Applied Mathematics, 2015, 6, 1891-1901 Published Online October 2015 in SciRes. http://www.scirp.org/journal/am http://dx.doi.org/10.4236/am.2015.611166
On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent Mohammed El Mokhtar Ould El Mokhtar Department of Mathematics, College of Science, Qassim University, Buraidah, Kingdom of Saudi Arabia Email:
[email protected],
[email protected] Received 5 September 2015; accepted 24 October 2015; published 27 October 2015 Copyright © 2015 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/
Abstract In this paper, we establish the existence of at least four distinct solutions to an elliptic problem with singular cylindrical potential, a concave term, and critical Caffarelli-Kohn-Nirenberg exponent, by using the Nehari manifold and mountain pass theorem.
Keywords Singular Cylindrical Potential, Concave Term, Critical Caffarelli-Kohn-Nirenberg Exponent, Nehari Manifold, Mountain Pass Theorem
1. Introduction In this paper, we consider the multiplicity results of nontrivial nonnegative solutions of the following problem ( λ , µ ) −2 b −c q−2 2 −2 Lµ , a u = 2∗ y ∗ h u ∗ u + λ y f u u , in N , y ≠ 0 1,2 u ∈ a
(
−div y where Lµ , a v :=
−2 a
)
∇v − µ y
−2 ( a +1)
v , where each point x in N is written as a pair
where k and N are integers such that N ≥ 3 and k belongs to
{1, , N } ,
( y, z ) ∈ k × N − k
−∞ < a < ( k − 2 ) 2, a ≤ b < a + 1,
2∗ 2 N ( N − 2 + 2 ( b − a ) ) is the critical Caffarelli-Kohn-Nirenberg exponent, 1 < q < 2,= How to cite this paper: El Mokhtar, M.E.M.O. (2015) On Elliptic Problem with Singular Cylindrical Potential, a Concave Term, and Critical Caffarelli-Kohn-Nirenberg Exponent. Applied Mathematics, 6, 1891-1901. http://dx.doi.org/10.4236/am.2015.611166
M. E. M. O. El Mokhtar
( ( k − 2 ( a + 1) ) 2 )
0 < c= q ( a + 1) + N (1 − q 2 ) , −∞ < µ < µa= ,k :
2
, λ is a real parameter, f ∈ µ′ C ( N ) ,
h is a bounded positive function on k . µ′ is the dual of µ , where µ and 01,2 will be defined later.
Some results are already available for ( λ , µ ) in the case k = N , see for example [1] [2] and the references therein. Wang and Zhou [1] proved that there exist at least two solutions for ( λ , µ ) with a = 0 ,
(( N − 2) 2)
0 < µ ≤ µ0, N =
2
and h ≡ 1 , under certain conditions on f. Bouchekif and Matallah [3] showed the
existence of two solutions of ( λ , µ ) under certain conditions on functions f and h, when 0 < µ ≤ µ0, N , λ ∈ ( 0, Λ∗ ) , −∞ < a < ( N − 2 ) 2 and a ≤ b < a + 1 , with Λ∗ a positive constant. Concerning existence results in the case k < N , we cite [4] [5] and the references therein. Musina [5] considered ( λ , µ ) with − a 2 instead of a and λ = 0 , also ( λ , µ ) with a = 0 , b = 0 , λ = 0 , with h ≡ 1 and a ≠ 2 − k . She established the existence of a ground state solution when 2 < k ≤ N and
0 < µ < µa , k =
(( k − 2 + a ) 2)
2
( λ µ )
for
with − a 2 instead of a and λ = 0 . She also showed that
,
( λ µ )
with a = 0 , b = 0 , λ = 0 does not admit ground state solutions. Badiale et al. [6] studied
,
( λ µ ) ,
with a = 0 ,
b = 0 , λ = 0 and h ≡ 1 . They proved the existence of at least a nonzero nonnegative weak solution u, satisfying u ( y, z ) = u ( y , z ) when 2 ≤ k < N and µ < 0 . Bouchekif and El Mokhtar [7] proved that ( λ , µ ) ad-
(
mits two distinct solutions when 2 < k ≤ N , b = N − p ( N − 2 ) 2 with p ∈ 2, 2∗ , µ < µ0,k , and λ ∈ ( 0, Λ∗ )
where Λ∗ is a positive constant. Terracini [8] proved that there is no positive solutions of ( λ , µ ) with b = 0 , λ = 0 when a ≠ 0 , h ≡ 1 and µ < 0 . The regular problem corresponding to a= b= µ= 0 and h ≡ 1 has been considered on a regular bounded domain Ω by Tarantello [9]. She proved that, for f ∈ H −1 ( Ω ) , the dual of H 01 ( Ω ) , not identically zero and satisfying a suitable condition, the problem considered admits two distinct solutions. Before formulating our results, we give some definitions and notation. and µ µ k \ {0} × N − k , the closure of We denote by a1,2 a1,2 k \ {0} × N − k = =
(
C0∞ k \ {0} × N − k
)
(
(
)
)
with respect to the norms
(∫
= u a ,0
−2 a
y
N
∇u dx 2
)
12
and
= u a,µ
(∫ ( y
−2 a
N
(
respectively, with µ < µa , k = ( k − 2 ( a + 1) ) 2
)
∇u − µ y 2
−2 ( a +1)
u
2
) dx )
12
,
for k ≠ 2 ( a + 1) .
2
From the Hardy-Sobolev-Maz’ya inequality, it is easy to see that the norm u More explicitly, we have
(1 − (1 µ ) max ( µ , 0 ) )
12
a,k
u
0, a
≤ u
µ ,a
(
≤ 1 − (1 µa , k ) min ( µ , 0 )
is equivalent to u
a,µ
)
12
u
0, a
a ,0
.
,
for all u ∈ µ . We list here a few integral inequalities. The starting point for studying ( λ , µ ) , is the Hardy-Sobolev-Maz’ya inequality that is particular to the cylindrical case k < N and that was proved by Maz’ya in [4]. It states that there exists positive constant Ca ,2∗ such that Ca ,2∗
for any v ∈ Cc∞
((
k
)
(∫
y
N
−2∗ b
v
2∗
dx
)
2 2∗
≤∫
N
)
\ {0} × N − k .
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(y
−2 a
∇v − µ y 2
−2 ( a +1)
)
v 2 dx ,
(1.1)
M. E. M. O. El Mokhtar
The second one that we need is the Hardy inequality with cylindrical weights [5]. It states that
µa , k ∫ N y
−2 ( a +1)
−2 a
v 2 dx ≤ ∫ N y
∇v dx, for all v ∈ µ , 2
(1.2)
It is easy to see that (1.1) hold for any u ∈ µ in the sense
(∫
N
y
−c
)
1 p
p
u dx
(∫
≤ Ca , p
y
N
−2 a
∇v dx 2
)
1 p
(1.3)
where Ca , p positive constant, 1 ≤ p ≤ 2 N ( N − 2 ) , c ≤ p ( a + 1) + N (1 − p 2 ) , and in [10], if p < 2 N ( N − 2 )
(
the embedding µ → L p N , y
−c
)
(
is compact, where L p N , y u
p ,c
=
(∫
y
N
−c
)
−c
)
is the weighted L p space with norm
1 p
p
u dx
.
Since our approach is variational, we define the functional J on µ by
J= (u ) :
(1 2 ) u µ ,a − P ( u ) − Q ( u ) , 2
with
(1 q ) ∫
∗ = P ( u ) : 2= h u ∗ dx, Q ( u ) : ∗ ∫ N y
−2 b
2
( λ µ )
A point u ∈ µ is a weak solution of the equation
N
y
−c
λ f u dx. q
if it satisfies
,
J ′ ( u ) , ϕ := R ( u ) ϕ − S ( u ) ϕ − T ( u ) ϕ = 0, for all ϕ ∈ µ
with
= R (u )ϕ :
∫
N
(y
−2 a
( ∇u ∇ϕ ) − µ
S ( u ) ϕ := 2∗ ∫
N
T ( u ) ϕ := ∫ N y Here
.,.
denotes the product in the duality µ′ ,
y −c
µ
−2∗ b
hu
(λ f u (
y
−2 ( a +1)
2∗
q −1
( uϕ ) )
ϕ
)
ϕ .
′ dual of
µ
µ
).
Let
u
S µ := inf
u∈µ \{0}
(∫
N
y
−2∗ b
2
µ ,a
u
2∗
dx
)
2 2∗
From [11], S µ is achieved. Throughout this work, we consider the following assumptions: (F) there exist ν 0 > 0 and δ 0 > 0 such that f ( x ) ≥ ν 0 , for all x in B ( 0, 2δ 0 ) . h0 > 0, h ( y ) ≥ h0 , y ∈ k . lim h ( y ) = (H) lim h ( y ) = y →0
y →∞
Here, B ( a, r ) denotes the ball centered at a with radius r. In our work, we research the critical points as the minimizers of the energy functional associated to the problem ( λ , µ ) on the constraint defined by the Nehari manifold, which are solutions of our system. Let Λ 0 be positive number such that
Λ 0 := ( Ca , q )
1 ( 2−q )
2 −2 where L ( q ) := ∗ 2∗ − q
−q
( h0 )
−1 ( 2∗ − 2 )
1 ( 2∗ − 2 )
2 − q 2∗ ( 2∗ − q )
.
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( Sµ )
2∗ 2 ( 2∗ − 2 )
L (q),
M. E. M. O. El Mokhtar
Now we can state our main results. Theorem 1. Assume that −∞ < a < ( k − 2 ) 2 , 0 < c= q ( a + 1) + N (1 − q 2 ) , − ∞ < µ < µ a, k , (F) satisfied and λ verifying 0 < λ < Λ 0 , then the system ( λ , µ ) has at least one positive solution. Theorem 2. In addition to the assumptions of the Theorem 1, if (H) hold and λ satisfying 0 < λ < (1 2 ) Λ 0 , then ( λ , µ ) has at least two positive solutions. Theorem 3. In addition to the assumptions of the Theorem 2, assuming N ≥ max ( 3, 6 ( a − b + 1) ) , there exists a positive real Λ1 such that, if λ satisfy 0 < λ < min ( (1 2 ) Λ 0 , Λ1 ) , then ( λ , µ ) has at least two positive solution and two opposite solutions. This paper is organized as follows. In Section 2, we give some preliminaries. Sections 3 and 4 are devoted to the proofs of Theorems 1 and 2. In the last Section, we prove the Theorem 3.
2. Preliminaries Definition 1. Let c ∈ , E a Banach space and I ∈ C1 ( E , ) . i) ( un )n is a Palais-Smale sequence at level c ( in short ( PS )c ) in E for I if
I ( un ) = c + on (1) and I ′ ( un ) = on (1) ,
where on (1) tends to 0 as n goes at infinity. ii) We say that I satisfies the ( PS )c condition if any ( PS )c sequence in E for I has a convergent subsequence. Lemma 1. Let X Banach space, and J ∈ C1 ( X , ) verifying the Palais-Smale condition. Suppose that J ( 0 ) = 0 and that: i) there exist R > 0 , r > 0 such that if u = R , then J ( u ) ≥ r ;
( u0 ) ∈ X such that c = inf max ( J ( γ ( t ) ) ) where γ ∈Γ t∈[ 0,1]
ii) there exist let
Γ=
u0 > R and J ( u0 ) ≤ 0 ;
{γ ∈ C ([0,1]; X ) such that γ ( 0 )=
}
0 et γ (1)= u0 ,
then c is critical value of J such that c ≥ r .
Nehari Manifold It is well known that J is of class C1 in µ and the solutions of ( λ , µ ) are the critical points of J which is not bounded below on µ . Consider the following Nehari manifold
{
}
2
− 2∗ P ( u ) − Q ( u ) = 0.
′ u∈ 0 , = µ \ {0} : J ( u ) , u = Thus, u ∈ if and only if
u
µ ,a
Note that contains every nontrivial solution of the problem
(2.1)
( λ µ ) . Moreover, we have the following ,
results. Lemma 2. J is coercive and bounded from below on . Proof. If u ∈ , then by (2.1) and the Hölder inequality, we deduce that
J ( u ) = ( ( 2∗ − 2 ) 2∗ 2 ) u
2
≥ ( ( 2∗ − 2 ) 2∗ 2 ) u
2
µ ,a
µ ,a
− ( ( 2∗ − q ) 2∗ q ) Q ( u ) (2 − q) − ∗ λ f 2∗ q
(
Thus, J is coercive and bounded from below on . Define
φ (u ) = J ′ (u ) , u .
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)
1 ( 2−q )
µ′
(C ) a, p
q
u
q
µ ,a
(2.2)
.
M. E. M. O. El Mokhtar
Then, for u ∈
φ ′ (u ) , u = 2u
2
µ ,a
− ( 2∗ ) P ( u ) − qQ ( u )
=− (2 q) u
2
2
µ ,a
− 2∗ ( 2∗ − q ) P ( u )
= ( 2∗ − q ) Q ( u ) − ( 2∗ − 2 ) u
2
µ ,a
(2.3)
.
Now, we split in three parts:
{ = {u ∈ : φ ′ ( u ) , u = {u ∈ : φ ′ ( u ) , u
} = 0} < 0} .
+ = u ∈ : φ ′ (u ) , u > 0 0 −
We have the following results. Lemma 3. Suppose that u0 is a local minimizer for J on . Then, if u0 ∉ 0 , u0 is a critical point of J. Proof. If u0 is a local minimizer for J on , then u0 is a solution of the optimization problem
min J ( u ) .
{u /φ ( u ) = 0}
Hence, there exists a Lagrange multipliers θ ∈ such that
J ′ ( u0 ) = θφ ′ ( u0 ) in ′
Thus, J ′ ( u0 ) , u0 = θ φ ′ ( u0 ) , u0 .
But
φ ′ ( u0 ) , u0 ≠ 0 , since u0 ∉ 0 . Hence θ = 0 . This completes the proof.
Lemma 4. There exists a positive number Λ 0 such that for all λ , verifying
0 < λ < Λ0 , we have = ∅ . Proof. Let us reason by contradiction. Suppose 0 ≠ ∅ such that 0 < λ < Λ 0 . Then, by (2.3) and for u ∈ 0 , we have 0
u
2
µ ,a
=2∗ ( 2∗ − q ) ( 2 − q ) P ( u ) =( ( 2∗ − q ) ( 2∗ − 2 ) ) Q ( u )
(2.4)
Moreover, by the Hölder inequality and the Sobolev embedding theorem, we obtain
u
µ ,a
≥ ( Sµ )
2∗ 2 ( 2∗ − 2 )
( 2 − q ) 2∗ ( 2∗ − q ) h0
−1 ( 2∗ − 2 )
(2.5)
and u
µ ,a
2 − q −1 ( 2 − q ) ≤ ∗ λ1 ( 2− q ) 2∗ − 2
(
) (C ) . q
a,q
From (2.5) and (2.6), we obtain λ ≥ Λ 0 , which contradicts an hypothesis. Thus = + − . Define
= c : inf = J ( u ) , c + : inf+ J (= u ) and c − : inf− J ( u ) . u∈
u∈
For the sequel, we need the following Lemma. Lemma 5. i) For all λ such that 0 < λ < Λ 0 , one has c ≤ c + < 0 .
1895
u∈
(2.6)
M. E. M. O. El Mokhtar
ii) For all λ such that 0 < λ < (1 2 ) Λ 0 , one has
(
c − > C0 = C0 λ1 , λ2 , S µ , f
µ′
)
( 2 − 2) ( 2 − q ) = ∗ 2∗ 2 2∗ ( 2∗ − q ) h0 (2 − q) − ∗ λ f 2∗ q
(
− 2 ( 2∗ − 2 )
)
1 ( 2−q )
µ′
( Sµ )
2∗ ( 2∗ − 2 )
q ( Ca , q ) .
Proof. i) Let u ∈ + . By (2.3), we have
( 2 − q ) 2∗ ( 2∗ − 1) u
2
µ ,a
> P (u )
and so
J (u ) =
( −1 2 )
u
2
µ ,a
2 ( 2 − q ) − 2 ( 2∗ − 1)( 2 − q ) + ( 2∗ − 1) P ( u ) < − ∗ ∗ u 2∗ 2 ( 2∗ − q )
2
µ ,a
.
We conclude that c ≤ c + < 0 . ii) Let u ∈ − . By (2.3), we get
( 2 − q ) 2∗ ( 2∗ − q ) u
2
µ ,a
< P (u ).
Moreover, by (H) and Sobolev embedding theorem, we have
P ( u ) ≤ ( Sµ )
− 2∗ 2
h0 u
2∗
µ ,a
.
This implies
u
µ ,a
> ( Sµ )
2∗ 2 ( 2∗ − 2 )
(2 − q) 2∗ ( 2∗ − q ) h0
−1 ( 2∗ − 2 )
, for all u ∈ − .
(2.7)
By (2.2), we get
J ( u ) ≥ ( ( 2∗ − 2 ) 2∗ 2 ) u
2
µ ,a
(2 − q) − ∗ λ f 2∗ q
(
)
1 ( 2−q )
µ′
(C ) a, p
q
u
q
µ ,a
.
Thus, for all λ such that 0 < λ < (1 2 ) Λ 0 , we have J ( u ) ≥ C0 . For each u ∈ with
∫
N
y
−2∗ b
hu
2∗
dx > 0 , we write
2 ( 2 − q ) u µ ,a = tm : tmax = (u ) 2∗ ( 2∗ − q ) ∫ N y −2∗ b h u 2∗ dx
( 2−q )
2∗ ( 2∗ − q )
> 0.
Lemma 6. Let λ real parameters such that 0 < λ < Λ 0 . For each u ∈ with one has the following: i) If Q ( u ) ≤ 0 , then there exists a unique t − > tm such that t − u ∈ − and
( )
∫
N
y
−2∗ b
hu
2∗
dx > 0 ,
J t − u = sup ( tu ) . ii) If Q ( u ) > 0 , then there exist unique t
( )
= J t +u
+
and t
−
t ≥0
such that 0 < t + < tm < t − ,
( )
= inf J ( tu ) and J t − u sup J ( tu ) .
0 ≤ t ≤ tm
t ≥0
Proof. With minor modifications, we refer to [12]. Proposition 1 (see [12]) i) For all λ such that 0 < λ < Λ 0 , there exists a ( PS )c+ sequence in + .
1896
(t u ) ∈ +
+
, t −u ∈ − ,
ii) For all λ such that 0 < λ < (1 2 ) Λ 0 , there exists a a
( PS )c
sequence in − .
−
M. E. M. O. El Mokhtar
3. Proof of Theorems 1 Now, taking as a starting point the work of Tarantello [13], we establish the existence of a local minimum for J on + . Proposition 2. For all λ such that 0 < λ < Λ 0 , the functional J has a minimizer u0+ ∈ + and it satisfies: i) J u0+ = c= c + ,
( ) ( )
ii) u0+ is a nontrivial solution of ( λ , µ ) . Proof. If 0 < λ < Λ 0 , then by Proposition 1 (i) there exists a
( un )n ( PS )c
+
sequence in + , thus it
bounded by Lemma 2. Then, there exists u0+ ∈ and we can extract a subsequence which will denoted by
( un ) n
such that
un u0+ weakly in
( strongly in L (
un u0+ weakly in L2∗ N , y
−2∗ b
un → u0+
−c
q
N
,y
)
)
(3.1)
un → u0+ a.e in N
( λ µ ) .
Thus, by (3.1), u0+ is a weak nontrivial solution of
Now, we show that un converges to u0+
,
strongly in . Suppose otherwise. By the lower semi-continuity of the norm, then either u0+
µ ,a
< lim inf un n →∞
µ ,a
and we obtain
( )
c ≤ J u0+ = ( ( 2∗ − 2 ) 2∗ 2 ) u0+
2
µ ,a
( )
− ( ( 2∗ − q ) 2∗ q ) Q u0+ < lim inf J ( un ) = c. n →∞
+ 0
We get a contradiction. Therefore, un converge to u strongly in . Moreover, we have u0+ ∈ + . If not, then by Lemma 6, there are two numbers t0+ and t0− , uniquely defined so that t0+ u0+ ∈ + and 1 . Since t − u0+ ∈ − . In particular, we have t0+ < t0− =
(
(
)
d d2 0 and J tu0+ J tu0+ = = t t0+= t dt dt 2
( )
( )
)
> 0,
t0+
( ) ( ) J (t u ) < J (t u ) < J (t u ) = J (u ) , which contradicts the fact that J ( u ) = c . Since J ( u ) = J ( u ) and u ∈ , then by Lemma 3, we may assume that u is a nontrivial nonnegative solution of ( ) . By the Harnack inequality, we conclude there exists t0+ < t − ≤ t0− such that J t0+ u0+ < J t − u0+ . By Lemma 6, we get + + 0 0
+ 0
−
+ 0
+ 0
+
+ 0
+ 0
− + 0 0
+ 0
+ 0
+ 0
+
λ ,µ
+ 0
that u > 0 and v > 0 , see for exanmple [14].
4. Proof of Theorem 2 Next, we establish the existence of a local minimum for J on − . For this, we require the following Lemma. Lemma 7. For all λ such that 0 < λ < (1 2 ) Λ 0 , the functional J has a minimizer u0− in − and it satisfies: i) J u0−= c − > 0,
( )
ii) u0− is a nontrivial solution of
( λ µ ) ,
in .
Proof. If 0 < λ < (1 2 ) Λ 0 , then by Proposition 1 ii) there exists a ( un )n , ( PS )c− sequence in − , thus it bounded by Lemma 2. Then, there exists u0− ∈ and we can extract a subsequence which will denoted by ( un )n such that
1897
M. E. M. O. El Mokhtar
un u0− weakly in
(
) )
−2∗ b
un u0− weakly in L2∗ N , y
(
un → u0− strongly in Lq N , y
−c
un → u0− a.e in N This implies
( )
P ( un ) → P u0− , as n goes to ∞.
Moreover, by (H) and (2.3) we obtain
P ( un ) > A ( q ) un
2
µ ,a
,
(4.1)
where, A ( q ) := ( 2 − q ) 2∗ ( 2∗ − q ) . By (2.5) and (4.1) there exists a positive number
C1 := A ( q )
2∗ ( 2∗ − 2 )
( Sµ )
2∗ ( 2∗ − 2 )
,
such that
P ( un ) > C1 .
(4.2)
This implies that
( )
P u0− ≥ C1 .
( un ) n
Now, we prove that
u0−
µ ,a
< lim inf un n →∞
µ ,a
converges to u0− strongly in . Suppose otherwise. Then, either
. By Lemma 6 there is a unique t0− such that
(t u ) ∈ − − 0 0
−
. Since
un ∈ − , J ( un ) ≥ J ( tun ) , for all t ≥ 0, we have
(
)
(
)
J t0− u0− < lim J t0− un ≤ lim J ( un ) = c− , n →∞
n →∞
and this is a contradiction. Hence,
( un )n → u0− strongly in . Thus,
( )
J ( un ) converges = to J u0− c − as n tends to + ∞.
( )
( )
Since J u0− = J u0− nonnegative solution of
and u0− ∈ − , then by (4.2) and Lemma 3, we may assume that u0− is a nontrivial
( λ µ ) . By the maximum principle, we conclude that ,
u0− > 0 .
Now, we complete the proof of Theorem 2. By Propositions 2 and Lemma 7, we obtain that ( λ , µ ) has two positive solutions u0+ ∈ + and u0− ∈ − . Since + − = ∅ , this implies that u0+ and u0− are distinct.
5. Proof of Theorem 3 In this section, we consider the following Nehari submanifold of
{
= u ∈
\ {0} : J ′ ( u ) , u = 0 and u
µ ,a
Thus, u ∈ if and only if
u
2
µ ,a
− 2∗ P ( u= ) − Q ( u ) 0 and u
1898
µ ,a
}
≥ >0 .
≥ > 0.
M. E. M. O. El Mokhtar
Firsly, we need the following Lemmas Lemma 8. Under the hypothesis of theorem 3, there exist 0 , Λ 2 > 0 such that is nonempty for any λ ∈ ( 0, Λ 2 ) and ∈ ( 0, 0 ) . Proof. Fix u0 ∈ \ {0} and let
g ( t ) =J ′ ( tu0 ) , tu0 = t 2 u0
2
µ ,a
− 2∗ t 2∗ P ( u0 ) − tQ ( u0 ) .
Clearly g ( 0 ) = 0 and g ( t ) → −∞ as n → +∞ . Moreover, we have g (1) =u0
2
µ ,a
≥ u0
If u0
= α0
µ ,a
( Sµ )
2
µ ,a
< 0 ≥ > 0 for 0 < =
2∗ 2
( 2∗ ( 2∗ − 1) )(
2∗ −1) 2∗
− 2∗ P ( u0 ) − Q ( u0 ) − 2∗ ( S µ )
− 2∗ 2
h0 u0
( h0 2∗ ( 2∗ − 1) )
(
−1 ( 2∗ − 2 )
( Sµ )
)
1 ( 2−q )
− λ f µ ,a
2∗
µ′
2∗ 2 ( 2∗ − 2 )
u0
µ ,a
.
, h0 ∈ ( 0, α 0 ) for
, then there exists
− 2∗ 2 = Λ 2 : ( h0 2∗ ( 2∗ − 1) ) ( S µ )
−1 ( 2∗ − 2 )
− Θ × Φ,
where
= Θ:
( 2∗ ( 2∗ − 1) )
2∗ −1
(( h0 )
2∗ 2
Sµ
)
− ( 2∗ )
2
2
and − 2∗ 2 = Φ : ( h0 2∗ ( 2∗ − 1) ) ( S µ )
−1 ( 2∗ − 2 )
and there exists t0 > 0 such that g ( t0 ) = 0 . Thus, ( t0 u0 ) ∈ and is nonempty for any λ ∈ ( 0, Λ 2 ) . Lemma 9. There exist M, Λ1 positive reals such that
φ ′ ( u ) , u < − M < 0, for u ∈ , and any λ verifying 0 < λ < min ( (1 2 ) Λ 0 , Λ1 ) .
Proof. Let u ∈ , then by (2.1), (2.3) and the Holder inequality, allows us to write
φ ′ ( u ) , u ≤ un where B (, q= ):
( 2∗ − 1) ( Ca , p )
q
2
µ ,a
(
λ f
)
1 ( 2−q )
µ′
B (, q ) − ( 2∗ − 2 ) ,
q − 2 . Thus, if 0 0 . Then, there exist r and η posi-
M. E. M. O. El Mokhtar
ii) there exists σ ∈ when σ
µ ,a
> r , with r = u
µ ,a
, such that J (σ ) ≤ 0 .
(u ) + 0
Proof. We can suppose that the minima of J are realized by
and u0− . The geometric conditions of the
mountain pass theorem are satisfied. Indeed, we have i) By (2.3), (5.1) and the fact that P ( u ) ≤ ( S µ )
− 2∗ 2
2∗
h0 u
µ ,a
, we get
J ( u ) ≥ (1 2 ) − ( 2∗ − 2 ) ( 2∗ − q ) q u
2
µ ,a
− ( Sµ )
− 2∗ 2
h0 u
2∗
µ ,a
,
Exploiting the function l= ( x ) x ( 2∗ − x ) and if N ≥ max ( 3, 6 ( a − b + 1) ) , we obtain that (1 2 ) − ( 2∗ − 2 ) ( 2∗ − q ) q > 0 for 1 < q < 2 . Thus, there exist η , r > 0 such that J ( u ) ≥ η > 0 when r = u
ii) Let t > 0 , then we have for all φ ∈
(
)
J ( tφ ) := t 2 2 φ
2
µ
( )
µ ,a
small.
)
(
− t 2∗ P (φ ) − t q q Q (φ ) .
Letting σ = tφ for t large enough. Since
= P (φ ) :
∫Ω y
−2∗ b
hφ
we obtain J (σ ) ≤ 0 . For t large enough we can ensure σ
2∗
µ ,a
dx > 0, >r.
Let Γ and c defined by = Γ:
: γ (0) {γ : [0,1] → =
= γ (1) u0+ u0− and
}
and
(
)
c := inf max inf J ( γ ( t ) ) . γ ∈Π t∈[ 0,1] γ ∈Π
Proof of Theorem 3. If
λ < min ( (1 2 ) Λ 0 , Λ1 ) , then, by the Lemmas 2 and Proposition 1 ii), J verifying the Palais-Smale condition in . Moreover, from the Lemmas 3, 9 and 10, there exists uc such that
= J ( uc ) c and uc ∈ . Thus uc is the third solution of our system such that uc ≠ u0+ and uc ≠ u0− . Since pect u, we obtain that −uc is also a solution of
( λ µ ) ,
is odd with res-
( λ µ ) . ,
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