Electronic Journal of Differential Equations, Vol. 2012 (2012), No. 169, pp. 1–18.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
EXISTENCE OF SOLUTIONS TO QUASILINEAR ELLIPTIC
SYSTEMS WITH COMBINED CRITICAL SOBOLEV-HARDY
TERMS
NEMAT NYAMORADI, MOHAMAD JAVIDI
Abstract. This article is devoted to the study of multiple positive solutions
to a singular elliptic system where the nonlinearity involves a combination of
concave and convex terms. Using the effect of the coefficient of the critical
nonlinearity, and a variational method, we establish the main result which is
based on a compactness argument.
1. Introduction
The aim of this paper is to establish the existence of nontrivial solution to the
elliptic system
∗
−∆p u − µ
|u|p−2 u
|u|p (s1 )−2 u
|u|q−2 u
α
|u|α−2 |v|β u
=
+
λh(x)
,
+
Q(x)
p
s
t
|x|
|x| 1
α+β
|x − x0 |
|x|s
∗
|v|q−2 v
|v|p−2 v
|v|p (s2 )−2 v
β
|u|α |v|β−2 v
+
λh(x)
−∆p v − µ
=
+
Q(x)
,
|x|p
|x|s2
α+β
|x − x0 |t
|x|s
x ∈ Ω,
u = v = 0,
x ∈ ∂Ω
(1.1)
where ∆p u = div(|∇u|p−2 ∇u), 0 ∈ Ω is a bounded domain in RN (N ≥ 3) with
smooth boundary
∂Ω, λ > 0 is a parameter, 1 ≤ q < p, 1 < p < N , 0 ≤ µ <
p
µ , N p−p ; Q(x) is nonnegative and continuous on Ω satisfying some additional
conditions which will be given later, Q(x0 ) = kQk∞ for 0 6= x0 6= Ω, h(x) ∈ C(Ω);
−t)
p(N −s)
∗
α, β > 1, α + β = p∗ (t) , p(N
N −p , p (s) , N −p (0 < s, s1 , s2 ≤ t < p) are critical
p
Sobolev-Hardy exponents. Note that p∗ (0) = p∗ := NN−p
is the critical Sobolev
exponent.
We denote by W01,p (Ω) the completion of C0∞ (Ω) with respect to the norm
1/p
R
|∇ · |p dx
.
Ω
2000 Mathematics Subject Classification. 35B33, 35J60, 35J65.
Key words and phrases. Ekeland variational principle; critical Hardy-Sobolev exponent;
concentration-compactness principle.
c
2012
Texas State University - San Marcos.
Submitted May 17, 2012. Published October 4, 2012.
1
2
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
Problem (1.1) is related to the well known Caffarelli-Kohn-Nirenberg inequality
in [3]:
Z
Z |u|r p/r
≤
C
|∇u|p dx, for all u ∈ W01,p (Ω),
(1.2)
dx
r,t,p
t
Ω |x|
Ω
where p ≤ r < p∗ (t). When t = r = p, the above inequality becomes the well
known Hardy inequality [3, 9, 10]:
Z
Z
|u|p
1
dx ≤
|∇u|p dx, for all u ∈ W01,p (Ω).
(1.3)
p
|x|
µ
Ω
Ω
In the space W01,p (Ω) we use the norm
Z |u|p 1/p
,
kukµ = kukD1,p (Ω) :=
|∇u|p − µ p dx
|x|
Ω
µ ∈ [0, µ).
By using the Hardy inequality (1.3) this norm is equivalent to the usual norm
1/p
R
p−2 |∇u|p dx
. The elliptic operator L := |∇ · |p−2 ∇ · −µ |·||x|p is positive in
Ω
W01,p (Ω) if 0 ≤ µ < µ.
Now, we define the space W = W01,p (Ω) × W01,p (Ω) with the norm
k(u, v)kp = kukpµ + kvkpµ .
Also, by Hardy inequality and Hardy-Sobolev inequality, for 0 ≤ µ < µ, 0 ≤ t < p
and p ≤ r ≤ p∗ (t) we can define the best Hardy-Sobolev constant:
R |u|p
p
dx
|∇u|
−
µ
p
|x|
Ω
.
(1.4)
Aµ,t,r (Ω) =
inf
p/r
R |u|r
u∈W01,p (Ω)\{0}
dx
Ω |x|t
In the important case when r = p∗ (t), we simply denote Aµ,t,p∗ (t) as Aµ,t .
For any 0 ≤ µ < µ, α, β > 1 and α+β = p∗ (t), by (1.2), (1.3), 0 < s1 , s2 ≤ t < p,
Set
R
p
|∇u|p − µ |u|
|x|p dx
Ω
(1.5)
Aµ,s :=
inf
R |u|p∗ (s) p∗p(s) ,
u∈W01,p (Ω)\{0}
dx
Ω |x|s
R
p
+|v|p |∇u|p + |∇v|p − µ |u||x|
dx
p
Ω
.
(1.6)
Ss,α,β :=
inf
p
R
α
β
|u| |v|
α+β
(u,v)∈W \{(0,0)}
dx
s
|x|
Ω
Then we have the following equality (whose proof is the same as that of Theorem
5 in [1])
Ss,α,β (µ) =
α β
α β α+β
α+β
+
Aµ,s .
β
α
Throughout this paper, let R0 be the positive constant such that Ω ⊂ B(0; R0 ),
where B(0; R0 ) = {x ∈ RN : |x| < R0 }. By Holder and Sobolev-Hardy inequalities,
EJDE-2012/169
EXISTENCE OF SOLUTIONS
3
for all u ∈ W01,p (Ω), we obtain
Z
Z |u|p∗ (s) p∗q(s)
Z
p∗p(s)−q
∗ (s)
|u|q
−s
≤
|x|
s
|x|s
Ω
Ω |x|
B(0;R0 )
∗
Z R0
p p(s)−q
∗ (s)
−q
≤
Aµ,sp kukq
rN −s+1 dr
(1.7)
0
≤
N ω RN −s p∗p(s)−q
∗ (s)
−q
N 0
Aµ,sp kukq ,
N −s
N/2
N
.
where ωN = N2π
Γ(N/2) is the volume of the unit ball in R
Existence of nontrivial non-negative solutions for elliptic equations with singular
potentials were recently studied by several authors, but, essentially, only with a
solely critical exponent. We refer, e.g., in bounded domains and for p = 2 to
[4, 10, 11, 13, 14, 15, 17], and for general p > 1 to [5, 6, 7, 8, 12, 16, 18, 19, 26] and
the references therein. For example, Han and Liu [17] studied the problem
−∆u − µ
∗
u
= λu + Q(x)|u|2 −2 u,
2
|x|
u = 0, x ∈ ∂Ω
x ∈ Ω,
(1.8)
where 0 ∈ Ω is a smooth bounded domain in RN (N ≥ 5), λ > 0, 0 ≤ µ < µ ,
N −2 2
, 2∗ = N2N
2
−2 and Q(x) is nonnegative and continuous on Ω satisfying some
suitable conditions. using critical point theory, the authors proved the existence
of nontrivial solutions to problem (1.8). Also, by investigating the effect of the
coefficient Q, Han [14] studied problem (1.8) and proved that there exists λ0 > 0
such that (1.8) has at least k positive solutions for λ ∈ (0, λ0 ).
Kang in [18] studied the following elliptic equation via the generalized MountainPass theorem [24],
∗
−∆p u − µ
|u|p−2 u
|u|p (t)−2 u
|u|p−2 u
=
+
λ
,
|x|p
|x|t
|x|s
u = 0, x ∈ ∂Ω
x ∈ Ω,
(1.9)
where Ω ⊂ RN is a bounded domain, 1 < p < N , 0 ≤ s, t < p and 0 ≤ µ < µ ,
N −p p
. Degiovanni and Lancelotti [6] studied problem (1.9) with µ = s = t = 0
p
and proved that (1.9) has at least one positive solutions for λ ≥ λ1 := A0,0 (A0,0 is
defined in (1.5)). Indeed, in [6] the much more difficult case λ ≥ λ1 is treated.
The authors in [8], via the Mountain-Pass Theorem of Ambrosetti and Rabinowitz [2], proved that
∗
∗
up (s)−1
up−1
,
−∆p u − µ p = |u|p −1 +
|x|
|x|s
in RN ,
p
admits a positive solution in RN , whenever µ < µ , N p−p and 0 < s < p.
Recently, in [26] the author studied the following equation via the Mountain-Pass
theorem,
− div
|Du|p−2 Du |x|ap
∗
∗
|u|p−2 u
|u|p (b)−2 u |u|p (c)−2 u
− µ (a+1)p =
+
,
|x|bp∗
|x|cp∗
|x|
in RN
4
N. NYAMORADI, M. JAVIDI
N −(a+1)p p
,
p
Np
N −(a+1−c)p .
where 1 < p < N , 0 ≤ µ < µ ,
Np
N −(a+1−b)p
∗
∗
EJDE-2012/169
0 ≤ a <
N −p
p ,
a ≤ b, c < a + 1,
and p (c) =
p (b) =
Zhang and Wei [27] studied the existence of multiple positive solutions for (1.1)
with t = s = 0, Q(x) = f (x) and h(x) = 1. Set s1 = s2 = t, s = t, x0 = 0 and
Q(x) = h(x) ≡ 1, then problem (1.1) reduces to the quasilinear elliptic system
∗
|u|p−2 u
|u|p (t)−2 u
ηα |u|α−2 |v|β u
|u|q−2 u
−∆p u − µ
=
+
+
λ
,
|x|p
|x|t
α+β
|x|t
|x|s
∗
−∆p v − µ
|v|p (t)−2 v
ηβ |u|α |v|β−2 v
|v|q−2 v
|v|p−2 v
=
+
+
θ
,
|x|p
|x|t
α+β
|x|t
|x|s
x ∈ Ω,
u = v = 0,
x ∈ ∂Ω
where λ > 0, θ > 0, 0 < η < ∞, 1 < p < N , 0 ≤ µ < µ ,
∗
(1.10)
N −p p
,
p
0 ≤ s, t < p,
p(N −t)
N −p
is the Hardy- Sobolev critical exponent. The
1 ≤ q < p, α + β = p (t) ,
author [23] have studied (1.10) via the Nehari manifold. In [20], Li et al. studied
the following quasilinear elliptic problem
∗
−∆p u − µ
∗
|u|p (s)−2 u
|u|p (t)−2
|u|p−2 u
=
K(x)
+
Q(x)
+ λf (x, u),
|x|p
|x|s
|x − x0 |t
u = 0, x ∈ ∂Ω
x ∈ Ω,
(1.11)
where 1 < p < N , K(x), Q(x) are nonnegative continuous functions on Ω, f satisfying some suitable conditions and obtained the existence of solutions via variational
methods. For p = 2, x0 = 0, K(x) ≡ 1 and Q(x) ≡ 0, the problem (1.11) has been
studied.
Motivated by the above works we study problem (1.1) by using the MountainPass Theorem of Ambrosetti and Rabinowitz. We shall show that system (1.1) has
at least two positive weak solutions.
In this article, we assume that 0 < s1 , s2 ≤ t < p, α, β > 1 and α + β = p∗ (t).
For 0 ≤ µ < µ, we set
N −s
p−s
p−s
,
Aµ,s
p(N − s)
N −t p−t
1
p−t
θ∗ := θ(µ, s1 ), θ(µ, s2 ),
N −p St,α,β .
p(N − t)
p−t
kQk∞
θ(µ, s) :=
Moreover, we assume that Q(x) satisfies some of the following assumptions:
(H1) Q ∈ C(Ω), Q(x) ≥ 0 and meas({x ∈ Ω, h(x) > 0}) > 0.
(H2) There exist ϑ > 0 such that Q(x0 ) = kQk∞ > 0 and Q(x) = Q(x0 )+O(|x−
x0 |% ), as x → x0 .
(H3) There exist β0 and ρ > 0 such that B2ρ0 (x0 ) ⊂ Ω and h(x) ≥ β0 for all
x ∈ B2ρ0 (x0 ).
Set
h+ := max{h, 0},
h− := max{−h, 0}.
The main results of this article are stated in the following two theorems.
EJDE-2012/169
EXISTENCE OF SOLUTIONS
5
Theorem 1.1. Assume that N ≥ 3, µ ∈ [0, µ), 1 < q < p and (H1). Then there
exists Λ∗11 > 0, such that for 0 < λ < Λ∗11 problem (1.1) has at lest one positive
solutions.
Theorem 1.2. Assume that N ≥ p2 , 0 ≤ µ < µ, θ∗ =
p−t
p(N −t)
N −t
1
N −p
p−t
kQk∞
p−t
St,α,β
,
−s
(H1)-(H3), Q(0) = 0, % > b(µ)p + p − N + t and N
b(µ) < q < p hold, and b(µ) is
the constant defined as in Lemma 2.4. Then there exists Λ∗∗ > 0, such that for
0 < λ < Λ∗∗ , problem (1.1) has at least two positive solutions.
This article is divided into three sections, organized as follows. In Section 2,
we establish some elementary results. In Section 3, we prove our main results
(Theorems 1.1 and 1.2).
2. Preliminary lemmas
The corresponding energy functional of problem (1.1) is defined by
Z Z
1
|u|p
|v|p λ
|u|q
|v|q
J(u, v) =
|∇u|p − µ p + |∇v|p − µ p dx −
h(x)( s + s )dx
p Ω
|x|
|x|
q Ω
|x|
|x|
Z
Z
p∗ (s1 )
p∗ (s2 )
1
|u|
1
|v|
− ∗
dx − ∗
dx
p (s1 ) Ω |x|s1
p (s2 ) Ω |x|s2
Z
1
|u|α |v|β
dx,
−
Q(x)
α+β Ω
|x − x0 |t
for each (u, v) ∈ W . Then J ∈ C 1 (W, R).
Lemma 2.1. Assume that N ≥ 3, 0 ≤ µ < µ, (H1), h+ 6= 0 and (u, v) is a weak
solution of problem (1.1). Then there exists a positive constant d depending on
N, |Ω|, |h+ |∞ , Aµ,s , s1 , s2 and q such that
p
J(u, v) ≥ −dλ p−q .
Proof. Since (u, v) is a weak solution of (1.1), then, Note that hJ 0 (u, v), (u, v)i = 0,
we have
hJ 0 (u, v), (u, v)i
Z
Z |u|q
|v|p |v|q
|u|p
p
p
h(x)( s + s )dx
=
|∇u| − µ p + |∇v| − µ p dx − λ
(2.1)
|x|
|x|
|x|
|x|
Ω
Ω
Z
Z
Z
p∗ (s1 )
p∗ (s2 )
α
β
|u|
|v|
|u| |v|
−
dx −
dx −
Q(x)
dx = 0.
s
s
1
2
|x|
|x − x0 |t
Ω
Ω |x|
Ω
Now, by using h+ 6= 0, (2.1), (1.7), the Hölder inequality and the Sobolev-Hardy
inequality, we have
Z Z
|u|q
1
|u|p
|v|p λ
|v|q
p
p
J(u, v) ≥
|∇u| − µ p + |∇v| − µ p dx −
h(x)( s + s )dx
p Ω
|x|
|x|
q Ω
|x|
|x|
Z
Z
Z
h
i
p∗ (s1 )
p∗ (s2 )
α
β
|v|
|u| |v|
|u|
1
dx
−
dx
−
Q(x)
dx
− ∗
s2
p (t) Ω |x|s1
|x − x0 |t
Ω |x|
Ω
Z Z
p
p
1
|u|
|v|
λ
|u|q
|v|q
=
|∇u|p − µ p + |∇v|p − µ p dx −
h(x)( s + s )dx
p Ω
|x|
|x|
q Ω
|x|
|x|
Z h
p
p
|u|
|v|
1
− ∗
|∇u|p − µ p + |∇v|p − µ p dx
p (t) Ω
|x|
|x|
6
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
i
|v|q
|u|q
+
)dx
|x|s
|x|s
Ω
Z 1
|u|p
1
|v|p |∇u|p − µ p + |∇v|p − µ p dx
− ∗
≥
p p (t) Ω
|x|
|x|
Z
1
|u|q
1 |v|q
h(x)( s + s )dx
−λ
− ∗
q
p (t) Ω
|x|
|x|
1
1 p
p
(kukµ + kvkµ )
≥
−
p p∗ (t)
p∗ (s)−q
1
1 N ωN R0N −s p∗ (s) − pq
−λ
− ∗
Aµ,s |h+ |∞ (kukqµ + kvkqµ )
q
p (t)
N −s
p∗ (s)−q
1
h 1
i
1 p
1 N ωN R0N −s p∗ (s) − pq
t −λ
− ∗
− ∗
Aµ,s |h+ |∞ tq
≥ 2 inf
t≥0
p p (s)
q
p (s)
N −s
Z
−λ
h(x)(
p
≥ −dλ p−q .
Here dΩ := supx,y∈Ω |x − y| is the diameter of Ω and d is a positive constant
depending on N, |Ω|, |h+ |∞ , Aµ,s , s1 , s2 and q.
Recall that a sequence (un , vn )n∈N is a (P S)c sequence for the functional J
if J(un , vn ) → c and J 0 (un , vn ) → 0. If any (P S)c sequence (un , vn )n∈N has a
convergent subsequence, we say that J satisfies the (P S)c condition.
Lemma 2.2. Assume that N ≥ 3, 0 ≤ µ < µ, (H1), h+ 6= 0 and Q(0) = 0. Then
J(u, v) satisfies the (P S)c condition with c satisfying
N −s1
N −s2
n p−s
p − s2
1
p−s1
p−s2
,
c < c∗ := min
Aµ,s
A
µ,s2 ,
1
p(N − s1 )
p(N − s2 )
(2.2)
N −t o
p
1
p−t
p−t
− dλ p−q .
N −p St,α,β
p(N − t)
p−t
kQk∞
Proof. It is easy to see that the (P S)c sequence (un , vn )n∈N of J(u, v) is bounded
in W . Then (un , vn ) * (u, v) weakly in W as n → ∞, which implies un * u
weakly and vn * v weakly in W01,p (Ω) as n → ∞. Passing to a subsequence we
may assume that
|∇un |p dx * α,
|un |p
dx * β,
|x|p
|∇vn |p dx * α
e,
|vn |p
e
dx * β,
|x|p
∗
∗
|un |p (s1 )
|vn |p (s2 )
dx * γ,
dx * γ
e,
s
1
|x|
|x|s2
|un |α |vn |β
Q(x)
dx * ν
|x − x0 |t
weakly in the sense of measures. Using the concentration-compactness principle in
[21], there exist an at most countable set I, a set of points {xi }i∈I ∈ Ω \ {0}, real
numbers axi , e
axi , dxi , i ∈ I, a0 , e
a0 , b0 , eb0 , c0 , e
c0 and d0 , such that
X
α ≥ |∇u|p dx +
axi δxi + a0 δ0 ,
(2.3)
i∈I
EJDE-2012/169
EXISTENCE OF SOLUTIONS
α
e ≥ |∇u|p dx +
X
e
axi δxi + e
a0 δ0 ,
7
(2.4)
i∈I
|u|p
dx + b0 δ0 ,
|x|p
|v|p
βe =
dx + eb0 δ0 ,
|x|p
β=
(2.5)
(2.6)
∗
γ=
|u|p (s1 )
+ c0 δ0 ,
|x|s1
(2.7)
∗
|v|p (s2 )
dx + e
c0 δ0 ,
|x|s2
X
|u|α |v|β
ν = Q(x)
dx
+
Q(xi )dxi δxi + Q(0)d0 δ0 ,
|x − x0 |t
γ
e=
(2.8)
(2.9)
i∈I
where δx is the Dirac-mass of mass 1 concentrated at the point x.
First, we consider the possibility of the concentration at {xi }i∈I ∈ Ω \ {0}.
Let > 0 be small enough, take ηxi ∈ Cc∞ (B2ε (xi )), such that ηxi |Bε (xi ) = 1,
0 ≤ ηxi ≤ 1 and |∇ηxi (x)| ≤ Cε . Then
o(1) = hJ 0 (un , vn ), (ηxpi un , ηxpi vn )i
Z
=
|∇un |p−2 ∇un ∇(ηxpi un ) + |∇vn |p−2 ∇vn ∇(ηxpi vn ) dx
Ω
Z Z
|un |α |vn |β p
|un |p p
|vn |p p −
Q(x)
η
dx
−
µ
ηxi +
η dx
xi
t
p
|x − x0 |
|x|
|x|p xi
Ω
Ω
Z
|u |q
|vn |q p n
p
η
+
η dx
−λ
h(x)
|x|s xi
|x|s xi
Ω
Z
Z
∗
∗
|vn |p (s2 ) p
|un |p (s1 ) p
η
dx
−
ηxi dx.
−
xi
|x|s1
|x|s2
Ω
Ω
From (2.5)-(2.9), one can obtain
Z Z
Z
|vn |p p |un |p p
p
p e
η
+
η
dx
=
lim
η
dβ
η
d
β
= 0, (2.10)
+
xi
xi
ε→0 n→∞ Ω
ε→0
|x|p xi
|x|p xi
Ω
Ω
Z Z
∗
∗
Z
|un |p (s1 ) p
|vn |p (s2 ) p p
p
η
+
η
dx
=
lim
η
dγ
+
η
de
γ
= 0,
lim lim
xi
xi
xi
xi
ε→0 n→∞ Ω
ε→0
|x|s1
|x|s2
Ω
Ω
Z
|u |q
|vn |q p n
p
h(x)
lim lim
η
+
η dx = 0,
ε→0 n→∞ Ω
|x|s xi
|x|s xi
Z
Z
|un |α |vn |β p
Q(x)
η
dx
=
lim
ηxpi dν = Q(xi )dxi .
lim lim
ε→0 n→∞ Ω
ε→0 Ω
|x − x0 |t xi
lim lim
Thus,
Z |∇un |p−2 ∇un ∇(ηxpi un )
+ |∇vn |p−2 ∇vn ∇(ηxpi vn ) dx − Q(xi )dxi .
0 = lim lim
ε→0 n→∞
Ω
(2.11)
8
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
Moreover, we have
lim lim Z
ε→0 n→∞
Ω
un |∇un |p−2 ∇un ∇ηxpi dx
Z
≤ lim lim
p
|∇ηxpi |p dx
1/p Z
|un |
ε→0 n→∞
Ω
Z
1/p
≤ C lim lim
|un |p |∇ηxpi |p dx
ε→0 n→∞
Ω
Z
1/p
p
= C lim
|u| |∇ηxpi |p dx
ε→0
Ω
Z
1/N Z
≤ C lim
|∇ηxpi |N dx
ε→0
≤ C lim
Bε (xi )
Z
ε→0
|∇un |p dx
p−1
p
Ω
(2.12)
∗
|u|p dx
1/P ∗
Bε (xi )
∗
|u|p dx
1/P ∗
= 0.
Bε (xi )
Similarly,
lim lim ε→0 n→∞
Z
Ω
vn |∇vn |p−2 ∇vn ∇ηxpi dx = 0.
Combining (2.11)-(2.13), there holds
Z
0 = lim lim
(|ηxi ∇un |p + |ηxi ∇vn |p )dx − Q(xi )dxi
ε→0 n→∞ Ω
Z
α − Q(xi )dxi .
= lim
ηxpi dα + ηxi de
ε→0
(2.13)
(2.14)
Ω
On the other hand, (1.6) implies
p
Z
1
|ηx un |α |ηxi vn |β p∗ (t)
St,α,β
Q(x) i
dx
p
p∗ (t)
|x − x0 |t
Ω
kQk∞
Z |ηx un |p + |ηxi vn |p ≤
|∇(ηxi un )|p + |∇(ηxi vn )|p − µ i
dx.
|x|p
Ω
(2.15)
Note that
Z
Z
p
|∇ηxi |p |vn |p dx = 0.
|∇ηxi | |un | dx = lim lim
lim lim
ε→0 n→∞
p
ε→0 n→∞
Ω
Ω
From this equality, (2.12) and (2.13), we obtain
Z
Z
|∇(ηxi un )|p dx,
lim lim
|ηxi ∇un |p dx = lim lim
ε→0 n→∞ Ω
ε→0 n→∞ Ω
Z
Z
p
lim lim
|ηxi ∇vn | dx = lim lim
|∇(ηxi vn )|p dx.
ε→0 n→∞
ε→0 n→∞
Ω
(2.16)
(2.17)
Ω
Relations (2.9), (2.10) and (2.15)-(2.17) imply
Z
Z
p
1
|ηxi |p dα + lim
|ηxi |p de
α.
St,α,β Q(xi )dxi p∗ (t) ≤ lim
p
ε→0 Ω
ε→0 Ω
p∗ (t)
kQk∞
(2.18)
Combining (2.14) and (2.18),
1
p
p∗ (t)
kQk∞
St,α,β Q(xi )dxi
p∗p(t)
≤ Q(xi )dxi ,
(2.19)
EJDE-2012/169
EXISTENCE OF SOLUTIONS
9
which implies that either
1
or Q(xi )dxi ≥
Q(xi )dxi = 0,
N −p
p−t
N −t
p−t
St,α,β
.
(2.20)
kQk∞
Now, we consider the possibility of the concentration at 0. For > 0 be small
enough, take η0 ∈ Cc∞ (B2ε (0)), such that η0 |Bε (0) = 1, 0 ≤ η0 ≤ 1 and |∇η0 (x)| ≤
C
ε . Then
o(1) = hJ 0 (un , vn ), (η0p un , 0)i
Z
Z
Z
|un |q p
|un |p p
η
dx
−
λ
η dx
h(x)
=
|∇un |p−2 ∇un ∇(η0p un )dx − µ
0
p
|x|s 0
Ω
Ω
Ω |x|
Z
Z
∗
|un |p (s1 ) p
α
|un |α |vn |β p
η dx.
−
η
dx
−
Q(x)
0
s
|x| 1
α+β Ω
|x − x0 |t 0
Ω
From (2.5), (2.7), (2.9) and Q(0) = 0, we obtain
Z
Z
∗
|un |p p
|un |p (s1 ) p
lim lim
η
dx
=
b
,
lim
lim
η0 dx = c0 ,
0
0
ε→0 n→∞ Ω |x|p
ε→0 n→∞ Ω
|x|s1
Z
Z
|un |α |vn |β p
|un |q p
lim lim
Q(x)
η
lim
h(x)
η dx = 0.
dx
=
lim
ε→0 n→∞ Ω
ε→0 n→∞ Ω
|x − x0 |t 0
|x|s 0
Thus,
Z
|∇un |p−2 ∇un ∇(η0p un )dx − µb0 − c0 .
0 = lim lim
ε→0 n→∞
(2.21)
Ω
Note that
Z
un |∇un |p−2 ∇un ∇η0p dx = 0.
lim lim
ε→0 n→∞
Ω
This equality and (2.21) yield
Z
η0p dα − µb0 = c0 .
lim
ε→0
(2.22)
Ω
On the other hand, (1.5) implies
Z p
Z |η u |p∗ (s1 ) p∗ (s
|η0 un |p 0 n
1)
p
≤
|∇(η
u
)|
−
µ
dx.
Aµ,s1
dx
0
n
s
|x| 1
|x|p
Ω
Ω
Thus
p
p∗ (s1 )
Aµ,s1 c0
Note that
Z
|∇(η0 un )|p dx − µb0 .
≤ lim lim
ε→0 n→∞
lim lim
ε→0 n→∞
Z
|η0 ∇un |p dx = lim lim
ε→0 n→∞
Ω
(2.23)
Ω
Z
|∇(η0 un )|p dx,
Ω
which together with (2.23) imply
p
p∗ (s1 )
Aµ,s1 c0
Z
≤ lim
ε→0
|η0 |p dα − µb0 .
(2.24)
≤ c0 ,
(2.25)
Ω
Therefore, from (2.22) and (2.24),
p
∗ (s )
1
Aµ,s1 c0p
which implies that either
N −s1
c0 = 0,
p−s1
or c0 ≥ Aµ,s
1 .
(2.26)
10
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
Similarly, either
N −s2
c0 = 0,
p−s2
or c0 ≥ Aµ,s
2 .
(2.27)
Recall that un * u weakly and vn * v weakly in W01,p (Ω), we have
c + o(1)
= J(un , vn )
Z |un − u|p
|vn − v|p 1
|∇un − ∇u|p − µ
dx
+ |∇vn − ∇v|p − µ
=
p
p Ω
|x|
|x|p
Z
Z
∗
∗
|un − u|p (s1 )
|vn − v|p (s2 )
1
1
− ∗
dx − ∗
dx
p (s1 ) Ω
|x|s1
p (s2 ) Ω
|x|s2
Z
|un − u|α |vn − v|β
1
dx + J(u, v).
Q(x)
− ∗
p (t) Ω
|x − x0 |t
(2.28)
On the other hand, from o(1) = J 0 (un , vn ), we obtain
J 0 (un , vn ) = 0.
(2.29)
Thus, 0 = hJ 0 (u, v), (u, v)i, which together with o(1) = hJ 0 (un , vn ), (un , vn )i imply
o(1) =
|un − u|p
|vn − v|p + |∇vn − ∇v|p − µ
dx
p
|x|
|x|p
Ω
Z
Z
∗
∗
|un − u|p (s1 )
|vn − v|p (s2 )
−
dx −
dx
|x|s1
|x|s2
Ω
Ω
Z
|un − u|α |vn − v|β
−
Q(x)
dx.
|x − x0 |t
Ω
Z |∇un − ∇u|p − µ
(2.30)
From (2.28)-(2.30) and Lemma 2.1,
c + o(1) ≥
Z
Z
∗
∗
|un − u|p (s1 )
|vn − v|p (s2 )
p − s2
p − s1
dx
+
dx
p(N − s1 ) Ω
|x|s1
p(N − s2 ) Ω
|x|s2
(2.31)
Z
p
|un − u|α |vn − v|β
p−t
p−q
Q(x)
dx − dλ
.
+
p(N − t) Ω
|x − x0 |t
Passing to the limit in (2.31) as n → ∞, we have
c≥
p
p−t X
p − s2
p − s1
e
c0 +
c0 +
Q(xi )dxi − dλ p−q .
2(N − s1 )
p(N − s2 )
p(N − t)
(2.32)
i∈I
By the assumption c < c∗ and in view of (2.20), (2.26) and (2.27), there holds
c0 = e
c0 = 0, Q(xi )dxi = 0, i ∈ I. Up to a subsequence, (un , vn ) → (u, v) strongly
in W as n → ∞.
When the restriction Q(0) = 0 is removed, we establish the following version of
Lemma 2.2.
EJDE-2012/169
EXISTENCE OF SOLUTIONS
11
Lemma 2.3. Assume that N ≥ 3, 0 ≤ µ < µ, (H1) and h+ 6= 0. Then J(u, v)
satisfies the (P S)c condition with c satisfying
−s1
−s2
Np−s
Np−s
n p − s 1
p − s2 1
1
1
2
c < c0 := min
Aµ,s1
,
Aµ,s2
,
p(N − s1 ) p
p(N − s2 ) p
−t
Np−t
1
o
S
t,α,β
p
p
p−t
− dλ p−q .
N −p
p(N − t)
p−t
kQk∞
(2.33)
The proof of the above lemma is similar to Lemma 2.2 and is omitted.
Lemma 2.4 ([18]). Assume that 1 < p < N , 0 ≤ t < p and 0 ≤ µ < µ. Then the
limiting problem
∗
−∆p u − µ
|u|p−1
|u|p (t)−1
=
,
p
|x|
|x|t
u ∈ D1,p (RN ),
in RN \ {0},
in RN \ {0},
u > 0,
has positive radial ground states
V (x) , that satisfy
Z Ω
p−N
p
p−N
x
|x|
Up,µ ( ) = p Up,µ ( ),
|V (x)|p |∇V (x)| − µ
dx =
|x|p
p
∗
Z
Ω
|V (x)|p
|x|t
∀ > 0,
(t)
(2.34)
N −t
dx = (Aµ,t ) p−t ,
where Up,µ (x) = Up,µ (|x|) is the unique radial solution of the limiting problem with
1
(N − t)(µ − µ) p∗ (t)−p
.
Up,µ (1) =
N −p
Furthermore, Up,µ has the following properties:
lim ra(µ) Up,µ (r) = C1 > 0,
r→0
lim rb(µ) Up,µ (r) = C2 > 0,
r→+∞
0
lim ra(µ)+1 |Up,µ
(r)| = C1 a(µ) ≥ 0,
r→0
0
lim rb(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
Lemma 2.5. Under the assumptions of Theorem 1.2, there exists (u1 , v1 ) ∈ W \
{(0, 0)} and Λ1 > 0, such that for 0 < λ < Λ1 , there holds
sup J(tu1 , tv1 ) <
t≥0
p−t
p(N − t)
1
N −p
p−t
kQk∞
N −t
p
p−t
St,α,β
(µ) − dλ p−q .
(2.35)
12
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
Proof. First, we give some estimates on the extremal function V (x) defined in
(2.34). For m ∈ N large, choose ϕ(x) ∈ C0∞ (RN ), 0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1 for
1
1
, ϕ(x) = 0 for |x| ≥ m
, k∇ϕ(x)kLp (Ω) ≤ 4m, set u (x) = ϕ(x)V (x). For
|x| ≤ 2m
→ 0, the behavior of u has to be the same as that of V , but we need precise
estimates of the error terms. For 1 < p < N , 0 ≤ t < p and 1 < q < p∗ (s), we have
the following estimates [4]:
Z N −t
|u |p |∇u |p − µ p dx = (Aµ,t ) p−t + O(b(µ)p+p−N ),
(2.36)
|x|
Ω
Z
∗
N −t
∗
|u |p (t)
dx = (Aµ,t ) p−t + O(b(µ)p (t)−N +t ),
(2.37)
t
|x|
Ω

N
−s

CN −s+(1− p )q ,
q>N

Z
b(µ) ,

q
N
|u |
−s
dx ≥ CN −s+(1− p )q | ln |, q = N
(2.38)
b(µ) ,
s

N
Ω |x|

N −s
Cq(b(µ)+1− p )q ,
q < b(µ) .
Now, we consider the functional I : W → R defined by
Z Z
|v|p 1
1
|u|p
|u|α |v|β
p
p
dx.
I(u, v) =
|∇u| − µ p + |∇v| − µ p dx − ∗
Q(x)
p Ω
|x|
|x|
p (t) Ω
|x − x0 |t
1
1
Let u1 = α p u , v1 = β p u and define the function g1 (t) := J(tu1 , tv1 ), t ≥ 0. Note
that limt→+∞ g1 (t) = −∞ and g1 (t) > 0 as t is close to 0. Thus supt≥0 g1 (t) is
attained at some finite t > 0 with g10 (t ) = 0. Furthermore, C 0 < t < C 00 ; where
C 0 and C 00 are the positive constants independent of . We have
Z
∗
∗
tp (t) α β
|u |p (t)
I(tu1 , tv1 ) = y(tu1 , tv1 ) − ∗ (α p β p ) (Q(x) − Q(x0 ))
dx. (2.39)
p (t)
|x − x0 |t
Ω
where
y(tu1 , tv1 )
Z Z
∗
∗
h tp
|u |p tp (t) αp βp
|u |p (t) i
p
:=
(α + β)
|∇u | − µ p dx − ∗ (α β )
dx .
Q(y0 )
p
|x|
p (t)
|x − x0 |t
Ω
Ω
Note that
∗
∗ (t)
tp
tp (t) 1
1 A p∗p(t)−p
A− ∗ B =
− ∗
,
p
p (t)
p p (t) B p∗ (t)
t≥0 p
From (H2), (2.36), (2.37) and (2.40) it follows that
sup
A, B > 0.
(2.40)
sup y(tu1 , tv1 )
t≥0
= y(t u1 , t v1 )
R |u |p
p
−t
(α
+
β)
|∇u
|
−
µ
dx Np−t
p
|x|
Ω
p−t
≤
∗
p(N − t) ((α αp β βp )kQk∞ R |u |p (t)t dx) NN−p
−t
Ω |x−x0 |
p−t
≤
p(N − t)
≤
p−t
p(N − t)
1
N −p
p−t
kQk∞
1
N −p
p−t
kQk∞
β
h α α+β
β
β
h α α+β
β
N −t
N −t
α β α+β
(Aµ,t ) p−t + O(b(µ)p+p−N ) i p−t
+
N −t
α
(Aµ,t ) p−t + O(b(µ)p∗ (t)−N +t )
+
α β α+β
α
N −t
(Aµ,t ) p−t
−t
i Np−t
+ O(b(µ)p+p−N )
EJDE-2012/169
≤
p−t
p(N − t)
EXISTENCE OF SOLUTIONS
1
N −p
p−t
St,α,β
−t
Np−t
+ O(b(µ)p+p−N ).
13
(2.41)
kQk∞
On the other hand, (H2) implies that there exists r1 < r, such that for x ∈ Br1 (y0 ),
|Q(x) − Q(x0 )| ≤ C|x − x0 |ϑ . Thus
Z
Z
∗
p∗ (t)
|u |p (t)
(Q(x) − Q(x0 )) |u |
≤C
dx
dx
|Q(x)
−
Q(x
)|
0
|x − x0 |t
|x − x0 |t
Ω
Ω
Z
∗
|x − x0 |ϑ |u |p (t)
=C
dx = O(ϑ−t )
t
|x
−
x
|
0
B2r (x0 )
(2.42)
From (2.39), (2.41) and (2.42), we conclude that
sup I(tu1 , tv1 ) = I(t u1 , t v1 )
t≥0
≤
p−t
p(N − t)
1
N −p
p−t
St,α,β
−t
Np−t
+ O(b(µ)p+p−N ).
(2.43)
kQk∞
Observe that there exists Λ∗1 > 0, such that for 0 < λ < Λ∗1 and
p−t
p(N − t)
1
N −p
p−t
St,α,β
−t
Np−t
p
− dλ p−q > 0.
kQk∞
Then for 0 < λ < Λ∗1 , there exists t1 ∈ (0, 1), such that
Z 1 p
sup J(tu1 , tv1 ) ≤ sup t
|∇u1 |p + |∇v1 |p dx
0≤t≤t1
0≤t≤t1 p
Ω
−t
Np−t
p
p−t
1
S
<
− dλ p−q .
t,α,β
N −p
p(N − t)
p−t
kQk∞
(2.44)
On the other hand,
sup J(tu1 , tv1 )
t≥t1
Z
Z
∗
h
∗
|u1 |q
1
|u1 |p (s1 ) i
λ
h(x) s dx − ∗
tp (s1 )
dx
≤ sup I(tu1 , tv1 ) − tq
q
|x|
p (s1 )
|x|s1
t≥t1
Ω
Ω
Z
Z
∗
|u1 |p (s1 ) i
1
λ
p∗ (s )
t1 1
≤ sup I(tu1 , tv1 ) − tq1
h(x)|u1 |q dx − ∗
dx
(2.45)
q
p (s1 )
|x|s1
t≥t1
Ω
Ω
−t
Np−t
p−t
1
≤
St,α,β
+ O(b(µ)p+p−N )
N −p
p(N − t)
p−t
kQk∞
Z
Z
∗
|u |p (s1 )
|u |q
−C
dx
−
λC
h(x) s dx
s
1
|x|
|x|
Ω
Ω
From (2.37),
Z
Ω
∗
∗
|u |p (s1 )
dx ≥ O(b(µ)p (s1 )−N +s1 ).
|x|s1
(2.46)
14
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
Also, from (2.38), it follows that
|u |q
h(x) s dx ≥ β0
|x|
Ω
Z
N −s
b(µ) ,
Since q ≥

N −s+(1− N
p )q ,


C
Z
Ω
q>
|u |q
N −s+(1− N
)q
p
dx ≥ C
| ln |, q =

|x|s

Cq(b(µ)+1− Np )q ,
q<
N −s
b(µ) ,
N −s
b(µ) ,
N −s
b(µ) .
(2.47)
by (2.45)-(2.47) we have
p−t
p(N − t)
sup J(tu1 , tv1 ) ≤
t≥t1
1
St,α,β
N −p
p−t
−t
Np−t
+ O(b(µ)p+p−N )
kQk∞
( N −s+(1− N )q
p
C
,
q=
N −s
b(µ) ,
N −s
b(µ) .
Note that b(µ)p + p − N < b(µ)p∗ (s1 ) − N + s1 , then we have
−t
Np−t
p−t
1
sup J(tu1 , tv1 ) ≤
N −p St,α,β
p(N − t)
p−t
t≥t1
kQk∞
( N −s+(1− N )q
p
C
,
q>
b(µ)p∗ (s1 )−N +s1
+ O(
)−λ
)q
N −s+(1− N
p
| ln |, q =
C
N −s
b(µ) ,
N −s
b(µ) .
b(µ)p∗ (s1 )−N +s1
+ O(
)−λ
C
N −s+(1− N
p )q
q>
| ln |,
(2.48)
Note that N > p2 , b(µ) ≥
N −s
q .
Thus
N p − q
N N − s + (1 − )q
< b(µ)p + p − N − N − s + (1 − )q .
p
q
p
p−q
Choose λ = τ , where N −s+(1− Np )q q < τ < b(µ)p+p−N − N −s+(1− Np )q .
Then
p
pτ
N
N
λO(N −s+(1− p )q ) = O(τ +N −s+(1− p )q ), dλ p−q = O( p−q ).
pτ
, τ + N − s + (1 − Np )q < b(µ)p + p − N , taking
Since τ + N − s + (1 − Np )q < p−q
small enough we deduce that there exists δ > 0, such that
∗
O(b(µ)p
(s1 )−N +s1
p
N
) − λO(N −s+(1− p )q ) < −dλ p−q ,
p
∀λ : 0 < λ p−q < δ. (2.49)
Choose Λ1 = min{Λ∗1 , p−q
p δ}. Then for all λ ∈ (0, Λ1 ) we have
sup J(tu1 , tv1 ) ≤
t≥t1
p−t
p(N − t)
1
N −p
p−t
St,α,β
−t
Np−t
p
− dλ p−q .
kQk∞
Together with (2.44), we get the conclusion of Lemma 2.5.
3. Proof of the main results
Proof of Theorem 1.1. Let
r := k(u, v)k ,
f (r) :=
p∗ (s )
− p1
µ,s1
∗
∗
p (s2 )
p (t)
∗
−
−
1
1
Aµ,s2 p rp (s2 ) − ∗ St,α,βp kQk∞ ,
∗
p (s2 )
p (t)
∗
N −s p (s)−q
q
−
λ N ωN R0
p∗ (s)
h(r) :=
Aµ,sp rq .
q
N −s
1 p
1
r − ∗
A
p
p (s1 )
∗
rp
(s1 )
−
EJDE-2012/169
EXISTENCE OF SOLUTIONS
15
From (1.5), (1.6) and (1.7),
J(u, v) ≥ f (r) − h(r).
∗
∗
Note that p < p (s1 ), p (s2 ), p∗ (t), it is easy to see that there exists % > 0 such
that f (r) achieves its maximum at % and f (%) > 0. Therefore, there exists Λ11 > 0,
such that for 0 < λ < Λ11 ,
inf
k(u,v)k=%
I(u, v) ≥ f (%) − h(%) > 0.
(3.1)
On the other hand, set B% = {(u, v); k(u, v)k ≤ %}. For (u, v) 6= (0, 0), we can
choose d > 0 small enough, such that (du, dv) ∈ B% and
I(du, dv) < 0.
(3.2)
Thus,
−∞<
inf
I(u, v) < 0.
(3.3)
(u,v)∈B%
Now we can apply the Ekeland variational principle in [22] and obtain {(un , vn )} ⊂
B% , such that
I(un , vn ) ≤
inf
I(u, v) +
(u,v)∈B%
I(un , vn ) ≤ I(u, v) +
1
,
n
1
k(un − u, vn − v)k,
n
(3.4)
(3.5)
for all (u, v) ∈ BR . Define
1
k(un − u, vn − v)k.
(3.6)
n
By (3.5), we have (un , vn ) is the minimizer of J1 (u, v) on B% . (3.1), (3.3) and (3.4)
imply that there exists > 0 and k ∈ N , such that for n ≥ k, {(u, v), k(u, v)k ≤
% − }. Therefore, for n ≥ k and (φ, ϕ) ∈ W , we can choose t > 0 small enough,
such that (un + tφ, vn + tϕ) ∈ B% and
J1 (u, v) := J(u, v) +
J1 (un + tφ, vn + tϕ) − J1 (un , vn )
≥ 0.
t
That is,
J(un + tφ, vn + tϕ) − J(un , vn )
1
+ k(φ, ϕ)k ≥ 0.
t
n
Passing to the limit in (3.7) as n → 0, one can obtain
1
hJ 0 (un , vn ), (φ, ϕ)i ≥ − k(φ, ϕ)k,
n
which implies
1
kJ 0 (un , vn )k ≤ .
n
Combining (3.4) and (3.8), there holds
lim J(un , vn ) =
n→∞
inf
J(u, v) < 0,
(3.8)
(3.9)
(u,v)∈B%
lim J 0 (un , vn ) = 0.
n→∞
(3.7)
(3.10)
We note that there exists Λ∗11 ∈ (0, Λ11 ), such that for 0 < λ < Λ∗11 , and c0 >
inf (u,v)∈B% I(u, v), where c0 is defined in Lemma 2.3. Thus, (3.9) and (3.10) and
Lemma Lemma 2.3 imply that for 0 < λ < Λ∗11 , (un , vn ) → (u, v) strongly in
16
N. NYAMORADI, M. JAVIDI
EJDE-2012/169
W . Therefore, (u, v) is a nontrivial solution of problem (1.1) satisfying J(u, v) =
inf (u,v)∈B% J(u, v) < 0. Note that J(u, v) = J(|u|, |v|) and
(|u|, |v|) ∈ {(u, v), k(u, v)k ≤ % − },
we have I(|u|, |v|) = inf (u,v)∈B% J(u, v) and J 0 (|u|, |v|) = 0. Then problem (1.1) has
a nontrivial nonnegative solution. By the strongly maximum principle, we get the
conclusion of Theorem 1.1.
Proof of Theorem 1.2. In view of the proof of Theorem 1.1, we know that for 0 <
λ < Λ11 , there exists % > 0, such that inf k(u,v)k=% I(u, v) ≥ ϑ∗ > 0. Moreover,
(3.9) and (3.10) hold. We note that there exists Λ12 ∈ (0, Λ11 ), such that for
0 < λ < Λ12 , c∗ > inf (u,v)∈B% J(u, v), where c∗ is defined in Lemma 2.2. Thus (3.9)
and (3.10) and Lemma 2.2 imply that (un , vn ) → (u, v) strongly in W . Standard
argument shows that for 0 < λ < Λ12 , problem (1.1) has at least one positive
solution satisfying J(u, v) < 0 and J 0 (u, v) = 0.
Now we prove a second positive solution. It is easy to see J(0, 0) = 0. Set
Λ∗∗ = min{Λ12 , Λ1 }, where Λ1 is given in Lemma 2.5. Then it follows from Lemma
2.5 there exists (u0 , v 0 ) ∈ W \ {0}, such that for 0 < λ < Λ∗∗ ,
sup J(tu0 , tv 0 ) < c∗ .
t≥0
On the other hand we obtain that liml→∞ J(lu0 , lv 0 ) = −∞. Thus there exists
l0 > 0 such that k(l0 u0 , l0 v 0 )k > % and J(l0 u0 , l0 v 0 ) < 0. Let
c := inf sup J(γ(t)),
γ∈Γ t∈[0,1]
where
Γ := {γ ∈ C 0 ([0, 1], W ) : γ(0) = (0, 0), γ(1) = (l0 u0 , l0 v 0 )}.
Thus, it follows from the mountain pass lemma in [2] that there exists a sequence
(un , vn ) ∈ W such that
lim J(un , vn ) = c,
n→∞
lim J 0 (un , vn ) = 0.
n→∞
Moreover, c ∈ (0, c∗ ). From Lemma 2.2, (un , vn ) → (u, v) strongly in W , which
implies that J(u, v) = c and J 0 (u, v) = 0, Therefore, (u, v) is a second nontrivial
solution of (1.1). Set u+ = max{u, 0}, v + = max{v, 0}. Replacing
Z
Z
Z
Z
Z
∗
∗
|u|q
|v|q
|u|p (s1 )
|v|p (s2 )
|u|α |v|β
dx,
dx,
dx,
dx,
Q(x)
dx
s
s
s
s
2
|x| 1
|x − x0 |t
Ω |x|
Ω |x|
Ω
Ω |x|
Ω
by
Z
Z
∗
(u+ )q
(v + )q
(u+ )p (s1 )
dx,
dx,
dx,
s
s
|x|s1
Ω |x|
Ω |x|
Ω
Z
Z
∗
(u+ )α (v + )β
(v + )p (s2 )
dx,
Q(x)
dx
|x|s2
|x − x0 |t
Ω
Ω
Z
and repeating the above process, we have a nonnegative solution (e
u, ve) of problem
(1.1) satisfying J(e
u, ve) > 0. Then by the strongly maximum principle, we have a
second positive solution.
EJDE-2012/169
EXISTENCE OF SOLUTIONS
17
Acknowledgments. The authors would like to thank the anonymous referees for
his/her valuable suggestions and comments.
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Nemat Nyamoradi
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran
E-mail address: nyamoradi@razi.ac.ir, neamat80@yahoo.com
Mohamad Javidi
Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah,
Iran
E-mail address: mo javidi@yahoo.com