On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents
We consider a quasilinear elliptic system involving the critical Hardy–Sobolev exponent and the Sobolev exponent. We use variational methods and analytic techniques to establish the existence of positive solutions of the system.
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nasplib_isofts_kiev_ua-123456789-1644152025-02-09T21:37:57Z On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents Про p-Лапласову систему з критичними показниками Хардi – Соболєва та критичними показниками Соболєва Nyamoradi, N. Статті We consider a quasilinear elliptic system involving the critical Hardy–Sobolev exponent and the Sobolev exponent. We use variational methods and analytic techniques to establish the existence of positive solutions of the system. Розглянуто квазiлiнiйну елiптичну систему з критичними показниками Хардi – Соболєва та Соболєва. Iз застосуванням варiацiйних методiв та аналiтичного пiдходу встановлено iснування додатних розв’язкiв системи. 2012 Article On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents / N. Nyamoradi // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 797-810. — Бібліогр.: 20 назв. — англ. 1027-3190 https://nasplib.isofts.kiev.ua/handle/123456789/164415 517.9 en Український математичний журнал application/pdf Інститут математики НАН України |
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Статті Статті Nyamoradi, N. On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents Український математичний журнал |
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We consider a quasilinear elliptic system involving the critical Hardy–Sobolev exponent and the Sobolev exponent. We use variational methods and analytic techniques to establish the existence of positive solutions of the system. |
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Article |
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Nyamoradi, N. |
| author_facet |
Nyamoradi, N. |
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Nyamoradi, N. |
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On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents |
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On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents |
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On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents |
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On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents |
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On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents |
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on a p-laplacian system with critical hardy–sobolev exponents and critical sobolev exponents |
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Інститут математики НАН України |
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2012 |
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| citation_txt |
On a p-Laplacian system with critical Hardy–Sobolev exponents and critical Sobolev exponents / N. Nyamoradi // Український математичний журнал. — 2012. — Т. 64, № 6. — С. 797-810. — Бібліогр.: 20 назв. — англ. |
| series |
Український математичний журнал |
| work_keys_str_mv |
AT nyamoradin onaplaplaciansystemwithcriticalhardysobolevexponentsandcriticalsobolevexponents AT nyamoradin proplaplasovusistemuzkritičnimipokaznikamihardisobolêvatakritičnimipokaznikamisobolêva |
| first_indexed |
2025-12-01T01:27:06Z |
| last_indexed |
2025-12-01T01:27:06Z |
| _version_ |
1850267335013695488 |
| fulltext |
UDC 517.9
N. Nyamoradi (Razi Univ., Iran)
ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV
EXPONENTS AND CRITICAL SOBOLEV EXPONENTS
ПРО p-ЛАПЛАСОВУ СИСТЕМУ З КРИТИЧНИМИ ПОКАЗНИКАМИ
ХАРДI – СОБОЛЄВА ТА КРИТИЧНИМИ ПОКАЗНИКАМИ СОБОЛЄВА
We consider a quasilinear elliptic system involving the critical Hardy – Sobolev exponent and Sobolev exponent. Using
variational methods and analytic techniques, we establish the existence of positive solutions of the system.
Розглянуто квазiлiнiйну елiптичну систему з критичними показниками Хардi – Соболєва та Соболєва. Iз застосуван-
ням варiацiйних методiв та аналiтичного пiдходу встановлено iснування додатних розв’язкiв системи.
1. Introduction. The aim of this paper is to establish the existence of nontrivial nonnegative solution
to the semilinear elliptic system
−∆pu1 − µ
|u1|p−2u1
|x|p
=
1
p∗
Fu1(u1, . . . , uk) +
|u1|p
∗(t)−2u1
|x|t
+ λ
|u1|p−2u1
|x|s
, x ∈ Ω,
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1)
−∆puk − µ
|uk|p−2uk
|x|p
=
1
p∗
Fuk(u1, . . . , uk) +
|uk|p
∗(t)−2uk
|x|t
+ λ
|uk|p−2uk
|x|s
, x ∈ Ω,
ui = 0, 1 ≤ i ≤ k, on ∂Ω,
where ∆pui = div(|∇ui|p−2∇ui), 0 ∈ Ω is a bounded domain in RN , N ≥ 3, with smooth
boundary ∂Ω, 1 < p < N, 0 ≤ µ < µ ,
(
N − p
p
)p
, λ > 0, 0 ≤ t < p, p∗(t) ,
p(N − t)
N − p
is the Hardy – Sobolev critical exponent, p∗ = p∗(0) =
pN
N − p
is the Sobolev critical exponent
and ∇F (u1, . . . , un) = (Fu1(u1, . . . , uk), . . . , Fuk(u1, . . . , uk)), where F : (R+)k → R+ are C1
function with positively homogeneous of degree p∗.
We denote by D1,p(Ω) the completion of C∞0 (Ω) with respect to the norm
(∫
Ω
|∇ · |pdx
)1/p
.
Problem () is related to the well known Caffarelli – Kohn – Nirenberg inequality in [5],∫
Ω
|u|r
|x|t
dx
p/r ≤ Cr,t,p ∫
Ω
|∇u|pdx for all u ∈ D1,p(Ω), (2)
where p ≤ r < p∗(t). If t = r = p, the above inequality becomes the well known Hardy inequality
[5, 10, 13] ∫
Ω
|u|p
|x|p
dx ≤ 1
µ
∫
Ω
|∇u|pdx for all u ∈ D1,p(Ω). (3)
In the space D1,p(Ω) we employ the following norm:
c© N. NYAMORADI, 2012
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6 797
798 N. NYAMORADI
‖u‖ = ‖u‖D1,p(Ω) :=
∫
Ω
(
|∇u|p − µ |u|
p
|x|p
)
dx
1/p
, µ ∈ [0, µ).
By using the Hardy inequality (3) this norm is equivalent to the usual norm
(∫
Ω
|∇u|pdx
)1/p
.
The operator L :=
(
|∇ · |p−2∇ · −µ | · |
p−2
|x|p
)
is positive in D1,p(Ω) if 0 ≤ µ < µ.
Now, we define the space Wk = D1,p(Ω)× . . .×D1,p(Ω) with the norm
∥∥(u1, . . . , uk)
∥∥p
k
=
k∑
i=1
‖ui‖p.
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:
Aµ,t,r(Ω) = inf
u∈D1,p(Ω)\{0}
∫
Ω
(
|∇u|p − µ |u|
p
|x|p
)
dx(∫
Ω
|u|r
|x|t
dx
)p/r . (4)
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,
Aµ,0(Ω) = inf
u∈D1,p(Ω)\{0}
∫
Ω
(
|∇u|p − µ |u|
p
|x|p
)
dx(∫
Ω
|u|p∗dx
)p/p∗ .
For any 0 ≤ µ < µ, by (2), (3), 0 ≤ t < p and the Minkowski’s inequality, the following best
constants are well defined:
SF,µ = SF,µ(Ω) = inf
(u1,...,uk)∈Wk\{(0,...,0)}
∑k
i=1
∫
Ω
(
|∇ui|p − µ
|ui|p
|x|p
)
dx(∫
Ω
F (u1, . . . , uk)dx
)p/p∗ . (5)
Another important parameter is Aµ,s,p(Ω), the (general) first eigenvalue of the operator L :
λ1 = Aµ,s,p(Ω) = inf
u∈D1,p(Ω)\{0}
∫
Ω
(
|∇u|p − µ |u|
p
|x|p
)
dx∫
Ω
|u|p
|x|t
dx
. (6)
Furthermore, λ1 is positive and simple, the corresponding eigenfunction φ1 does not change sign, the
operator L admits a sequence of eigenvalues diverging to +∞ [18, 19]. Without loss of generality,
we can assume that φ1 > 0. Setting
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 799
E ,
u ∈ D1,p(Ω);
∫
Ω
φp−1
1 u
|x|t
= 0
, (7)
and
λ∗ = λ∗µ,s(Ω) = inf
u∈E\{0}
∫
Ω
(
|∇u|p − µ |u|
p
|x|p
)
dx∫
Ω
|u|p
|x|t
dx
, (8)
then we have λ1 < λ∗ (see [15], Lemma 2.1).
Existence of nontrivial nonnegative 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 [6, 11, 12, 16], and for general p > 1 to [7, 8, 13 – 15]
and the references therein. For example, Kang in [15] studied the following elliptic equation via the
generalized Mountain – Pass theorem [17]:
−∆pu− µ
|u|p−2u
|x|p
=
|u|p∗(t)−2u
|x|t
+ λ
|u|p−2u
|x|s
, x ∈ Ω,
u = 0, x ∈ ∂Ω,
where Ω ⊂ RN is a bounded domain, 1 < p < N, 0 ≤ s, t < p and 0 ≤ µ < µ ,
(
N − p
p
)p
. Also,
the authors in [9] via the Mountain – Pass theorem of Ambrosetti and Rabinowitz [2], proved that
−∆pu− µ
up−1
|x|p
= |u|p∗−1 +
up
∗(s)−1
|x|s
in RN
admits a positive solution in RN , whenever µ < µ ,
(
N − p
p
)p
.
In this work, motivated by the above works we are interested to study the problem (1) by using
the Mountain – Pass theorem of Ambrosetti and Rabinowitz [2]. We shall show that the system () has
a positive weak solution.
This paper 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 2 and 3).
2. Local (PS)c condition. The corresponding energy functional of problem () is defined by
J(u) =
1
p
‖u‖pWk
− 1
p∗
∫
Ω
F (u)dx− 1
p∗(t)
k∑
i=1
∫
Ω
|ui|p
∗(t)
|x|t
dx− λ
p
k∑
i=1
∫
Ω
|ui|p
|x|s
dx,
for each u = (u1, . . . , uk) ∈Wk. Then J ∈ C1(Wk,R).
Before proving the main results, we stat several lemmas.
The following lemma is well know, where we have employed the equivalent norm in W 1,p(Ω),
see [13] for the case when µ = 0.
Lemma 1. Assume that 0 ≤ s < p, p ≤ q ≤ p∗(s) and 0 ≤ µ < µ. Then:
(i) there exists a constant C > 0 such that
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
800 N. NYAMORADI∫
Ω
|u|q
|x|s
dx
1/q
≤ C‖u‖ for all u ∈ D1,p(Ω),
(ii) the map u→ u
xs/q
from D1,p(Ω) into Lq(Ω) is compact if p ≤ q < p∗(s).
Also, we need the following version of Brèzis – Lieb lemma [4].
Lemma 2. Consider F ∈ C1((R+)k,R+) with F (0, . . . , 0) = 0 and |Fui(u1, . . . , uk)| ≤
≤ C1
(∑n
j=1
|uj |p−1
)
for i = 1, . . . , k and some 1 ≤ p < ∞, C1 > 0. Let un = (un1 , . . . , u
n
k) be
bounded sequence in Lp(Ω, (R+)k), and such that un ⇀ u = (u1, . . . , uk) weakly in Wk. Then one
has ∫
Ω
F (un)dx→
∫
Ω
F (un − u)dx+
∫
Ω
F (u)dx as n→∞.
Lemma 3. Assume that 0 ≤ s < p and 0 ≤ µ < µ. Then the functional J satisfies the (PS)c
condition for all
0 < c < c∗ := min
{
1
N
S
N
p
F,µ,
p− t
p(N − t)
(Aµ,t)
(N−t)/(p−t)
}
. (9)
Proof. Suppose {un = (un1 , . . . , u
n
k)} ⊂ Wk satisfies J(un) → c and J ′(un) → 0 with c < c∗.
It is easy to show that {un} is bounded in Wk and there exists u = (u1, . . . , uk) such that un ⇀ u
up to a subsequence. Moreover, for 1 ≤ i ≤ k, we may assume
uni ⇀ ui weakly in D1,p(Ω),
uni ⇀ ui weakly in Lp
∗(t)(Ω, |x|t), 0 < t ≤ p,
uni → ui strongly in Lq(Ω), 1 ≤ q < p∗,
uni → ui a.e. on Ω.
Hence, we have J ′(u) = 0 by the weak continuity of J. Let ũni = uni − ui for 1 ≤ i ≤ k. Then we
have ∫
Ω
|ũni |p = o(1) for 1 ≤ i ≤ k (10)
and by Brèzis – Lieb lemma [4], we obtain
‖ũn‖pk → ‖un‖
p
k − ‖u‖
p
k as n→∞, (11)
k∑
i=1
∫
Ω
|ũni |p
∗(t)
|x|t
dx =
k∑
i=1
∫
Ω
|uni |p
∗(t)
|x|t
dx−
k∑
i=1
∫
Ω
|ui|p
∗(t)
|x|t
dx+ o(1), (12)
and by Lemma 2, ∫
Ω
F (ũn)dx−
∫
Ω
F (un)dx→
∫
Ω
F (u)dx as n→∞. (13)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 801
Since, J(un) = c+ o(1), J ′(un) = o(1) and (10) – (13), we can deduce that
1
p
‖ũn‖p −
1
p∗
∫
Ω
F (ũn)dx− 1
p∗(t)
k∑
i=1
∫
Ω
|ũni |p
∗(t)
|x|t
dx = c− J(u) + o(1),
and
‖ũn‖p −
∫
Ω
F (ũn)dx−
k∑
i=1
∫
Ω
|ũni |p
∗(t)
|x|t
dx = o(1).
Now, we define
α := lim
n→∞
∫
Ω
F (ũn)dx,
β := lim
n→∞
k∑
i=1
∫
Ω
|ũni |p
∗(t)
|x|t
dx, (14)
γ := lim
n→∞
k∑
i=1
∫
Ω
(
|∇ũni |p − µ
|ũni |p
|x|p
)
dx = lim
n→∞
‖ũn‖p.
Let ξ ∈ C∞0 (Ω) be such that ξ|Ω ≡ 1. Since ξun ∈Wk, and since limn→∞〈J ′(un), ξun〉 = 0, using
(10) – (13) and the definitions of α, β and γ in (14), we get that γ ≤ α+ β.
By (14), we obtain
αp/p
∗
= lim
n→∞
∫
Ω
F (ũn)dx
p/p∗ ≤ 1
SF,µ
lim
n→∞
‖ũn‖p =
1
SF,µ
γ, (15)
and by definition of Aµ,t,
βp/p
∗(t) = lim
n→∞
k∑
i=1
∫
Ω
|ũni |p
∗(t)
|x|t
dx
p/p∗(t)
≤
≤ lim
n→∞
( 1
Aµ,t
)p∗(t)/p k∑
i=1
∫
Ω
(
|∇ũni |p − µ
|ũni |p
|x|p
)
dx
p∗(t)/p
p/p∗(t)
=
=
1
Aµ,t
lim
n→∞
[
k∑
i=1
‖ũni ‖p
∗(t)
]p/p∗(t)
=
=
1
Aµ,t
lim
n→∞
‖ũn‖p =
1
Aµ,t
γ. (16)
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
802 N. NYAMORADI
From γ ≤ α+ β, (15) and (16), we can obtain
αp/p
∗ ≤ 1
SF,µ
γ ≤ 1
SF,µ
α+
1
SF,µ
β,
αp/p
∗
(
1− 1
SF,µ
α
p∗−p
p∗
)
≤ 1
SF,µ
β.
(17)
On the other hand, J(un) − 1
p
〈J ′(un), un〉 = c + o(‖un‖) = c + o(1) as n → ∞ since (‖un‖)n∈N
is bounded, which yields,(
1
p
− 1
p∗
)∫
Ω
F (un)dx+
(
1
p
− 1
p∗(t)
) k∑
i=1
∫
Ω
|uni |p
∗(t)
|x|t
dx = c+ o(1) (18)
as n→∞. Therefore ∫
Ω
F (un)dx ≤ cN + o(1) (19)
as n→∞.
Moreover, by (19), we obtain
α ≤ cN. (20)
Plugging (20) into (17), we have
αp/p
∗
(
1− 1
SF,µ
(cN)p/N
)
≤ 1
SF,µ
β.
By the upper bound 9 on c there exists δ1, depending on N, p, µ and c, such that αp/p
∗ ≤ δ1β.
Similarly, there exists δ2, depending on N, p, µ, t and c, such that βp/p
∗(t) ≤ δ2α. In particular, it
follows from these two latest inequalities that there exists ε0 = ε0(N, p, µ, s, c) > 0 such that either
α = β = 0 or {α ≥ ε0 and β ≥ ε0}. (21)
Up to a subsequence, from (12) and (13) it follows that
c = J(un)− 1
p
〈J ′(un), un〉+ o(1) ≥
≥ 1
N
∫
Ω
F (un)dx+
p− t
p(N − t)
k∑
i=1
∫
Ω
|uni |p
∗(t)
|x|t
dx+ o(1) =
=
1
N
∫
Ω
F (u)dx+ α
+
p− t
p(N − t)
(
k∑
i=1
∫
Ω
|ui|p
∗(t)
|x|t
dx+ β
)
as n → ∞. From (21) and by assumption c < c∗ we get α = β = 0. Up to a subsequence, un → u
strongly in Wk.
Lemma 3 is proved.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 803
Lemma 4 [15]. Assume that 1 < p < N, 0 ≤ t < p and 0 ≤ µ < µ. Then the limiting problem
−∆pu− µ
|u|p−1
|x|p
=
|u|p∗(t)−1
|x|t
in RN \ {0},
u ∈ D1,p(RN ), u > 0 in RN \ {0},
has positive radial ground states
Vε(x) , ε(p−N)/pUp,µ
(x
ε
)
= ε(p−N)/pUp,µ
(
|x|
ε
)
∀ε > 0, (22)
that satisfy ∫
Ω
(
|∇Vε(x)|p − µ |Vε(x)|p
|x|p
)
dx =
∫
Ω
|Vε(x)|p∗(t)
|x|t
dx = (Aµ,t)
(N−t)/(p−t),
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→0
ra(µ)Up,µ(r) = C1 > 0,
lim
r→+∞
rb(µ)Up,µ(r) = C2 > 0,
lim
r→0
ra(µ)+1|U ′p,µ(r)| = C1a(µ) ≥ 0,
lim
r→+∞
rb(µ)+1|U ′p,µ(r)| = C2b(µ) > 0,
where Ci, i = 1, 2, are positive constants and a(µ) and b(µ) are zeros of the function
f(ζ) = (p− 1)ζp − (N − p)ζp−1 + µ, ζ ≥ 0, 0 ≤ µ < µ,
that satisfy
0 ≤ a(µ) <
N − p
p
< b(µ) ≤ N − p
p− 1
.
Now, the p∗-homogeneity of F yields
F (u) ≤M
(
k∑
i=1
|ui|p
)p∗/p
for some constant M > 0. (23)
Recall that p < p∗ since 1 < p < N.
Theorem 1. Suppose 0 ≤ s < p and 0 ≤ µ < µ. Then:
(i) SF,µ = M−p/p
∗
Aµ,0;
(ii) SF,µ has the minimizers (e1Vε(x), . . . , ekVε(x)) ∀ε > 0, where Σk
i=1e
p
i = 1 and Vε(x) are
the extremal functions of Aµ,0 defined as in (22) (by plugging t = 0 in Lemma 4).
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
804 N. NYAMORADI
Proof. (i) By (23) and the Minkowski’s inequality
∫
Ω
F (u)dx
p/p∗ ≤Mp/p∗
∫
Ω
(
k∑
i=1
|ui|p
)p∗/p
dx
p/p
∗
≤
≤Mp/p∗
k∑
i=1
∫
Ω
|ui|p
∗
dx
p/p∗ ≤
≤Mp/p∗A−1
µ,0
k∑
i=1
‖ui‖p = Mp/p∗A−1
µ,0‖u‖
p
Wk
,
where u = (u1, . . . , uk) ∈Wk. So that
Mp/p∗ ‖u‖p(∫
Ω
F (u)dx
)p/p∗ ≥ Aµ,0.
Consider now the map u0 = e0v0 where e0 = (e1, . . . , ek) ∈ (R+)k satisfies
∑k
i=1
epi = 1 and
v0 ∈ D1,p(Ω) is an extremal function for Aµ,0. Then
∫
Ω
F (u0)dx
p/p∗ = Mp/p∗
∫
Ω
|v0|p
∗
dx
p/p∗ =
= Mp/p∗A−1
µ,0‖v0‖p = Mp/p∗A−1
µ,0‖tv0‖pWk
= Mp/p∗A−1
µ,0‖u0‖pWk
.
So that
SF,µ = M−p/p
∗
Aµ,0. (24)
(ii) From (5), (22) and (24) the desired result follows.
Theorem 1 is proved.
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ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 805
3. Main results. The main conclusions of this paper are summarized in the following theorems.
Theorem 2. Assume that N + ps− s− p2 > 0, λ ∈ (λ1, λ
∗) and 0 ≤ µ ≤ µ1, where
µ1 ,
N + ps− s− p2
p
(
N − s
p
)p−1
.
Then the problem () has a positive solution.
Theorem 3. Assume that 0 ≤ µ < µ, λ > 0 and λ ∈ (λ̃, λ1), where
λ̃ = λ1
1 +
MC
− p∗
p (N − p)
N
− c∗p−t/N−t
k(p− t)
p(N − t)
∫
Ω
|x|Nt+st−Ns−pt/p−t
−p−t/N−t .
Then the problem () has a positive solution.
In the following, we will give some estimates on the extremal function Vε(x) defined in (22).
For m ∈ N large, choose ϕ(x) ∈ C∞0 (RN ), 0 ≤ ϕ(x) ≤ 1, ϕ(x) = 1 for |x| ≤ 1
2m
, ϕ(x) = 0 for
|x| ≥ 1
m
, ‖∇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 [15]:∫
Ω
(
|∇uε|p − µ
|uε|p
|x|p
)
dx = (Aµ,t)
N−t/p−t +O
(
εb(µ)p+p−N), (25)
∫
Ω
|uε|p
∗(t)
|x|t
dx = (Aµ,t)
N−t/p−t +O
(
εb(µ)p∗(t)−N+t
)
, (26)
∫
Ω
|uε|q
|x|s
dx ≥
Cε
N−s+
(
1−Np
)
q
, q >
N − s
b(µ)
,
Cε
N−s+
(
1−Np
)
q| ln ε|, q =
N − s
b(µ)
,
Cε
q
(
b(µ)+1−Np
)
q
, q <
N − s
b(µ)
.
(27)
Lemma 5. Assume thatN+ps−s−p2 > 0, λ ∈ (λ1, λ
∗), ε > 0 small enough and 0 ≤ µ ≤ µ1,
where
µ1 ,
N + ps− s− p2
p
(
N − s
p
)p−1
.
Then, there exist a function u = (u1, . . . , uk) ∈Wk such that
sup
t≥0
J(tu) < c∗ := min
{
1
N
S
N/p
F,µ ,
p− t
p(N − t)
(Aµ,t)
N−t/p−t
}
.
Proof. We divide the proof into two steps.
Step 1. We prove that under the assumptions of this lemma, there exists (u1, . . . , uk) ∈Wk such
that
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
806 N. NYAMORADI
sup
t≥0
J(tu1, . . . , tuk) <
1
N
SF,µ.
We consider the functional I : W → R defined by
I(u) =
1
p
‖u‖pWk
− 1
p∗
∫
Ω
F (u)dx for all u ∈Wk
and by (25), (26) and (27) in case t = 0,∫
Ω
(
|∇uε|p − µ
|uε|p
|x|p
)
dx = (Aµ,0)N/p +O
(
εb(µ)p+p−N),
∫
Ω
|uε|p
∗
dx = (Aµ,0)N/p +O
(
εb(µ)p∗−N),
∫
Ω
|uε|q
|x|s
dx ≥
CεN−s+(1−N/p)q, q >
N − s
b(µ)
,
CεN−s+
(
1−N/p
)
q | ln ε|, q =
N − s
b(µ)
,
Cεq
(
b(µ)+1−N/p
)
q, q <
N − s
b(µ)
.
Set u0 = (e1uε, . . . , ekuε) ∈Wk where (e1, . . . , ek) ∈ (R+)k and
∑k
i=1
epi = 1.
Also, we define the function g1(t) := J(te1uε, . . . , tekuε), t ≥ 0. Note that limt→+∞ g1(t) =
= −∞ and g1(t) > 0 as t is close to 0. Thus supt≥0g1(t) is attained at some finite tε > 0 with
g′1(tε) = 0. Furthermore, C ′ < tε < C ′′; where C ′ and C ′′ are the positive constants independent
of ε.
Then, by the definition of SF,µ, we obtain
I(tεe1uε, . . . , tεekuε) ≤
1
N
(∑k
i=1
epi
)∫
Ω
(
|∇uε|p − µ
|uε|p
|x|p
)
dx(∫
Ω
F (e1uε, . . . , ekuε)dx
)p/p∗
N/p
≤
≤ 1
N
∫
Ω
(
|∇uε|p − µ
|uε|p
|x|p
)
dx
Mp/p∗
(∫
Ω
|uε|p
∗
dx
)p/p∗
N/p
≤
≤ 1
N
(
1
Mp/p∗
)N/p (Aµ,0)N/p +O
(
εb(µ)p+p−N)
(Aµ,0)N/p∗ +O
(
ε
(
b(µ)p∗−N
)
p/p∗)
N/p ≤
≤ 1
N
(
1
Mp/p∗
)N/p((
Aµ,0
)N/p
+O(εb(µ)p+p−N )
)
=
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ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 807
=
1
N
(
Aµ,0
Mp/p∗
)N/p
+O
(
εb(µ)p+p−N) =
1
N
S
N/p
F,µ +O
(
εb(µ)p+p−N),
where the following fact has bas used:
sup
t≥0
(
tp
p
A− tp
∗
p∗
B
)
=
1
N
(
A
B
p
p∗
)N/p
, A, B > 0.
Consequently,
J(tεe1uε, . . . , tεekuε) ≤
1
N
S
N/p
F,µ +O(εb(µ)p+p−N )− kλm
p
∫
Ω
|tεuε|p
|x|s
dx,
where m := min{ep1, . . . , e
p
n}.
If p >
N − s
b(µ)
, from (27) we have
∫
Ω
|tεuε|p
|x|s
dx ≥ CεN−s+p−N = O
(
εN−s+p−N
)
.
Furthermore, N − s+ p−N < pb(µ) + p−N.
If p =
N − s
b(µ)
, then N − s+ p−N = pb(µ) + p−N. From (27) we have∫
Ω
|tεuε|p
|x|s
dx ≥ CεN−s+p−N | ln ε| = O
(
εN−s+p−N | ln ε|
)
.
Hence, if ε > 0 small and pb(µ)−N + s ≥ 0, then we have
sup
t≥0
J(te1uε, . . . , tekuε) <
1
N
SF,µ.
On the other hand, it is easy to verify that the function
f(ζ) = (p− 1)ζp − (N − p)ζp−1 + µ, ζ ≥ 0,
has the only minimal point ζ ′′ =
N − p
p
. Moreover, f(ζ) is decreasing in (0, ζ ′′) and is increasing
in (ζ ′′,+∞). Hence,
pb(µ)−N + s ≥ 0⇐⇒ b(µ) ≥ N − s
p
⇐⇒
⇐⇒ 0 = f(b(µ)) ≥ f
(
N − s
p
)
⇐⇒ 0 ≤ µ ≤ µ1
for N + ps− s− p2 > 0.
Step 2. We prove that under the assumptions of this lemma, there exists (u1, . . . , uk) ∈Wk such
that
sup
t≥0
J(tu1, . . . , tuk) < c∗.
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
808 N. NYAMORADI
In case
1
N
S
N/p
F,µ ≤
p− t
p(N − t)
(Aµ,t)
N−t/p−t,
we take (u1, . . . , uk) ∈ Wk \ {(0, . . . , 0)} as in step 1 to get the result. Otherwise we take
(u1, . . . , uk) = (e1uε, . . . , ekuε) ∈ \{(0, . . . , 0)} where (e1, . . . , ek) ∈ (R+)k and
∑k
i=1
epi = 1
and uε satisfy (25) – (27).
Now, we using arguments similar to the first step, with I replace by:
Ĩ(u) =
1
p
‖u‖pWk
− 1
p∗(t)
k∑
i=1
∫
Ω
|ui|p
∗(t)
|x|t
dx for all u ∈Wk.
Which gives the step 2.
Lemma 5 is proved.
Proof of Theorem 2. Set c = infh∈Γ maxt∈[0,1] J(h(t)), where
Γ =
{
h ∈ C([0, 1],Wk) | h(0) = (0, 0), J(h(1)) < 0
}
.
For any u = (u1, . . . , uk) ∈ Wk \ {(0, . . . , 0)}, from Lemma 1 (i) and the Minkowski’s inequality,
one can get∫
Ω
F (u)dx ≤M
∫
Ω
(
k∑
i=1
|ui|p
)p∗/p
dx
(p/p∗)·(p∗/p)
≤MC−(p∗/p)‖u‖pWk
. (28)
Then it follows that
J(u) =
1
p
‖u‖pWk
− 1
p∗
∫
Ω
F (u)dx− 1
p∗(t)
k∑
i=1
∫
Ω
|ui|p
∗(t)
|x|t
dx− λ
p
k∑
i=1
∫
Ω
|ui|p
|x|s
dx ≥
≥ C
(
‖u‖pWk
− ‖u‖p
∗
Wk
− ‖u‖p
∗(t)
Wk
)
≥ C‖u‖pWk
− C‖u‖p
∗
Wk
.
Hence, there exists a constant ρ > 0 small such that
b := inf
‖u‖Wk
=ρ
J(u) > 0 = J(0, . . . , 0).
Since J(tu) → −∞ as t → ∞, there exists t0 > 0 such that ‖t0u‖ > ρ and J(t0u) < 0. By
the Mountain – Pass theorem [2], there exists a sequence {un} ⊂ Wk such that J(un) → c and
J ′(un)→ 0 as n→∞. From Lemma 5 it follows that
0 < c ≤ sup
t∈[0,1]
J(tt0e1uε, . . . , tt0ekuε)) ≤ sup
t≥0
J(te1uε, . . . , tekuε)) < c∗.
By Lemma 3 there exists a subsequence of {un}, still denoted by {un}, such that un → u strongly
in Wk. Thus we get a critical point u = (u1, . . . , uk) of J satisfying () and c is a critical value. Set
u+ = max{u, 0}. Replacing the terms∫
Ω
|ui|p
∗(t)
|x|t
dx,
∫
Ω
F (u)dx,
∫
Ω
|up|
|x|s
dx for 1 ≤ i ≤ k
ISSN 1027-3190. Укр. мат. журн., 2012, т. 64, № 6
ON A p-LAPLACIAN SYSTEM WITH CRITICAL HARDY – SOBOLEV . . . 809
in J(u) by ∫
Ω
|u+
i |p
∗(t)
|x|t
dx,
∫
Ω
F (u+)dx,
∫
Ω
|u+
i |p
|x|s
dx for 1 ≤ i ≤ k
respectively and repeating the above process, we get a nonnegative solution u = (u1, . . . , uk) to ().
Also, by the maximum principle we deduce that ui > 0 in Ω for 1 ≤ i ≤ k.
Theorem 2 is proved.
Proof of Theorem 3. The proof follows the same lines as that in [3]. Let w = u1 = · · · = uk =
= τφ1, τ > 0. Then by (28) and the Hölder inequality we obtain
J(u1, . . . , uk) =
λ1 − λ
p
k∑
i=1
∫
Ω
|w|p
|x|s
dx− 1
p∗
∫
Ω
F (w, . . . , w)dx− 1
p∗(t)
k∑
i=1
∫
Ω
|w|p∗(t)
|x|t
dx ≤
≤ λ1 − λ
p
k∑
i=1
∫
Ω
|w|p
|x|s
dx+
λ1MC−(p∗/p)
p∗
k∑
i=1
∫
Ω
|w|p
|x|s
dx− 1
p∗(t)
k∑
i=1
∫
Ω
|w|p∗(t)
|x|t
dx =
=
k
p
(
λ1
(
1 +
MC−(p∗/p)(N − p)
N
)
− λ
)∫
Ω
|w|p
|x|s
dx− k
p∗(t)
∫
Ω
|w|p∗(t)
|x|t
dx ≤
≤ k
p
(
λ1
(
1 +
MC−p
∗/p(N − p)
N
)
− λ
)∫
Ω
|w|p∗(t)
|x|t
dx
p/(p∗(t))
×
×
∫
Ω
|x|(Nt+st−Ns−pt)/(p−t)dx
(p−t)/(N−t)
− k
p∗(t)
∫
Ω
|w|p∗(t)
|x|t
dx ≤
≤
λ1
(
1 +
MC
− p∗
p (N − p)
N
)
− λ
(N−t)/(p−t)
k(p− t)
p(N − t)
∫
Ω
|x|(Nt+st−Ns−pt)/(p−t)dx,
where we have used the fact that
max
τ≥0
(
c1τ
p − c2τ
p∗(t)
)
=
c1(p− t)
N − t
(
c1(N − p)
c2(N − t)
)(N−p)/(p−t)
∀ c1, c2 > 0.
If λ ∈ (λ̃, λ1), then
max
τ≥0
J(τφ1, . . . , τφ1) ≤ c∗.
Hence, we can obtain a PS-sequence in the cone of nonnegative functions, which has a weak limit
(u1, . . . , uk) with ui ≥ 0 and ui 6= 0 for 1 ≤ i ≤ k. By the maximum principle [20], we obtain that
ui > 0 in Ω and (u1, . . . , uk) is a positive solution of ().
Theorem 3 is proved.
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810 N. NYAMORADI
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Received 01.06.11
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