nonzero solution of (1). Define ′ Φλ,θ (u, v) = ⟨Jλ,θ (u, v), (u, v)⟩,
then for (u, v) ∈ Nλ,θ ′ ⟨Φλ,θ (u, v), (u, v)⟩ = p∥(u, v)∥p − p∗ (t )B(u, v) − qKλ,θ (u, v)
(19)
= (p − q)∥(u, v)∥ − (p (t ) − q)B(u, v)
(20)
= (p − p (t ))∥(u, v)∥ − (q − p (t ))Kλ,θ (u, v).
(21)
∗
p
∗
∗
p
Now, we split Nλ,θ into three parts: + ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ > 0 ,
0 ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ = 0 ,
− ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ < 0 .
+ − 0 To state our main result, we now present some important properties of Nλ,θ , Nλ,θ and Nλ,θ .
Lemma 2. There exists a positive number C = C (p, q, N , S ) > 0 such that if p
p
0 < λ p−q + θ p−q < C , 0 then Nλ,θ = ∅.
Proof. Suppose otherwise, let
C =
p−q
p p∗ (t )−p
(1 + η)(p∗ (t ) − q)
p∗ (t ) − p
p−p q
p∗ (t ) − q
N ωN RN0 −s N −s
− p(∗p∗ (s)−q) p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
.
Then, there exists (λ, θ ) with p
p
0 < λ p−q + θ p−q < C , 0 0 such that Nλ,θ ̸= ∅. Then for (u, v) ∈ Nλ,θ , by (20) and (21) we have
∥(u, v)∥p =
(p∗ (t ) − q) B(u, v). p−q
Moreover, by the Young inequality and (6), it follows that
(p∗ (t ) − q) B(u, v) p−q ∗ ∗ ∗ ∗ p∗ (t ) − q |u|p (t ) + |v|p (t ) α |u|p (t ) β |v|p (t ) ≤ dx ∗ dx + ∗ dx p−q |x|t p (t ) Ω |x|t p (t ) Ω |x|t Ω
∥(u, v)∥p =
∗
(1 + η)(p∗ (t ) − q) − p p(t ) ∗ ≤ Aµ,t ∥(u, v)∥p (t ) . p−q It follows that
∥(u, v)∥ ≥
p−q
(1 + η)(p∗ (t ) − q)
p∗ (t ) p
p∗ (t1)−p ,
Aµ,t
and p∗ (t ) − p
|u|q |v|q ∥( u , v)∥ = K ( u , v) = λ f dx + θ g dx λ,θ s s p∗ ( t ) − q Ω |x| Ω |x| ∗ p (∗s)−q p (s) p p−p q p − pq N ωN RN0 −s Aµ,s λ p−q + θ p−q ≤ ∥(u, v)∥q . N −s p
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N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
Thus
∥(u, v)∥ ≤
p (t ) − q ∗
p∗ ( t )
p−1 p
Nω
−p
N −s N R0
p∗ (s)−q p∗ (s)(p−p)
− p(pq−p)
Aµ,s
N −s
p
p
λ p−q + θ p−q
1p
.
This implies p
p
λ p−q + θ p−q ≥ C . This is a contradiction! Here
C =
p−q
p p∗ (t )−p
(1 + η)(p∗ (t ) − q)
p∗ (t ) − p
p−p q
N ωN R0N −s
p∗ (t ) − q
− p(∗p∗ (s)−q)
p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
N −s
.
Lemma 3. The energy functional Jλ,θ is coercive and bounded below on Nλ,θ . Proof. If (u, v) ∈ Nλ,θ , then by (6), Jλ,θ (u, v) =
1 p
∥(u, v)∥p −
p (t ) − p
1 p∗ (t )
B(u, v) −
∗
≥
pp∗ (t )
∥(u, v)∥p −
1 q
Kλ,θ (u, v)
p (t ) − q ∗
Nω
p∗ (t )q
N −s N R0
p∗ (∗s)−q p (s)
N −s
p
p
λ p−q + θ p−q
p−p q
− pq
Aµ,s ∥(u, v)∥q .
Since 0 ≤ s, t < N , 1 < q < p < p∗ (t ), we see that Jλ,θ is coercive and bounded below on Nλ,θ .
Furthermore, similar to the argument in [23, Theorem 2.3] (or see [28]), we can conclude the following result. 0 ′ Lemma 4. Assume that (u0 , v0 ) is a local minimizer for Jλ,θ on Nλ,θ and that (u0 , v0 ) ̸∈ Nλ,θ , then Jλ,θ (u0 , v0 ) = 0 in W −1 .
Now, by Lemma 2, we let
p p ΘC0 = (λ, θ ) ∈ R2 \ {(0, 0)} : 0 < λ p−q + θ p−q < C0 , where C0 =
p−p q
ξλ,θ = + ξλ,θ = − ξλ,θ =
q p
+ − C < C . If (λ, θ ) ∈ ΘC0 , we have Nλ,θ = Nλ,θ ∪ Nλ,θ . Define
inf
Jλ,θ (u, v)
inf
Jλ,θ (u, v)
inf
Jλ,θ (u, v).
(u,v)∈Nλ,θ + (u,v)∈Nλ,θ − (u,v)∈Nλ,θ
Lemma 5. There exists a positive number C0 such that if (λ, θ ) ∈ ΘC0 , then + (i) ξλ,θ ≤ ξλ,θ < 0; − (ii) there exists d0 = d0 (p, q, N , K , S , λ, θ ) > 0 such that ξλ,θ > d0 . + Proof. (i) For (u, v) ∈ Nλ,θ , by (21), we have
Kλ,θ (u, v) ≥
p∗ (t ) − p p∗ (t ) − q
∥(u, v)∥p
and so Jλ,θ (u, v) =
1 p
1
−
1
p∗ (t ) 1
∥(u, v)∥ − p
1 q
1
−
1 p∗ (t ) 1
∥(u, v)∥p − − ∗ p q p (t ) p∗ (t ) − p 1 1 = − ∥(u, v)∥p < 0. p∗ (t ) p q ≤
−
p∗ (t )
Kλ,θ (u, v) p∗ (t ) − p p∗ (t ) − q
∥(u, v)∥p
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
287
+ + Thus, from the definition of ξλ,θ and ξλ,θ , we can deduce that ξλ,θ < ξλ,θ < 0. − (ii) For (u, v) ∈ Nλ,θ , by Lemma 2,
∥(u, v)∥ ≥
p∗ (t ) p
p−q
(1 + η)(p∗ (t ) − q)
p∗ (t1)−p
Aµ,t
.
Moreover, by Lemma 3, Jλ,θ (u, v) ≥
p∗ ( t ) − p pp∗ (t )
−
∥(u, v)∥p
p∗ (t ) − q
N ωN RN0 −s
p∗ (t )q
= ∥(u, v)∥q −
pp∗ (t )
p∗ (t )q
N ωN RN0 −s
−
p∗ (∗s)−q p (s)
N −s p∗ (t ) p
(1 + η)(p∗ (t ) − q) p∗ (t ) − q
p (s)
p∗ (t )q
p−p q − q p p p λ p−q + θ p−q Aµ,s ∥(u, v)∥q
∥(u, v)∥p−q
p−q
p∗ (∗s)−q
N −s
p∗ (t ) − p
p∗ (t ) − q
≥
p∗ (tq)−p
Aµ,t
N ωN RN0 −s
− pq Aµ,s
λ p−q + θ p−q
p∗ (t ) − p pp∗ (t )
p∗ (∗s)−q p (s)
N −s
− pq Aµ,s
p
p
p−p q
∥(u, v)∥p−q
p
p
λ p−q + θ p−q
p−p q
.
Thus, if p
p
0 < λ p−q + θ p−q < C0 , − we have then for each (u, v) ∈ Nλ,θ
Jλ,θ (u, v) ≥ d0 = d0 (p, q, N , K , S , λ, θ ) > 0. For each (u, v) ∈ W \ {(0, 0)} such that B(u, v) > 0, let
tmax =
(p − q)∥(u, v)∥p (p∗ (t ) − q)B(u, v)
1 p∗ (t )−p
.
Lemma 6. Assume that p
p
0 < λ p−q + θ p−q < C0 . Then, for every (u, v) ∈ W with B(u, v) > 0 there exists tmax > 0 such that there are unique t + and t − with 0 < t + < tmax < t − ± such that (t ± u, t ± v) ∈ Nλ,θ and Jλ,θ (t + u, t + v) =
inf
0≤t ≤tmax
Jλ,θ (tu, t v),
Jλ,θ (t − u, t − v) = sup Jλ,θ (tu, t v). t ≥tmax
Proof. The proof is similar to [18, Lemma 2.6] and is omitted.
Remark 1. If p
p
0 < λ p−q + θ p−q < C0 . Then, by Lemmas 5 and 6 for every (u, v) ∈ W with B(u, v) > 0, we can easily deduce that there exists tmax > 0 such that − and there are unique t − with tmax < t − such that (t − u, t − v) ∈ Nλ,θ − Jλ,θ (t − u, t − v) = sup Jλ,θ (tu, t v) ≥ ξλ,θ > 0. t ≥0
288
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
3. Proof of the main result We begin by stating some results which will be required later in the proof of Theorem 1. First, we get the following result. p
p
Lemma 7. (i) If 0 < λ p−q + θ p−q < C , then there exists a (PS )ξλ,θ -sequence {(un , vn )} ⊂ Nλ,θ in W for Jλ,θ . p
p
− (ii) If 0 < λ p−q + θ p−q < C0 , then there exists a (PS )ξ − -sequence {(un , vn )} ⊂ Nλ,θ in W for Jλ,θ , where C is the positive
constant given in Lemma 2, and C0 =
p−p q q p
λ,θ
C.
Proof. The proof is similar to [20, Proposition 9] and is omitted.
p
p
Theorem 3. Assume that 0 ≤ s, t < p, N ≥ 3, 0 ≤ µ < µ and 1 ≤ q < p. If 0 < λ p−q + θ p−q < C , then there exists + + (u+ 0 , v0 ) ∈ Nλ,θ such that + + (i) Jλ,θ (u+ 0 , v0 ) = ξλ,θ = ξλ,θ , + + (ii) (u0 , v0 ) is a positive solution of (1), + + + (iii) Jλ,θ (u+ 0 , v0 ) → 0 as λ → 0 , θ → 0 .
Proof. By Lemma 7, there exists a minimizing sequence {(un , vn )} for Jλ,θ on Nλ,θ such that Jλ,θ (un , vn ) = ξλ,θ + o(1) and
′ Jλ,θ (un , vn ) = o(1)
in W −1 .
(22)
Since Jλ,θ is coercive on Nλ,θ (see Lemma 3), we get {(un , vn )} bounded in W . Thus, there is a subsequence {(un , vn )} and + (u+ 0 , v0 ) ∈ W such that 1 ,p
+ u ⇀ u+ 0 , vn ⇀ v0 , n + un ⇀ u0 , vn ⇀ v0+ , + u → u+ 0 , vn → v0 , n + un → u+ , v → v n 0 0 ,
weakly in D0 (Ω ), ∗ weakly in Lp (t ) (Ω , |x|−t ), strongly in Lq (Ω , |x|−s ), for 1 ≤ q < p∗ (s), a.e. in Ω .
(23)
This implies that + Kλ,θ (un , vn ) → Kλ,θ (u+ 0 , v0 ),
as n → ∞.
+ By (22) and (23), it is easy to prove that (u+ 0 , v0 ) is a weak solution of problem (1). Since
Jλ,θ (un , vn ) =
p∗ (t ) − p pp∗ (t )
≥−
∥(un , vn )∥p −
p∗ (t ) − q qp∗ (t )
p∗ (t ) − q qp∗ (t )
Kλ,θ (un , vn )
Kλ,θ (un , vn ),
and by Lemma 5(i), Jλ,θ (un , vn ) → ξλ,θ < 0
as n → ∞. 1 ,p
+ + + Letting n → ∞, we see that Kλ,θ (u+ 0 , v0 ) > 0. Now, we prove that un → u0 , vn → v0 strongly in D0 (Ω ) and + + + + Jλ,θ (u0 , v0 ) = ξλ,θ . By applying Fatou’s lemma and (u0 , v0 ) ∈ Nλ,θ , we get
p∗ (t ) − q p∗ (t ) − p + + + p + ξλ,θ ≤ Jλ,θ (u+ , v ) = ∥( u , v )∥ − Kλ,θ (u+ 0 0 0 0 0 , v0 ) p∗ (t )p qp∗ (t ) ∗ p∗ (t ) − p p ( t ) − q ≤ lim inf ∥(un , vn )∥p − Kλ,θ (un , vn ) n→∞ p∗ (t )p qp∗ (t )
≤ lim inf Jλ,θ (un , vn ) = ξλ,θ . n→∞
This implies that + Jλ,θ (u+ 0 , v0 ) = ξλ,θ , + p lim ∥(un , vn )∥p = ∥(u+ 0 , v0 )∥ .
n→∞
1,p + Then, un → u+ 0 and vn → v0 strongly in D0 (Ω ).
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
289
+ + + + − + − Moreover, we have (u+ 0 , v0 ) ∈ Nλ,θ . In fact, if (u0 , v0 ) ∈ Nλ,θ , by Lemma 6, there are unique t0 and t0 such that
+ − + − + − (t0 u0 , t0+ v0+ ) ∈ Nλ,θ , (t0− u+ 0 , t0 v0 ) ∈ Nλ,θ and t0 < t0 = 1. Since + +
d dt
+ + Jλ,θ (t0+ u+ 0 , t0 v0 ) = 0
d2
and
+ + Jλ,θ (t0+ u+ 0 , t0 v0 ) > 0,
dt 2
+ + + + there exist t0+ < t ≤ t0− such that Jλ,θ (t0+ u+ 0 , t0 v0 ) < Jλ,θ (t 0 u0 , t 0 v0 ). By Lemma 6, we have + + + + Jλ,θ (t0+ u+ 0 , t0 v0 ) < Jλ,θ (t 0 u0 , t 0 v0 ) − + ≤ Jλ,θ (t0− u+ 0 , t0 u0 ) + = Jλ,θ (u+ 0 , v0 ) + + + + + + + + + which contradicts Jλ,θ (u+ 0 , v0 ) = ξλ,θ . Since Jλ,θ (u0 , v0 ) = Jλ,θ (|u0 |, |v0 |) and (|u0 |, |v0 |) ∈ Nλ,θ , by Lemma 4, we may + assume that (u+ 0 , v0 ) is a non-negative solution of problem (1). Moreover, by Lemmas 3 and 5, we have + 0 > ξλ,θ = Jλ,θ (u+ 0 , v0 )
≥−
p∗ (t ) − q
N ωN RN0 −s
p∗ (t )q
q p∗p(∗s()− s)
p
p
p−p q
λ p−q + θ p−q
N −s
+ + + This implies that Jλ,θ (u+ 0 , v0 ) → 0 as λ → 0 , θ → 0 .
− pq
+ q Aµ,s ∥(u+ 0 , v0 )∥ .
Lemma 8. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). If {(un , vn )} ⊂ W is a
(PS )c -sequence for Jλ,θ for all 0 < c < c ∗ := weakly to a nonzero solution of (1).
p−t p(N −t )
N −t
(Sη,α,β (µ)) p−t , then there exists a subsequence of {(un , vn )} converging
′ Proof. Suppose {(un , vn )} ⊂ W satisfies Jλ,θ (un , vn ) → c and Jλ,θ (un , vn ) → 0 with c < c ∗ . Since {(un , vn )} is bounded in W , then there exists (u, v) such that (un , vn ) ⇀ (u, v) up to a subsequence. Moreover, we may assume
u ⇀ u, n un ⇀ u, u ⇀ u, n un → u,
vn ⇀ v, vn ⇀ v, vn ⇀ v, un → u,
1 ,p
weakly in D0 (Ω ), weakly in Lq (Ω , |x|−s ) for all 1 ≤ q < p, ∗ weakly in Lp (t ) (Ω , |x|−t ), a.e. on Ω .
′ Hence, we have Jλ,θ (u, v) = 0 by the weak continuity of J. Let un = un − u, vn = vn − v . Then we have
∥( un , vn )∥p = ∥(un , vn )∥p − ∥(u, v)∥p = o(1),
(24)
and by the Brèzis–Lieb lemma [29], we obtain
| un |p (t ) dx = |x|t ∗
Ω
|un |p (t ) dx − |x|t ∗
Ω
|u|p (t ) dx + o(1), |x|t ∗
Ω
∗ ∗ |vn |p (t ) |v|p (t ) dx − dx + o(1), |x|t |x|t Ω Ω Ω | un |α | vn |β |un |α |vn |β |u|α |v|α dx = dx − dx + o(1). t t |x| |x| |x|t Ω Ω Ω
| vn |p (t ) dx = |x|t ∗
(25)
Hence, as n → ∞, ′ ′ ′ ⟨Jλ,θ ( un , vn ), ( un , vn )⟩ = ⟨Jλ,θ (un , vn ), (un , vn )⟩ − ⟨Jλ,θ (u, v), (u, v)⟩ + o(1) = o(1).
Consequently,
∥( un , vn )∥p → l,
∗ (t )
|un |p
Ω
+ |vn |p (t ) + | un |α | vn |β dx → l. |x|t ∗
From definition of Sη,α,β (µ) and (26), we obtain p
Sη,α,β (µ)l p∗ (t ) = Sη,α,β (µ) lim
n→∞
∗ (t )
|un |p Ω
≤ lim ∥( un , vn )∥ = l, p
n→∞
p∗p(t ) ∗ + |vn |p (t ) + | un |α | vn | β dx |x|t
(26)
290
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
which implies that either l=0 Note that
N −t
l ≥ (Sη,α,β (µ)) p−t .
or
′ ⟨Jλ,θ (u, v), (u, v)⟩
Jλ,θ (u, v) = Jλ,θ (u, v) −
(27)
= 0 and 1 ′ ⟨Jλ,θ (u, v), (u, v)⟩ ≥ 0. p
(28)
From (26) and (28), we get c = Jλ,θ (un , vn ) + o(1)
= Jλ,θ ( un , vn ) + Jλ,θ (u, v) + o(1) ∗ ∗ 1 1 |un |p (t ) + |vn |p (t ) + | un |α | vn |β ≥ ∥( un , vn )∥p − ∗ dx p p (t ) Ω |x|t p∗ ( t ) − p = l + o(1) pp∗ (t ) p−t l + o(1). = p(N − t ) By (27)–(29) and the assumption c < c ∗ we deduce that l = 0. Up to a subsequence, (un , vn ) → (u, v) strongly in W .
(29)
Now, we will give some estimates on the extremal function Vϵ (x) defined in (8). For m ∈ N large, choose ϕ(x) ∈ 1 C0∞ (RN ), 0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1 for |x| ≤ 2m , ϕ(x) = 0 for |x| ≥ m1 , ∥∇ϕ(x)∥Lp (Ω ) ≤ 4m, set uϵ (x) = ϕ(x)Vϵ (x). For ϵ → 0, the behavior of uϵ has to be the same as that of Vϵ , but we need precise estimates of the error terms. For 1 < p < N , 0 ≤ s, t < p and 1 < q < p∗ (s), we have the following estimates [25]:
Ω
N −t |uϵ |p |∇ uϵ | − µ p dx = (Aµ,t ) p−t + O(ϵ b(µ)p+p−N ), |x| p
(30)
N −t |uϵ |p (t ) ∗ dx = (Aµ,t ) p−t + O(ϵ b(µ)p (t )−N +t ), t |x| Ω N −s N −s+ 1− Np q , C ϵ , q> b(µ) N −s |uϵ |q N −s+ 1− Np q , | ln ϵ|, q = dx ≥ C ϵ s b(µ) Ω |x| N N −s C ϵ q b(µ)+1− p q , . q< b(µ)
∗
(31)
(32)
Lemma 9. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). Then, there exists δ1 > 0 such p
p
that for 0 < λ p−q + θ p−q < δ1 , we have sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t p(N − t )
Np−−tt
In particular, ξλ,θ < p(N −t ) Sη,α,β (µ) p−t
Sη,α,β (µ)
Np−−tt
p
.
(33) p
for all 0 < λ p−q + θ p−q < δ1 .
Proof. We consider the functions G(τ ) = Jλ,θ (τ uϵ , τ (τmin uϵ )) ≤ ∗ (t )
τp p
(1 + (τmin )p )
Ω
p τq | u | ϵ |∇ uϵ |p − µ p dx − Kλ,θ (τ uϵ , τ (τmin uϵ )) |x| q
|uϵ |p (t ) − ∗ (1 + η(τmin )β + (τmin )α+β ) dx, p (t ) |x|t Ω τp |uϵ |p g1 (τ ) = (1 + (τmin )p ) |∇ uϵ |p − µ p dx p |x| Ω ∗ ∗ τ p (t ) |uϵ |p (t ) − ∗ (1 + η(τmin )β + (τmin )α+β ) dx. p (t ) |x|t Ω τp
∗
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
291
By (4), (34) and the fact that
sup
τp
τ ≥0
p
∗ (t )
τp
A− ∗ B p (t )
p−t
=
p(N − t )
Np−−tt
A
,
p
B p∗ (t )
A, B > 0
(34)
we conclude that,
Np−−tt |uϵ |p p |∇ u | − µ dx ϵ p Ω |x| p−t sup g1 (τ ) ≤ |uϵ |p∗ (t ) p∗p(t ) p(N − t ) τ ≥0 (1 + η(τmin )β + (τmin )α+β ) Ω |x|t dx
≤
(1 + (τmin )p )
N −t
p−t
(S (µ)) p−t + O(ϵ b(µ)p+p−N )
g (τmin )
p(N − t )
Np−−tt (35)
N −t
(S (µ)) p−t + O(ϵ b(µ)p∗ (t )−N +t ) Np−−tt p−t ≤ g (τmin )S (µ) + O(ϵ b(µ)p+p−N ) p(N − t ) Np−−tt p−t Sη,α,β (µ) ≤ + O(ϵ b(µ)p+p−N ). p(N − t )
(36)
(37)
On the other hand, using the definitions of g and uϵ , we get
τp
G(τ ) = Jλ,θ (τ uϵ , τ (τmin uϵ )) ≤
p
∥(uϵ , τmin uϵ )∥p ,
for all τ ≥ 0 and λ > 0, θ > 0
combining this with (30), let ϵ ∈ (0, 1), then there exists τ0 ∈ (0, 1) independent of ϵ such that sup G(τ ) < 0≤τ ≤τ0
p−t
p(N − t )
p
Sη,α,β (µ)
Np−−tt
p
p
for all 0 < λ p−q + θ p−q < δ1 .
,
(38)
p
Hence, as 0 < λ p−q + θ p−q < δ1 , 1 ≤ q < p, by (35), we have that
sup G(τ ) = sup g1 (τ ) −
τ ≥τ0
τq
τ ≥τ0
≤
p−t
q
Kλ,θ (τ uϵ , τ (τmin uϵ ))
p(N − t )
Sη,α,β (µ)
Np−−tt
+ o(ϵ b(µ)p+p−N ) −
τ0q q
q (λ + τmin θ)
Ω
|uϵ |q dx. |x|s
(39)
−s (i) If 1 ≤ q < bN(µ) , then by (32), we have that
Ω
|uϵ |q q dx ≥ C ϵ |x|s
and since b(µ) >
N −p , p
b(µ)p+1− Np
then
(b(µ)p + p − N ) > q b(µ)p + 1 −
N
.
p
p
p
Combining this with (38) and (39), for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we can choose ϵ small enough such that sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t
p(N − t )
−s (ii) If bN(µ) ≤ q < p, then by (32) and b(µ) >
N −s+ 1− Np q , C ϵ
Ω
|uϵ |q dx ≥ |x|s N −s+ 1− Np q C ϵ | ln ϵ|,
Sη,α,β (µ)
N −p p
Np−−tt
.
we have that
q>
N −s
q=
b(µ)
b(µ) N −s
, ,
292
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
and
(b(µ)p + p − N ) > N − s + 1 −
N
p
q. p
p
Combining this with (38) and (39), for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we can choose ϵ small enough such that sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t
p(N − t )
Sη,α,β (µ)
Np−−tt
.
From (i) and (ii), (33) holds.
p
p
− From Lemma 6, (33) and the definitions of ξλ,θ , for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we obtain that there exists
− − − − τλ,θ such that (τλ,θ uϵ , τλ,θ τmin uϵ ) ∈ Nλ,θ and
− − − ξλ,θ ≤ Jλ,θ (τλ,θ uϵ , τλ,θ τmin uϵ ) ≤ sup Jλ,θ (τ uϵ , τ τmin uϵ ) < τ ≥0
The proof is complete.
p−t p(N − t )
Np−−tt
Sη,α,β (µ)
.
Theorem 4. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). There exists Λ > 0 such that p
p
− for any λ, θ > 0 satisfy 0 < λ p−q + θ p−q < Λ the functional Jλ,θ has a minimizer (U , V ) in Nλ,θ and satisfies the following: − (i) Jλ,θ (U , V ) = ξλ,θ , (ii) (U , V ) is a positive solution of (1), where Λ = min{C0 , δ1 }.
p−p q
p
p
Proof. If 0 < λ p−q + θ p−q < C0 =
q p
C , then by Lemma 5(ii), Lemmas 7 and 9, there exists a (PS )ξλ,θ -sequence
Np−−tt
0, p(N −t ) Sη,α,β (µ) . By Lemma 3, {(un , vn )} is bounded in W . From Lemma 8, there exists a subsequence denoted by {(un , vn )} and a nontrivial solution (U , V ) ∈ W of (1) such that un ⇀ 1 ,p U , vn ⇀ V weakly in D0 (Ω ). − + − First, we prove that (U , V ) ∈ Nλ,θ . Arguing by contradiction, we assume that (U , V ) ∈ Nλ,θ . Since Nλ,θ is closed in − − ∈ in W for Jλ,θ with ξλ,θ {(un , vn )} ⊂ Nλ,θ
p−t
1,p
W0 (Ω ), we have ∥(U , V )∥ < lim infn→∞ ∥(un , vn )∥. Thus, by Lemma 6, there exists a unique τ − such that (τ − U , τ − V ) ∈ − − Nλ,θ . If, (u, v) ∈ Nλ,θ then it is easy to see that Jλ,θ (u, v) =
p−t p(N − t )
∥(u, v)∥p −
p∗ (t ) − q qp∗ (t )
Kλ,θ (u, v).
(40)
− , ∥(U , V )∥ < lim infn→∞ ∥(un , vn )∥ and (40), we can get From Remark 1 (un , vn ) ∈ Nλ,θ − − ξλ,θ ≤ Jλ,θ (τ − U , τ − V ) ≤ lim Jλ,θ (τ − un , τ − vn ) < lim Jλ,θ (un , vn ) = ξλ,θ . n→∞
This is a contradiction. Thus, (U , V ) ∈
n→∞
− Nλ,θ ,
− Next, by the same argument as that in Theorem 3, we get that (un , vn ) → (U , V ) strongly in W and Jλ,θ (U , V ) = ξλ,θ >0
for all 0 < λ
p p−q
+θ
p p−q
< C0 =
p−p q q p
− C . Since Jλ,θ (U , V ) = Jλ,θ (|U |, |V |) and (|U |, |V |) ∈ Nλ,θ , by Lemma 4 we may assume
that (U , V ) is a nontrivial non-negative solution of (1). Finally, by the maximum principle [30], we obtain that (U , V ) is a non positive solution of (1). The proof is complete. p
p
Proof of Theorem 1. The part (i) of Theorem 1 immediately follows from Theorem 3. When 0 < λ p−q + θ p−q < C0 =
p−p q q p
C < C , by Theorems 3 and 4, we obtain that (1) has at least two positive solutions (u0 , v0 ) and (U , V ) such that
− + − + − + . Since Nλ,θ ∩ Nλ,θ = ∅, this implies that Nλ,θ and Nλ,θ are distinct. This completes the (u0 , v0 ) ∈ Nλ,θ and (U , V ) ∈ Nλ,θ
proof of Theorem 1.
Acknowledgments The author would like to thank the anonymous referees for his/her valuable suggestions and comments. N.N. thanks Razi University for support.
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
293
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. J. Garcia Azorero, I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441–476. N. Ghoussoub, F. Robert, The effect of curvature on the best constant in the HardySobolev inequality, Geom. Funct. Anal. 16 (2006) 897–908. D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004) 521–537. N. Ghoussoub, F. Robert, Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, Int. Math. Res. Pap. IMRP (2006) 1–85. Article ID 21867. D. Kang, S. Peng, Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential, Appl. Math. Lett. 18 (2005) 1094–1100. D. Cao, D. Kang, Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms, J. Math. Anal. Appl. 333 (2007) 889–903. F. Demengel, E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth, Adv. Differential Equations 3 (1998) 533–574. N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000) 5703–5743. P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal. 61 (2005) 735–758. D. Kang, On the quasilinear elliptic problem with a critical HardySobolev exponent and a Hardy term, Nonlinear Anal. 69 (2008) 2432–2444. P. Rabinowitz, Minimax Methods in Critical Points Theory with Applications to Differential Equations, in: CBMS Regional Conference Series in Mathematics, vol. 65, Providence, RI, 1986. R. Filippucci, P. Pucci, F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl. 91 (2009) 156–177. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. G.A. Afrouzi, S.H. Rasouli, The Nehari manifold for a class of concave–convex elliptic systems involving the p-Laplacian and nonlinear boundary condition, Nonlinear Anal. 73 (2010) 3390–3401. C.O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal. 60 (2005) 611–624. K.J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calc. Var. 22 (2005) 483–494. K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336. K.J. Brown, T.F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations 69 (2007) 1–9. K.J. Brown, T.F. Wu, On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006) 253–270. K.J. Brown, T.F. Wu, Multiplicity of positive solution of p-Laplacian problems with sign-changing weight function, Int. J. Math. Anal. 1 (12) (2007) 557–563. T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities, Nonlinear Anal. 71 (2009) 2688–2698. K.J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign changing weight function, J. Differential Equations 193 (2003) 481–499. T.S. Hsu, Multiple positive solutions for a quasilinear elliptic problem involving critical SobolevHardy exponents and concave–convex nonlinearities, Nonlinear Anal. 74 (2011) 3934–3944. D. Kang, On the quasilinear elliptic problems with critical HardySobolev exponents and Hardy terms, Nonlinear Anal. 68 (2008) 1973–1985. C. Alves, D. Filho, M. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000) 771–787. Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960) 101–123. P.A. Binding, P. Drabek, Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations 5 (1997) 1–11. H. Brèzis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202.
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Existence and multiplicity of solutions to a singular elliptic system with critical Sobolev–Hardy exponents and concave–convex nonlinearities Nemat Nyamoradi Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran
article
abstract
info
Article history: Received 5 July 2011 Available online 23 June 2012 Submitted by Manuel del Pino
This paper is concerned with a singular elliptic system, which involves the Caffarelli– Kohn–Nirenberg inequality and critical Sobolev–Hardy exponents. The existence and multiplicity results of positive solutions are obtained by variational methods. © 2012 Elsevier Inc. All rights reserved.
Keywords: Nehari manifold Critical Hardy–Sobolev exponent Elliptic system Multiple positive solutions Concave–convex nonlinearities
1. Introduction The aim of this paper is to establish the existence of nontrivial solutions to the following quasilinear elliptic system
∗ |u|p (t )−2 u ηα |u|α−2 |v|β u |u|q−2 u |u|p−2 u = + +λ , − ∆ u − µ p p t t | x| |x| α+β |x| |x|s ∗ |v|p−2 v |v|p (t )−2 v ηβ |u|α |v|β−2 v |v|q−2 v − ∆ v − µ = + + θ , p |x|p |x|t α+β |x|t |x|s u = v = 0, where ∆p u = div(|∇ u|
(1)
x ∈ Ω, x ∈ ∂Ω
∇ u), 0 ∈ Ω is a bounded domain in R (N ≥ 3) with a smooth boundary ∂ Ω , λ > p p > 0, 0 < η < ∞, 1 < p < N , 0 ≤ µ < µ , N − , 0 ≤ s, t < p, 1 ≤ q < p, p p−2
0, θ
x ∈ Ω,
α + β = p∗ (t ) ,
p(N −t ) N −p
N
is the Hardy–Sobolev critical exponent.
1,p
We denote by D0 (Ω ) the completion of C0∞ (Ω ) with respect to the norm
Ω
|∇ · |p dx
1p
.
Problem (1) is related to the well known Caffarelli–Kohn–Nirenberg inequality in [1]:
pr |u|r dx ≤ Cr ,t ,p |∇ u|p dx, for all u ∈ D10,p (Ω ), t | x | Ω Ω where p ≤ r < p∗ (t ). If t = r = p, the above inequality becomes the well known Hardy inequality [1–3]: |u|p 1 dx ≤ |∇ u|p dx, for all u ∈ D10,p (Ω ). p µ Ω Ω | x|
E-mail addresses: [email protected], [email protected]. 0022-247X/$ – see front matter © 2012 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2012.06.026
(2)
(3)
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
281
1,p
In the space D0 (Ω ) we employ the following norm:
∥u∥ = ∥u∥D1,p (Ω ) :=
Ω
0
|∇ u|p − µ
|u|p |x|p
1p
dx
,
µ ∈ [0, µ).
By using the Hardy inequality (3) this norm is equivalent to the usual norm
|·|p−2
L := |∇ · |p−2 ∇ · −µ |x|p
Ω
1p |∇ u|p dx . The elliptic operator
1,p
is positive in D0 (Ω ) if 0 ≤ µ < µ. 1 ,p
1 ,p
Now, we define the space W = D0 (Ω ) × D0 (Ω ) with the norm
∥(u, v)∥p = ∥u∥p + ∥v∥p . Also, by the Hardy inequality and the Hardy–Sobolev inequality, for 0 ≤ µ < µ, 0 ≤ t < p and p ≤ r ≤ p∗ (t ) we can define the best Hardy–Sobolev constant: Aµ,t ,r (Ω ) =
inf
1,p u∈D0 (Ω )\{0}
p |∇ u|p − µ ||ux||p dx Ω . pr r |u| Ω | x| t
dx
In the important case when r = p∗ (t ), we simply denote Aµ,t ,p∗ (t ) as Aµ,t . Note that Aµ,0 is the best constant in the Sobolev inequality, namely,
p |∇ u|p − µ ||ux||p dx Ω . Aµ,0 (Ω ) = inf p 1,p p∗ dx p∗ u∈D0 (Ω )\{0} | u | Ω For any 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ), by (2), (3), 0 ≤ t < p and the Young inequality, the following best constants are well defined S (µ) :=
inf
1,p u∈D0 (Ω )\{0}
|u|p p dx |∇ u | − µ p Ω | x| = Aµ,t p ∗ p∗ (t ) |u| |x|t
Ω
S˜α,β (µ) :=
Ω
inf
(u,v)∈W \{(0,0)}
p (t )
dx
p
|∇ u|p + |∇ u|p − µ |u| |x+|v| |p p ∗ α β |u| |v| Ω |x|t
dx
p
dx (4)
p (t )
Sη,α,β (µ) :=
p p |∇ u|p + |∇v|p − µ |u| |x+|v| dx |p p ∗ . ∗ ∗ p (t ) |u|p (t ) +|v|p (t ) +η|u|α |v|β Ω
inf
(u,v)∈W \{(0,0)}
Ω
(5)
dx
|x|t
Throughout this paper, let R0 be the positive constant such that Ω ⊂ B(0; R0 ), where B(0; R0 ) = {x ∈ RN : |x| < R0 }. By 1,p Holder and Sobolev–Hardy inequalities, for all u ∈ W0 (Ω ), we obtain
Ω
|u|q ≤ |x|s
|x|−s
p∗ (∗s)−q
B(0;R0 ) R0
≤
Ω
r N −s+1 dr
q p∗p(∗s()− s)
|u|p (s) |x|s ∗
p (s)
q p∗ (s)
− pq
Aµ,s ∥u∥q
0
≤ where ωN =
N ωN RN0 −s N −s
p∗ (∗s)−q p (s)
− pq
Aµ,s ∥u∥q ,
(6)
N
2π 2 NΓ ( N ) 2
is the volume of the unit ball in RN .
The existence of nontrivial non-negative solutions for elliptic equations with singular potentials has been recently studied by several authors, but, essentially, only with a solely critical exponent. We refer, e.g., in bounded domains and for p = 2
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N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
to [4,3,5,6], and for general p > 1 to [7–11] and the references therein. For example, Kang in [11] studied the following elliptic equation via the generalized Mountain-Pass Theorem [12]:
|u|p−2 u |u|p (t )−2 u |u|p−2 u = +λ , p t | x| |x| |x|s ∗
−∆p u − µ u = 0,
x ∈ Ω,
(7)
x ∈ ∂Ω
where Ω ⊂ RN is a bounded domain, 1 < p < N , 0 ≤ s, t < p and 0 ≤ µ < µ , Mountain-Pass Theorem of Ambrosetti and Rabinowitz [14], proved that
−∆ p u − µ
up−1
|x|p
∗ −1
= |u|p
+
∗ up (s)−1
,
|x|s
N −p p p
. Also, the authors in [13] via the
in RN
N −p p
admits a positive solution in RN , whenever µ < µ , . p Also, in recent years, several authors have used the Nehari manifold to solve semilinear and quasilinear problems (see [15–22] and references therein). Brown and Zhang [23] have studied a subcritical semi-linear elliptic equation with a sign-changing weight function and a bifurcation real parameter in the case p = 2 and Dirichlet boundary conditions. In [24] the author studied Eq. (7) via the Nehari manifold. Exploiting the relationship between the Nehari manifold and fibering maps (i.e., maps of the form t → Jλ (tu) where Jλ is the Euler function associated with the equation), they gave an interesting explanation of the well-known bifurcation result. In fact, the nature of the Nehari manifold changes as the parameter λ crosses the bifurcation value. In this work, we give a variational method which is similar to the fibering method (see [17,23,24]) to prove the existence and multiplicity of nontrivial non-negative solutions of problem (1). Now, we will begin to state the main theorem. First, we define the following constants
p−q
C =
p p∗ (t )−p
(1 + η)(p∗ (t ) − q) p−p q q C0 = C.
p∗ (t ) − p
p−p q
N ωN R0N −s
p∗ (t ) − q
− p(∗p∗ (s)−q)
N −s
p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
,
p
Then, we have the following result. Theorem 1. Assume that 0 ≤ s, t < p, N ≥ 3, 0 ≤ µ < µ and 1 ≤ q < p. Then we have the following results. p
p
(i) If λ, θ > 0 satisfy λ p−q + θ p−q < C , then (1) has at least one positive solution in W . p
p
(ii) If λ, θ > 0 satisfy 0 < λ p−q + θ p−q < C0 , then (1) has at least two positive solutions in W . This paper is divided into three sections, organized as follows. In Section 2, we establish some elementary results. Finally, in Section 3, we prove our main result (Theorem 1). 2. Notations and preliminaries First, we conclude the relationship between S (µ) and Sη,α,β (µ). To continue, we recall a recent result on the extremal functions of Aµ,t [25]. Lemma 1 ([25]). Assume that 1 < p < N , 0 ≤ t < p and 0 ≤ µ < µ. Then the limiting problem
∗ p−1 |u|p (t )−1 −∆ u − µ |u| = , p |x|p |x|t u ∈ D1,p (RN ), u > 0,
in RN \ {0}, in RN \ {0},
has positive radial ground states Vϵ (x) , ϵ
p−N p
Up,µ
x ϵ
=ϵ
p−N p
Up,µ
|x| , ϵ
∀ϵ > 0,
(8)
that satisfy
Ω
|∇ Vϵ (x)|p − µ
|Vϵ (x)|p |x|p
∗ (t )
dx = Ω
|Vϵ (x)|p |x|t
N −t
N −t
dx = (Aµ,t ) p−t = (S (µ)) p−t ,
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
283
where Up,µ (x) = Up,µ (|x|) is the unique radial solution of the limiting problem with
Up,µ (1) =
(N − t )(µ − µ) N −p
1 p∗ (t )−p
.
Furthermore, Up,µ have the following properties: lim r a(µ) Up,µ (r ) = C1 > 0,
r →0
lim r b(µ) Up,µ (r ) = C2 > 0,
r →+∞
lim r a(µ)+1 |Up′ ,µ (r )| = C1 a(µ) ≥ 0,
r →0
lim r b(µ)+1 |Up′ ,µ (r )| = C2 b(µ) > 0,
r →+∞
where Ci (i = 1, 2) are positive constants and a(µ) and b(µ) are zeros of the function f (ζ ) = (p − 1)ζ p − (N − p)ζ p−1 + µ,
ζ ≥ 0, 0 ≤ µ < µ,
that satisfy 0 ≤ a(µ) <
N −p
N −p
< b(µ) ≤
p
p−1
.
Now, for any α, β > 1 and α + β = p∗ (t ), we define g (τ ) :=
1 + τp
,
p
(1 + ητ β + τ α+β ) α+β
τ ≥ 0,
(9)
g (τmin ) := min g (τ ) > 0,
(10)
τ ≥0
where τmin > 0 is a minimal point of g (τ ) and therefore a root of the equation ∗ p∗ (t )τ p (t )−2 ηβτ β−2 − ηατ β − p∗ (t ) = 0,
τ ≥ 0.
(11)
Theorem 2. Suppose N ≥ 3, 0 ≤ t < p, 0 < η < ∞ and 0 ≤ µ < µ. Then: (i) Sη,α,β (µ) = g (τmin )S (µ); (ii) Sη,α,β (µ) has the minimizers (Vϵ (x), τmin Vϵ (x)), ∀ϵ > 0, where Vϵ (x) are the extremal functions of S (µ) defined as in (8). Proof. (i) Note that limτ →0+ g (τ ) = limτ →+∞ g (τ ) = 1. Thus minτ ≥0 g (τ ) must be achieved at finite τmin ≥ 0. Furthermore, direct calculation shows that there exists a positive constant C such that 0 < C ≤ g (τmin ) = min g (τ ) ≤ 1, τ ≥0
0 < τmin < ∞.
From the fact that g ′ (τmin ) = 0 we deduce that τmin is a root of Eq. (11). 1,p To continue, we follow the argument in [26]. Suppose 0 ≤ µ < µ and {wn } ⊂ D0 (Ω ) \ {0} is a minimizing sequence for S (µ). Let e1 , e2 > 0 be chosen later. Taking un = e1 wn and vn = e2 wn in (5) we have p e1
α+β
(e1
+
α β
|wn |p p |∇w | − µ dx n p Ω | x| ≥ Sη,α,β (µ). p ∗ p∗ (t )
p e2 p α+β α+β )
+ η e1 e2 + e2
|wn | Ω |x|t
dx
(12)
p (t )
Note that
g
e2 e1
p
=
p
e1 + e2 α+β
(e1
p α+β α+β )
β
+ ηeα1 e2 + e2
Choose e1 and e2 in (12) such that
e2 e1
.
= τmin . Passing to the limit as n → ∞ we have
g (τmin )S (µ) ≥ Sη,α,β (µ). Let {(un , vn )} be a minimizing sequence of Sη,α,β (µ) and define zn = sn vn , where
sn =
Ω
(13)
1 α+β −1 |vn |α+β |un |α+β dx dx . |x|t |x|t Ω
(14)
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N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
Then
Ω
|zn |α+β dx = |x|t
Ω
|un |α+β dx. |x|t
(15)
From the Young inequality it follows that
Ω
|un |α |zn |β α dx ≤ |x|t α+β
Ω
|un |α+β β dx + |x|t α+β
|zn |α+β dx. |x|t
Ω
By (15) we have
Ω
|un |α |zn |β dx ≤ |x|t
Ω
|un |α+β dx = |x|t
Ω
|zn |α+β dx. |x|t
(16)
Consequently,
p p p p n| n| |∇ un |p + |∇vn |p − µ |un | |x+|v |∇ un |p + |∇vn |p − µ |un | |x+|v dx dx Ω |p |p ≥ p p α+β α+β −β −(α+β) |un |α+β |un |α+β +|vn |α+β +η|un |α |vn |β dx dx ( 1 + η s + s ) n n Ω Ω |x|t | x| t p |un | p |∇ u | − µ dx n p Ω |x| ≥ p α+β −β −(α+β) |un |α+β (1 + ηsn + sn ) Ω |x|t dx
Ω
−p
sn
Ω
p
|∇ zn | −
p µ ||zxn||p
+ −β
−(α+β)
(1 + ηsn + sn
)
dx
|zn |α+β Ω |x|t
p α+β
dx
1 ≥ g ( s− n )S (µ) ≥ g (τmin )S (µ).
As n → ∞ we have Sη,α,β (µ) ≥ g (τmin )S (µ).
(17)
From (13) and (17) it follows that Sη,α,β (µ) = g (τmin )S (µ),
∀µ ∈ [0, µ).
(18)
(ii) From (5), (8) and (18) the desired result follows.
The corresponding energy functional of problem (1) is defined by Jλ,θ (u, v) =
1 p
∥(u, v)∥p −
1 p∗ (t )
B(u, v) −
1 q
Kλ,θ (u, v),
for each (u, v) ∈ W , where
∗ |u|α |v|β + |v|p (t ) dx + η dx, |x|t |x|t Ω Ω |u|q |v|q Kλ,θ (u, v) = λ dx + θ dx. s s Ω |x| Ω |x|
B(u, v) =
∗ (t )
|u|p
Then Jλ,θ ∈ C 1 (W , R). Now, we consider the problem on the Nehari manifold. Define the Nehari manifold (cf. [27]): ′ Nλ,θ = (u, v) ∈ W \ {(0, 0)}|⟨Jλ,θ (u, v), (u, v)⟩ = 0 ,
where ′ ⟨Jλ,θ (u, v), (u, v)⟩ = ∥(u, v)∥p − B(u, v) − Kλ,θ (u, v).
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
285
Note that Nλ,θ contains every nonzero solution of (1). Define ′ Φλ,θ (u, v) = ⟨Jλ,θ (u, v), (u, v)⟩,
then for (u, v) ∈ Nλ,θ ′ ⟨Φλ,θ (u, v), (u, v)⟩ = p∥(u, v)∥p − p∗ (t )B(u, v) − qKλ,θ (u, v)
(19)
= (p − q)∥(u, v)∥ − (p (t ) − q)B(u, v)
(20)
= (p − p (t ))∥(u, v)∥ − (q − p (t ))Kλ,θ (u, v).
(21)
∗
p
∗
∗
p
Now, we split Nλ,θ into three parts: + ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ > 0 ,
0 ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ = 0 ,
− ′ Nλ,θ = (u, v) ∈ Nλ,θ : ⟨Φλ,θ (u, v), (u, v)⟩ < 0 .
+ − 0 To state our main result, we now present some important properties of Nλ,θ , Nλ,θ and Nλ,θ .
Lemma 2. There exists a positive number C = C (p, q, N , S ) > 0 such that if p
p
0 < λ p−q + θ p−q < C , 0 then Nλ,θ = ∅.
Proof. Suppose otherwise, let
C =
p−q
p p∗ (t )−p
(1 + η)(p∗ (t ) − q)
p∗ (t ) − p
p−p q
p∗ (t ) − q
N ωN RN0 −s N −s
− p(∗p∗ (s)−q) p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
.
Then, there exists (λ, θ ) with p
p
0 < λ p−q + θ p−q < C , 0 0 such that Nλ,θ ̸= ∅. Then for (u, v) ∈ Nλ,θ , by (20) and (21) we have
∥(u, v)∥p =
(p∗ (t ) − q) B(u, v). p−q
Moreover, by the Young inequality and (6), it follows that
(p∗ (t ) − q) B(u, v) p−q ∗ ∗ ∗ ∗ p∗ (t ) − q |u|p (t ) + |v|p (t ) α |u|p (t ) β |v|p (t ) ≤ dx ∗ dx + ∗ dx p−q |x|t p (t ) Ω |x|t p (t ) Ω |x|t Ω
∥(u, v)∥p =
∗
(1 + η)(p∗ (t ) − q) − p p(t ) ∗ ≤ Aµ,t ∥(u, v)∥p (t ) . p−q It follows that
∥(u, v)∥ ≥
p−q
(1 + η)(p∗ (t ) − q)
p∗ (t ) p
p∗ (t1)−p ,
Aµ,t
and p∗ (t ) − p
|u|q |v|q ∥( u , v)∥ = K ( u , v) = λ f dx + θ g dx λ,θ s s p∗ ( t ) − q Ω |x| Ω |x| ∗ p (∗s)−q p (s) p p−p q p − pq N ωN RN0 −s Aµ,s λ p−q + θ p−q ≤ ∥(u, v)∥q . N −s p
286
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
Thus
∥(u, v)∥ ≤
p (t ) − q ∗
p∗ ( t )
p−1 p
Nω
−p
N −s N R0
p∗ (s)−q p∗ (s)(p−p)
− p(pq−p)
Aµ,s
N −s
p
p
λ p−q + θ p−q
1p
.
This implies p
p
λ p−q + θ p−q ≥ C . This is a contradiction! Here
C =
p−q
p p∗ (t )−p
(1 + η)(p∗ (t ) − q)
p∗ (t ) − p
p−p q
N ωN R0N −s
p∗ (t ) − q
− p(∗p∗ (s)−q)
p (s)(p−q)
q p−q
p∗ (t ) p∗ (t )−p
Aµ,s Aµ,t
N −s
.
Lemma 3. The energy functional Jλ,θ is coercive and bounded below on Nλ,θ . Proof. If (u, v) ∈ Nλ,θ , then by (6), Jλ,θ (u, v) =
1 p
∥(u, v)∥p −
p (t ) − p
1 p∗ (t )
B(u, v) −
∗
≥
pp∗ (t )
∥(u, v)∥p −
1 q
Kλ,θ (u, v)
p (t ) − q ∗
Nω
p∗ (t )q
N −s N R0
p∗ (∗s)−q p (s)
N −s
p
p
λ p−q + θ p−q
p−p q
− pq
Aµ,s ∥(u, v)∥q .
Since 0 ≤ s, t < N , 1 < q < p < p∗ (t ), we see that Jλ,θ is coercive and bounded below on Nλ,θ .
Furthermore, similar to the argument in [23, Theorem 2.3] (or see [28]), we can conclude the following result. 0 ′ Lemma 4. Assume that (u0 , v0 ) is a local minimizer for Jλ,θ on Nλ,θ and that (u0 , v0 ) ̸∈ Nλ,θ , then Jλ,θ (u0 , v0 ) = 0 in W −1 .
Now, by Lemma 2, we let
p p ΘC0 = (λ, θ ) ∈ R2 \ {(0, 0)} : 0 < λ p−q + θ p−q < C0 , where C0 =
p−p q
ξλ,θ = + ξλ,θ = − ξλ,θ =
q p
+ − C < C . If (λ, θ ) ∈ ΘC0 , we have Nλ,θ = Nλ,θ ∪ Nλ,θ . Define
inf
Jλ,θ (u, v)
inf
Jλ,θ (u, v)
inf
Jλ,θ (u, v).
(u,v)∈Nλ,θ + (u,v)∈Nλ,θ − (u,v)∈Nλ,θ
Lemma 5. There exists a positive number C0 such that if (λ, θ ) ∈ ΘC0 , then + (i) ξλ,θ ≤ ξλ,θ < 0; − (ii) there exists d0 = d0 (p, q, N , K , S , λ, θ ) > 0 such that ξλ,θ > d0 . + Proof. (i) For (u, v) ∈ Nλ,θ , by (21), we have
Kλ,θ (u, v) ≥
p∗ (t ) − p p∗ (t ) − q
∥(u, v)∥p
and so Jλ,θ (u, v) =
1 p
1
−
1
p∗ (t ) 1
∥(u, v)∥ − p
1 q
1
−
1 p∗ (t ) 1
∥(u, v)∥p − − ∗ p q p (t ) p∗ (t ) − p 1 1 = − ∥(u, v)∥p < 0. p∗ (t ) p q ≤
−
p∗ (t )
Kλ,θ (u, v) p∗ (t ) − p p∗ (t ) − q
∥(u, v)∥p
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
287
+ + Thus, from the definition of ξλ,θ and ξλ,θ , we can deduce that ξλ,θ < ξλ,θ < 0. − (ii) For (u, v) ∈ Nλ,θ , by Lemma 2,
∥(u, v)∥ ≥
p∗ (t ) p
p−q
(1 + η)(p∗ (t ) − q)
p∗ (t1)−p
Aµ,t
.
Moreover, by Lemma 3, Jλ,θ (u, v) ≥
p∗ ( t ) − p pp∗ (t )
−
∥(u, v)∥p
p∗ (t ) − q
N ωN RN0 −s
p∗ (t )q
= ∥(u, v)∥q −
pp∗ (t )
p∗ (t )q
N ωN RN0 −s
−
p∗ (∗s)−q p (s)
N −s p∗ (t ) p
(1 + η)(p∗ (t ) − q) p∗ (t ) − q
p (s)
p∗ (t )q
p−p q − q p p p λ p−q + θ p−q Aµ,s ∥(u, v)∥q
∥(u, v)∥p−q
p−q
p∗ (∗s)−q
N −s
p∗ (t ) − p
p∗ (t ) − q
≥
p∗ (tq)−p
Aµ,t
N ωN RN0 −s
− pq Aµ,s
λ p−q + θ p−q
p∗ (t ) − p pp∗ (t )
p∗ (∗s)−q p (s)
N −s
− pq Aµ,s
p
p
p−p q
∥(u, v)∥p−q
p
p
λ p−q + θ p−q
p−p q
.
Thus, if p
p
0 < λ p−q + θ p−q < C0 , − we have then for each (u, v) ∈ Nλ,θ
Jλ,θ (u, v) ≥ d0 = d0 (p, q, N , K , S , λ, θ ) > 0. For each (u, v) ∈ W \ {(0, 0)} such that B(u, v) > 0, let
tmax =
(p − q)∥(u, v)∥p (p∗ (t ) − q)B(u, v)
1 p∗ (t )−p
.
Lemma 6. Assume that p
p
0 < λ p−q + θ p−q < C0 . Then, for every (u, v) ∈ W with B(u, v) > 0 there exists tmax > 0 such that there are unique t + and t − with 0 < t + < tmax < t − ± such that (t ± u, t ± v) ∈ Nλ,θ and Jλ,θ (t + u, t + v) =
inf
0≤t ≤tmax
Jλ,θ (tu, t v),
Jλ,θ (t − u, t − v) = sup Jλ,θ (tu, t v). t ≥tmax
Proof. The proof is similar to [18, Lemma 2.6] and is omitted.
Remark 1. If p
p
0 < λ p−q + θ p−q < C0 . Then, by Lemmas 5 and 6 for every (u, v) ∈ W with B(u, v) > 0, we can easily deduce that there exists tmax > 0 such that − and there are unique t − with tmax < t − such that (t − u, t − v) ∈ Nλ,θ − Jλ,θ (t − u, t − v) = sup Jλ,θ (tu, t v) ≥ ξλ,θ > 0. t ≥0
288
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
3. Proof of the main result We begin by stating some results which will be required later in the proof of Theorem 1. First, we get the following result. p
p
Lemma 7. (i) If 0 < λ p−q + θ p−q < C , then there exists a (PS )ξλ,θ -sequence {(un , vn )} ⊂ Nλ,θ in W for Jλ,θ . p
p
− (ii) If 0 < λ p−q + θ p−q < C0 , then there exists a (PS )ξ − -sequence {(un , vn )} ⊂ Nλ,θ in W for Jλ,θ , where C is the positive
constant given in Lemma 2, and C0 =
p−p q q p
λ,θ
C.
Proof. The proof is similar to [20, Proposition 9] and is omitted.
p
p
Theorem 3. Assume that 0 ≤ s, t < p, N ≥ 3, 0 ≤ µ < µ and 1 ≤ q < p. If 0 < λ p−q + θ p−q < C , then there exists + + (u+ 0 , v0 ) ∈ Nλ,θ such that + + (i) Jλ,θ (u+ 0 , v0 ) = ξλ,θ = ξλ,θ , + + (ii) (u0 , v0 ) is a positive solution of (1), + + + (iii) Jλ,θ (u+ 0 , v0 ) → 0 as λ → 0 , θ → 0 .
Proof. By Lemma 7, there exists a minimizing sequence {(un , vn )} for Jλ,θ on Nλ,θ such that Jλ,θ (un , vn ) = ξλ,θ + o(1) and
′ Jλ,θ (un , vn ) = o(1)
in W −1 .
(22)
Since Jλ,θ is coercive on Nλ,θ (see Lemma 3), we get {(un , vn )} bounded in W . Thus, there is a subsequence {(un , vn )} and + (u+ 0 , v0 ) ∈ W such that 1 ,p
+ u ⇀ u+ 0 , vn ⇀ v0 , n + un ⇀ u0 , vn ⇀ v0+ , + u → u+ 0 , vn → v0 , n + un → u+ , v → v n 0 0 ,
weakly in D0 (Ω ), ∗ weakly in Lp (t ) (Ω , |x|−t ), strongly in Lq (Ω , |x|−s ), for 1 ≤ q < p∗ (s), a.e. in Ω .
(23)
This implies that + Kλ,θ (un , vn ) → Kλ,θ (u+ 0 , v0 ),
as n → ∞.
+ By (22) and (23), it is easy to prove that (u+ 0 , v0 ) is a weak solution of problem (1). Since
Jλ,θ (un , vn ) =
p∗ (t ) − p pp∗ (t )
≥−
∥(un , vn )∥p −
p∗ (t ) − q qp∗ (t )
p∗ (t ) − q qp∗ (t )
Kλ,θ (un , vn )
Kλ,θ (un , vn ),
and by Lemma 5(i), Jλ,θ (un , vn ) → ξλ,θ < 0
as n → ∞. 1 ,p
+ + + Letting n → ∞, we see that Kλ,θ (u+ 0 , v0 ) > 0. Now, we prove that un → u0 , vn → v0 strongly in D0 (Ω ) and + + + + Jλ,θ (u0 , v0 ) = ξλ,θ . By applying Fatou’s lemma and (u0 , v0 ) ∈ Nλ,θ , we get
p∗ (t ) − q p∗ (t ) − p + + + p + ξλ,θ ≤ Jλ,θ (u+ , v ) = ∥( u , v )∥ − Kλ,θ (u+ 0 0 0 0 0 , v0 ) p∗ (t )p qp∗ (t ) ∗ p∗ (t ) − p p ( t ) − q ≤ lim inf ∥(un , vn )∥p − Kλ,θ (un , vn ) n→∞ p∗ (t )p qp∗ (t )
≤ lim inf Jλ,θ (un , vn ) = ξλ,θ . n→∞
This implies that + Jλ,θ (u+ 0 , v0 ) = ξλ,θ , + p lim ∥(un , vn )∥p = ∥(u+ 0 , v0 )∥ .
n→∞
1,p + Then, un → u+ 0 and vn → v0 strongly in D0 (Ω ).
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
289
+ + + + − + − Moreover, we have (u+ 0 , v0 ) ∈ Nλ,θ . In fact, if (u0 , v0 ) ∈ Nλ,θ , by Lemma 6, there are unique t0 and t0 such that
+ − + − + − (t0 u0 , t0+ v0+ ) ∈ Nλ,θ , (t0− u+ 0 , t0 v0 ) ∈ Nλ,θ and t0 < t0 = 1. Since + +
d dt
+ + Jλ,θ (t0+ u+ 0 , t0 v0 ) = 0
d2
and
+ + Jλ,θ (t0+ u+ 0 , t0 v0 ) > 0,
dt 2
+ + + + there exist t0+ < t ≤ t0− such that Jλ,θ (t0+ u+ 0 , t0 v0 ) < Jλ,θ (t 0 u0 , t 0 v0 ). By Lemma 6, we have + + + + Jλ,θ (t0+ u+ 0 , t0 v0 ) < Jλ,θ (t 0 u0 , t 0 v0 ) − + ≤ Jλ,θ (t0− u+ 0 , t0 u0 ) + = Jλ,θ (u+ 0 , v0 ) + + + + + + + + + which contradicts Jλ,θ (u+ 0 , v0 ) = ξλ,θ . Since Jλ,θ (u0 , v0 ) = Jλ,θ (|u0 |, |v0 |) and (|u0 |, |v0 |) ∈ Nλ,θ , by Lemma 4, we may + assume that (u+ 0 , v0 ) is a non-negative solution of problem (1). Moreover, by Lemmas 3 and 5, we have + 0 > ξλ,θ = Jλ,θ (u+ 0 , v0 )
≥−
p∗ (t ) − q
N ωN RN0 −s
p∗ (t )q
q p∗p(∗s()− s)
p
p
p−p q
λ p−q + θ p−q
N −s
+ + + This implies that Jλ,θ (u+ 0 , v0 ) → 0 as λ → 0 , θ → 0 .
− pq
+ q Aµ,s ∥(u+ 0 , v0 )∥ .
Lemma 8. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). If {(un , vn )} ⊂ W is a
(PS )c -sequence for Jλ,θ for all 0 < c < c ∗ := weakly to a nonzero solution of (1).
p−t p(N −t )
N −t
(Sη,α,β (µ)) p−t , then there exists a subsequence of {(un , vn )} converging
′ Proof. Suppose {(un , vn )} ⊂ W satisfies Jλ,θ (un , vn ) → c and Jλ,θ (un , vn ) → 0 with c < c ∗ . Since {(un , vn )} is bounded in W , then there exists (u, v) such that (un , vn ) ⇀ (u, v) up to a subsequence. Moreover, we may assume
u ⇀ u, n un ⇀ u, u ⇀ u, n un → u,
vn ⇀ v, vn ⇀ v, vn ⇀ v, un → u,
1 ,p
weakly in D0 (Ω ), weakly in Lq (Ω , |x|−s ) for all 1 ≤ q < p, ∗ weakly in Lp (t ) (Ω , |x|−t ), a.e. on Ω .
′ Hence, we have Jλ,θ (u, v) = 0 by the weak continuity of J. Let un = un − u, vn = vn − v . Then we have
∥( un , vn )∥p = ∥(un , vn )∥p − ∥(u, v)∥p = o(1),
(24)
and by the Brèzis–Lieb lemma [29], we obtain
| un |p (t ) dx = |x|t ∗
Ω
|un |p (t ) dx − |x|t ∗
Ω
|u|p (t ) dx + o(1), |x|t ∗
Ω
∗ ∗ |vn |p (t ) |v|p (t ) dx − dx + o(1), |x|t |x|t Ω Ω Ω | un |α | vn |β |un |α |vn |β |u|α |v|α dx = dx − dx + o(1). t t |x| |x| |x|t Ω Ω Ω
| vn |p (t ) dx = |x|t ∗
(25)
Hence, as n → ∞, ′ ′ ′ ⟨Jλ,θ ( un , vn ), ( un , vn )⟩ = ⟨Jλ,θ (un , vn ), (un , vn )⟩ − ⟨Jλ,θ (u, v), (u, v)⟩ + o(1) = o(1).
Consequently,
∥( un , vn )∥p → l,
∗ (t )
|un |p
Ω
+ |vn |p (t ) + | un |α | vn |β dx → l. |x|t ∗
From definition of Sη,α,β (µ) and (26), we obtain p
Sη,α,β (µ)l p∗ (t ) = Sη,α,β (µ) lim
n→∞
∗ (t )
|un |p Ω
≤ lim ∥( un , vn )∥ = l, p
n→∞
p∗p(t ) ∗ + |vn |p (t ) + | un |α | vn | β dx |x|t
(26)
290
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
which implies that either l=0 Note that
N −t
l ≥ (Sη,α,β (µ)) p−t .
or
′ ⟨Jλ,θ (u, v), (u, v)⟩
Jλ,θ (u, v) = Jλ,θ (u, v) −
(27)
= 0 and 1 ′ ⟨Jλ,θ (u, v), (u, v)⟩ ≥ 0. p
(28)
From (26) and (28), we get c = Jλ,θ (un , vn ) + o(1)
= Jλ,θ ( un , vn ) + Jλ,θ (u, v) + o(1) ∗ ∗ 1 1 |un |p (t ) + |vn |p (t ) + | un |α | vn |β ≥ ∥( un , vn )∥p − ∗ dx p p (t ) Ω |x|t p∗ ( t ) − p = l + o(1) pp∗ (t ) p−t l + o(1). = p(N − t ) By (27)–(29) and the assumption c < c ∗ we deduce that l = 0. Up to a subsequence, (un , vn ) → (u, v) strongly in W .
(29)
Now, we will give some estimates on the extremal function Vϵ (x) defined in (8). For m ∈ N large, choose ϕ(x) ∈ 1 C0∞ (RN ), 0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1 for |x| ≤ 2m , ϕ(x) = 0 for |x| ≥ m1 , ∥∇ϕ(x)∥Lp (Ω ) ≤ 4m, set uϵ (x) = ϕ(x)Vϵ (x). For ϵ → 0, the behavior of uϵ has to be the same as that of Vϵ , but we need precise estimates of the error terms. For 1 < p < N , 0 ≤ s, t < p and 1 < q < p∗ (s), we have the following estimates [25]:
Ω
N −t |uϵ |p |∇ uϵ | − µ p dx = (Aµ,t ) p−t + O(ϵ b(µ)p+p−N ), |x| p
(30)
N −t |uϵ |p (t ) ∗ dx = (Aµ,t ) p−t + O(ϵ b(µ)p (t )−N +t ), t |x| Ω N −s N −s+ 1− Np q , C ϵ , q> b(µ) N −s |uϵ |q N −s+ 1− Np q , | ln ϵ|, q = dx ≥ C ϵ s b(µ) Ω |x| N N −s C ϵ q b(µ)+1− p q , . q< b(µ)
∗
(31)
(32)
Lemma 9. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). Then, there exists δ1 > 0 such p
p
that for 0 < λ p−q + θ p−q < δ1 , we have sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t p(N − t )
Np−−tt
In particular, ξλ,θ < p(N −t ) Sη,α,β (µ) p−t
Sη,α,β (µ)
Np−−tt
p
.
(33) p
for all 0 < λ p−q + θ p−q < δ1 .
Proof. We consider the functions G(τ ) = Jλ,θ (τ uϵ , τ (τmin uϵ )) ≤ ∗ (t )
τp p
(1 + (τmin )p )
Ω
p τq | u | ϵ |∇ uϵ |p − µ p dx − Kλ,θ (τ uϵ , τ (τmin uϵ )) |x| q
|uϵ |p (t ) − ∗ (1 + η(τmin )β + (τmin )α+β ) dx, p (t ) |x|t Ω τp |uϵ |p g1 (τ ) = (1 + (τmin )p ) |∇ uϵ |p − µ p dx p |x| Ω ∗ ∗ τ p (t ) |uϵ |p (t ) − ∗ (1 + η(τmin )β + (τmin )α+β ) dx. p (t ) |x|t Ω τp
∗
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
291
By (4), (34) and the fact that
sup
τp
τ ≥0
p
∗ (t )
τp
A− ∗ B p (t )
p−t
=
p(N − t )
Np−−tt
A
,
p
B p∗ (t )
A, B > 0
(34)
we conclude that,
Np−−tt |uϵ |p p |∇ u | − µ dx ϵ p Ω |x| p−t sup g1 (τ ) ≤ |uϵ |p∗ (t ) p∗p(t ) p(N − t ) τ ≥0 (1 + η(τmin )β + (τmin )α+β ) Ω |x|t dx
≤
(1 + (τmin )p )
N −t
p−t
(S (µ)) p−t + O(ϵ b(µ)p+p−N )
g (τmin )
p(N − t )
Np−−tt (35)
N −t
(S (µ)) p−t + O(ϵ b(µ)p∗ (t )−N +t ) Np−−tt p−t ≤ g (τmin )S (µ) + O(ϵ b(µ)p+p−N ) p(N − t ) Np−−tt p−t Sη,α,β (µ) ≤ + O(ϵ b(µ)p+p−N ). p(N − t )
(36)
(37)
On the other hand, using the definitions of g and uϵ , we get
τp
G(τ ) = Jλ,θ (τ uϵ , τ (τmin uϵ )) ≤
p
∥(uϵ , τmin uϵ )∥p ,
for all τ ≥ 0 and λ > 0, θ > 0
combining this with (30), let ϵ ∈ (0, 1), then there exists τ0 ∈ (0, 1) independent of ϵ such that sup G(τ ) < 0≤τ ≤τ0
p−t
p(N − t )
p
Sη,α,β (µ)
Np−−tt
p
p
for all 0 < λ p−q + θ p−q < δ1 .
,
(38)
p
Hence, as 0 < λ p−q + θ p−q < δ1 , 1 ≤ q < p, by (35), we have that
sup G(τ ) = sup g1 (τ ) −
τ ≥τ0
τq
τ ≥τ0
≤
p−t
q
Kλ,θ (τ uϵ , τ (τmin uϵ ))
p(N − t )
Sη,α,β (µ)
Np−−tt
+ o(ϵ b(µ)p+p−N ) −
τ0q q
q (λ + τmin θ)
Ω
|uϵ |q dx. |x|s
(39)
−s (i) If 1 ≤ q < bN(µ) , then by (32), we have that
Ω
|uϵ |q q dx ≥ C ϵ |x|s
and since b(µ) >
N −p , p
b(µ)p+1− Np
then
(b(µ)p + p − N ) > q b(µ)p + 1 −
N
.
p
p
p
Combining this with (38) and (39), for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we can choose ϵ small enough such that sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t
p(N − t )
−s (ii) If bN(µ) ≤ q < p, then by (32) and b(µ) >
N −s+ 1− Np q , C ϵ
Ω
|uϵ |q dx ≥ |x|s N −s+ 1− Np q C ϵ | ln ϵ|,
Sη,α,β (µ)
N −p p
Np−−tt
.
we have that
q>
N −s
q=
b(µ)
b(µ) N −s
, ,
292
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
and
(b(µ)p + p − N ) > N − s + 1 −
N
p
q. p
p
Combining this with (38) and (39), for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we can choose ϵ small enough such that sup Jλ,θ (τ uϵ , τ τmin uϵ ) < c ∗ := τ ≥0
p−t
p(N − t )
Sη,α,β (µ)
Np−−tt
.
From (i) and (ii), (33) holds.
p
p
− From Lemma 6, (33) and the definitions of ξλ,θ , for any λ, θ > 0 where 0 < λ p−q + θ p−q < δ1 we obtain that there exists
− − − − τλ,θ such that (τλ,θ uϵ , τλ,θ τmin uϵ ) ∈ Nλ,θ and
− − − ξλ,θ ≤ Jλ,θ (τλ,θ uϵ , τλ,θ τmin uϵ ) ≤ sup Jλ,θ (τ uϵ , τ τmin uϵ ) < τ ≥0
The proof is complete.
p−t p(N − t )
Np−−tt
Sη,α,β (µ)
.
Theorem 4. Assume that 0 ≤ s, t < p, 1 ≤ q < p, 0 ≤ µ < µ, α, β > 1 and α + β = p∗ (t ). There exists Λ > 0 such that p
p
− for any λ, θ > 0 satisfy 0 < λ p−q + θ p−q < Λ the functional Jλ,θ has a minimizer (U , V ) in Nλ,θ and satisfies the following: − (i) Jλ,θ (U , V ) = ξλ,θ , (ii) (U , V ) is a positive solution of (1), where Λ = min{C0 , δ1 }.
p−p q
p
p
Proof. If 0 < λ p−q + θ p−q < C0 =
q p
C , then by Lemma 5(ii), Lemmas 7 and 9, there exists a (PS )ξλ,θ -sequence
Np−−tt
0, p(N −t ) Sη,α,β (µ) . By Lemma 3, {(un , vn )} is bounded in W . From Lemma 8, there exists a subsequence denoted by {(un , vn )} and a nontrivial solution (U , V ) ∈ W of (1) such that un ⇀ 1 ,p U , vn ⇀ V weakly in D0 (Ω ). − + − First, we prove that (U , V ) ∈ Nλ,θ . Arguing by contradiction, we assume that (U , V ) ∈ Nλ,θ . Since Nλ,θ is closed in − − ∈ in W for Jλ,θ with ξλ,θ {(un , vn )} ⊂ Nλ,θ
p−t
1,p
W0 (Ω ), we have ∥(U , V )∥ < lim infn→∞ ∥(un , vn )∥. Thus, by Lemma 6, there exists a unique τ − such that (τ − U , τ − V ) ∈ − − Nλ,θ . If, (u, v) ∈ Nλ,θ then it is easy to see that Jλ,θ (u, v) =
p−t p(N − t )
∥(u, v)∥p −
p∗ (t ) − q qp∗ (t )
Kλ,θ (u, v).
(40)
− , ∥(U , V )∥ < lim infn→∞ ∥(un , vn )∥ and (40), we can get From Remark 1 (un , vn ) ∈ Nλ,θ − − ξλ,θ ≤ Jλ,θ (τ − U , τ − V ) ≤ lim Jλ,θ (τ − un , τ − vn ) < lim Jλ,θ (un , vn ) = ξλ,θ . n→∞
This is a contradiction. Thus, (U , V ) ∈
n→∞
− Nλ,θ ,
− Next, by the same argument as that in Theorem 3, we get that (un , vn ) → (U , V ) strongly in W and Jλ,θ (U , V ) = ξλ,θ >0
for all 0 < λ
p p−q
+θ
p p−q
< C0 =
p−p q q p
− C . Since Jλ,θ (U , V ) = Jλ,θ (|U |, |V |) and (|U |, |V |) ∈ Nλ,θ , by Lemma 4 we may assume
that (U , V ) is a nontrivial non-negative solution of (1). Finally, by the maximum principle [30], we obtain that (U , V ) is a non positive solution of (1). The proof is complete. p
p
Proof of Theorem 1. The part (i) of Theorem 1 immediately follows from Theorem 3. When 0 < λ p−q + θ p−q < C0 =
p−p q q p
C < C , by Theorems 3 and 4, we obtain that (1) has at least two positive solutions (u0 , v0 ) and (U , V ) such that
− + − + − + . Since Nλ,θ ∩ Nλ,θ = ∅, this implies that Nλ,θ and Nλ,θ are distinct. This completes the (u0 , v0 ) ∈ Nλ,θ and (U , V ) ∈ Nλ,θ
proof of Theorem 1.
Acknowledgments The author would like to thank the anonymous referees for his/her valuable suggestions and comments. N.N. thanks Razi University for support.
N. Nyamoradi / J. Math. Anal. Appl. 396 (2012) 280–293
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References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]
L. Caffarelli, R. Kohn, L. Nirenberg, First order interpolation inequality with weights, Compos. Math. 53 (1984) 259–275. J. Garcia Azorero, I. Peral, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations 144 (1998) 441–476. N. Ghoussoub, F. Robert, The effect of curvature on the best constant in the HardySobolev inequality, Geom. Funct. Anal. 16 (2006) 897–908. D. Cao, P. Han, Solutions for semilinear elliptic equations with critical exponents and Hardy potential, J. Differential Equations 205 (2004) 521–537. N. Ghoussoub, F. Robert, Concentration estimates for Emden–Fowler equations with boundary singularities and critical growth, Int. Math. Res. Pap. IMRP (2006) 1–85. Article ID 21867. D. Kang, S. Peng, Solutions for semilinear elliptic problems with critical Sobolev–Hardy exponents and Hardy potential, Appl. Math. Lett. 18 (2005) 1094–1100. D. Cao, D. Kang, Solutions of quasilinear elliptic problems involving a Sobolev exponent and multiple Hardy-type terms, J. Math. Anal. Appl. 333 (2007) 889–903. F. Demengel, E. Hebey, On some nonlinear equations involving the p-Laplacian with critical Sobolev growth, Adv. Differential Equations 3 (1998) 533–574. N. Ghoussoub, C. Yuan, Multiple solutions for quasi-linear PDEs involving the critical Sobolev and Hardy exponents, Trans. Amer. Math. Soc. 352 (2000) 5703–5743. P. Han, Quasilinear elliptic problems with critical exponents and Hardy terms, Nonlinear Anal. 61 (2005) 735–758. D. Kang, On the quasilinear elliptic problem with a critical HardySobolev exponent and a Hardy term, Nonlinear Anal. 69 (2008) 2432–2444. P. Rabinowitz, Minimax Methods in Critical Points Theory with Applications to Differential Equations, in: CBMS Regional Conference Series in Mathematics, vol. 65, Providence, RI, 1986. R. Filippucci, P. Pucci, F. Robert, On a p-Laplace equation with multiple critical nonlinearities, J. Math. Pures Appl. 91 (2009) 156–177. A. Ambrosetti, P.H. Rabinowitz, Dual variational methods in critical point theory and applications, J. Funct. Anal. 14 (1973) 349–381. G.A. Afrouzi, S.H. Rasouli, The Nehari manifold for a class of concave–convex elliptic systems involving the p-Laplacian and nonlinear boundary condition, Nonlinear Anal. 73 (2010) 3390–3401. C.O. Alves, A. El Hamidi, Nehari manifold and existence of positive solutions to a class of quasilinear problems, Nonlinear Anal. 60 (2005) 611–624. K.J. Brown, The Nehari manifold for a semilinear elliptic equation involving a sublinear term, Calc. Var. 22 (2005) 483–494. K.J. Brown, T.F. Wu, A semilinear elliptic system involving nonlinear boundary condition and sign changing weight function, J. Math. Anal. Appl. 337 (2008) 1326–1336. K.J. Brown, T.F. Wu, A fibering map approach to a semilinear elliptic boundary value problem, Electron. J. Differential Equations 69 (2007) 1–9. K.J. Brown, T.F. Wu, On semilinear elliptic equations involving concave–convex nonlinearities and sign-changing weight function, J. Math. Anal. Appl. 318 (2006) 253–270. K.J. Brown, T.F. Wu, Multiplicity of positive solution of p-Laplacian problems with sign-changing weight function, Int. J. Math. Anal. 1 (12) (2007) 557–563. T.S. Hsu, Multiple positive solutions for a critical quasilinear elliptic system with concave–convex nonlinearities, Nonlinear Anal. 71 (2009) 2688–2698. K.J. Brown, Y. Zhang, The Nehari manifold for a semilinear elliptic problem with a sign changing weight function, J. Differential Equations 193 (2003) 481–499. T.S. Hsu, Multiple positive solutions for a quasilinear elliptic problem involving critical SobolevHardy exponents and concave–convex nonlinearities, Nonlinear Anal. 74 (2011) 3934–3944. D. Kang, On the quasilinear elliptic problems with critical HardySobolev exponents and Hardy terms, Nonlinear Anal. 68 (2008) 1973–1985. C. Alves, D. Filho, M. Souto, On systems of elliptic equations involving subcritical or critical Sobolev exponents, Nonlinear Anal. 42 (2000) 771–787. Z. Nehari, On a class of nonlinear second-order differential equations, Trans. Amer. Math. Soc. 95 (1960) 101–123. P.A. Binding, P. Drabek, Y.X. Huang, On Neumann boundary value problems for some quasilinear elliptic equations, Electron. J. Differential Equations 5 (1997) 1–11. H. Brèzis, E. Lieb, A relation between pointwise convergence of functions and convergence of functionals, Proc. Amer. Math. Soc. 88 (1983) 486–490. J.L. Vazquez, A strong maximum principle for some quasilinear elliptic equations, Appl. Math. Optim. 12 (1984) 191–202.