On Elliptic Problem with Singular Cylindrical Potential, a Concave ...

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 .

1892

(y

−2 a

∇v − µ y 2

−2 ( a +1)

)

v 2 dx ,

(1.1)


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 )  

.

1893

( Sµ )

2∗ 2 ( 2∗ − 2 )

L (q),


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 .

1894

)

1 ( 2−q )

µ′

(C ) a, p

q

u

q

µ ,a

(2.2)

.


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)


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  − .

−


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


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.


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-


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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