Multiple Solutions for a Class of Concave Convex Quasilinear Elliptic Systems with Nonlinear Boundary Condition

(1)

Multiple Solutions for a Class of Concave-Convex

Quasilinear Elliptic Systems with Nonlinear Boundary

Condition

Li Wang

School of Basic Science, East China Jiaotong University, Nanchang, China Email: [email protected]

Received June 7, 2012; revised February 6, 2013; accepted February 13, 2013

ABSTRACT

In this paper, a quasilinear elliptic system is investigated, which involves concave-convex nonlinearities and nonlinear boundary condition. By Nehari manifold, fibering method and analytic techniques, the existence of multiple nontrivial nonnegative solutions to this equation is verified.

Keywords: Multiple Solutions; Quasilinear Elliptic Systems; Nehari Manifold; Fibering Method

1. Introduction

In this article, we are interested in the existence of two nontrivial nonnegative solutions of the following problem:

 

2 2

2 2 2

,

| | ,

,

.

p p

p q p

u m x u u u u v x

v m x v v u v v x

u v

u f x u u v g x v

x

 

  

  

 

 

  

    

 _



    



 

 _ _

    

 

   

n n

2

,

q v



(1.1) where _{ }_RN is a bounded domain with smooth boun-

dary, 2 p p Np

N p

  

    

 is the critical Sobolev

exponent for the embedding _W1,p

 

_RN __{L R}p



N



.

1 q p, 

n is the outer normal derivative,



_{ }_,



_ 2



 



R  0,0 , the weight m(x) is a positive bounded function and f x g x

   

, C

 

 are smooth functions which may change sign in Ω. By Nehari manifold, fibering method and analytic techniques, the existence of multiple positive solutions to this equation is verified.

In recent years, there have been many papers con- cerned with the existence and multiplicity of positive solutions for semilinear elliptic problems. Some interest- ing results can be found in Garcia-Azorero et al. [1], Wu [2-4] and the references therein. More recently, Hsu [5]

has considered the following elliptic system:

2 2

2 _, _,

2 _{| |} _, _,

0, .

q p

u u u v u u x

v u v v v v x

u v x

 

 _

 

 _

 

 

    

 _



_{ } _ _{ }

 _



   

 

(1.2)

By variational methods, he proved that problem (1.2) has at least two positive solutions if the pair of the parameters



 ,



belongs to a certain subset of 2

R . How-

ever, as far as we know, there are few results of problem (1.1) in addition to concave-convex nonlinearities, i.e., 1 q p, including nonlinear boundary condition. We focus on the existence of at least two nontrivial nonnegative solutions for problems (1.1) in the present paper.

Set

1

,

p p p

p q p q

p q p q

S q p

S q

   

 

   

 _{ }

 _

 _ 

  _ _  



     _{ }  





 _(1.3)

where S S, satisfy









d d

p p

p

q _q p

u x S u

u x S u x

  _{ }_

 

 

 



,

x

(1.4)

(2)

Theorem 1.1. If the parameters  , satisfy









0, 1



0,



,

p

p p _{p q}

p q p q q

f g

p

    

 

 

 

 _ _{ }  _

 

 

 

1





then problem (1.1) has at least two solutions



u v, and



U V,



satisfy u v U V, , , 0 in Ω and u v, , ,U V0.

It should be mentioned that the similar results about the existence of multiplicity of positive solutions for the Laplace problem with critical growth and sublinear per- turbation have been discussed in the recent paper [6-8] and the reference therein.

This paper is organized as follows. Some preliminaries and properties of the Nehair manifold are established in Sections 2, and Theorems 1.1 is proved in Sections 3.

2. Preliminaries

Let _W1,p

 

_ _{denotes the usual Sobolev space. In the}

Banach space we introduce

the norm which is equivalent to the standard one:

 

1, 1,

: p p

W W  W 

 





 





1, 1,

,

d d

p p p

p p

W

p p

u v u v

u m x u

v m x v



 

  

  



x

First, we give the definition of the weak solution of (1.1).

Definition 2.1. We say that

 

u v, W is a weak solution to (1.1) if for all



 1, 2



W ,we have

 





 

2 2

1 1

2 2

1 1

d d

p p

q

u u m x u u x

d

f x u u s u u v x

 



  

 

 



 

 

   

 





 





 

2 2

1 2

2 2

d

d d .

p p

q

v v m x v v x

g x v v s u v  v x

 



  

 

 



 

 

   

 





It is clear that problem (1.1) has a variational structure. Let be the corresponding energy functional of problem (1.1), and it is defined by

 

, :

I u v WR

 

, 1

 

, p 1

 

, 1

 

W

I u v u v R u v G u v

p q  

  

 , ,

where

 

,

 

qd

 

q

R u v 

_

_f x u s

_

_g x v d ,s

 

, d

G u v 

_

_ u v  x.

It is not difficult to verify that the functional I is not bounded neither from below nor from above. So it is convenient to consider I restricted to a natural constraint,

the Nehari manifold, that contains all the critical points of I. First we introduce the following notation: for any functional F W: R we denote by F u v

 

,  ₁, ₂



the Gateaux derivative of F at

 

u v, W in the direc- tion of



 1, 2



W,

 

and

 





1

1 1

, ,

F u v  F u  v 0,

 2

_{ }

_

_

2 2

, ,

F u v  F u v  _0.

Define the Nehari manifold

 



 



   



, 0,0 , , ,



N u v W I u v u v 0 . Note that

N contains all solutions of (1.1) and if and

only if

 

,

u v N

 

u v, _Wp R u v

 

, G u v

 

, . (2.1) Lemma 2.1. I u v

 

, is coercive and bounded below

on N.

Proof. Suppose

 

u v, N. From (2.1), the Holder

inequality and the Sobolev embedding theorem, it follows that

 



  





 





,

1 1 1

, , ,

1 1 1 1

, ,

,

, .

p

W

p

W

p

W

q

q p

W I u v

u v R u v G u v

p q

u v R u v

p q

p u v p

q

S u v f g

q

 

   

  _ _

 



 

  



   

_  _ _  _

 

   

  

  

 



(2.2)

Thus I is coercive and bounded below on N since .

q p    Define 

 

u v,  I u v

 

, ,



u,v



.Then for all

 

u v, N we have

   

 

  

  



  



  



  



 



  



  

, , ,

, ,

p W

u v u v

p u v qR u v G u v

p q u v q G u v

p u v q R

p q R u v p G u v

 

   

 

 

   

    

 

_   _   

 

  _   _



.

u v

(2.3)

Arguing as that in [9,10], we split N into three parts:

 

   



, , , ,



N u v N  u v u v 0 ,

 

   





0 _, _{, , ,}

N  u v N  u v u v 0 ,

 

   



, , , ,



N  u v N  u v u v 0 .

(3)

on N and

 

_{u v}_, __N0_. Then _{I u v}_

 

_, _₀_in_W1_.

Proof.If is a local minimizer for I on N, then

is a solution of the optimization problem mini-mize subject to

 

u v,

 

u v,

I

 

u v, 0.

 

Hence, by the theory of Lagrange multipliers, there exists R such that

 

, 

 

u v,

I u v  

 

1 _

   

in _W1



_



.

Here is the dual space of the Sobolev space

. Thus, W W

   

, , ,

I u v u v

   

, , ,

u v u

    v

But  u v, , ,u v 0 since

 

u v, _N0_. Hence 0.



Lemma 2.3. N0  for all



 f



p q



 g





0, 1



p p

p q

 

    

Proof.We argue by contradiction. Suppose that for all











0, 1



p p

p q p q

f g

   

     there is

 

_{u v}_, __N0_,_{then (2.3) and the Sobolev embedding the-}

orem imply that



  



  





 

, ,

p W

p

W

p q u v q G u

q S u v

 

   

 



 _

   

  

v

(2.4)

and



  



  





















 

1, 1,

, ,

p W

q

q q

p

p p

q

q p

W

q u v

q R u v

q f u x g x

q S f u g v

q S f g u v

   

   

  



 









 

  

   



d d

, .

q q

v

 

 

 

(2.5) Thus from (2.4), (2.5) we have

 

1

,

p p W

p q

u v S

q

   

 

 _{ }

 _ 

   

 

  (2.6)

and

 









1

,

.

q p q _{p p q}

W

p p _p

p q p q

q

u v S

p

f g

   

 

  _

 

 



      _{ } 

 

 



 

 

Consequently,









1,

p p

p q p q

p _p

q

p _{p q}

p p q

f g

p q _S p _S

q q

  _{ }

 

 

   

 

 

   _





 _  _ _{ } _

 _ _ _ _

  _   _

 

 

which is a contradiction.

By Lemma 2.3, we can write N N N for all











0, 1



p p

p q p q

f g

   

    

Define

 u v N,inf I u v

 

, ,  u v N,inf I u v

 

,

 _  _

 

  .

Lemma 2.4. (i) _0 _{for all}











0, 1



p p

p q p q

f g

   

    

(ii) There exists a positive constant d0 depending on

, ,

  q N S S f g, , , , , , ,  such that  d0 for all

_









0, 1 .

p

p p _{p q}

p q p q q

f g

p

    

 

 

 

 _ _{ }  _

 

 

 

Proof.(i) Suppose

 

u v, N,then we have

 











 

,

1 1 _, 1 1 _,

1 1 1 1

, ,

, 0

p

W

p p

W W

p

W I u v

u v G u v

p q q

p q

u v u v

p q q q

p p q u v pq

 

   

   

_  _ _  _ 

   

     

_  _ _  _ _

  

    

  

  



for 1   q p  .



Thus we get that _0.

(ii) Suppose









0, 1 ,

p

p p _{p q}

p q p q q

f g

p

    

 

 

 

 _ _{ } _

 

 

 



and

 

u v, _N_._{Then (2.4) implies that}

 

1

, ,

p p W

p q

u v S

q

   

 

 _{ }

 _ 

  

 

 

  (2.7)

and (2.5) implies that

 









, ,

.

q

q p

W

p q

p p _p

p q p q

R u v S u v

f g

 



 

 



 



 

 



(4)

From (2.7) and (2.8) it follows that

 



  





 

1 1 1 1

, , , , p W q

p _p q

W , W

I u v u v R u v

p q

u v S u v

p q                  _  _ _  _              









 



  





























, , – – – – p q

p p _p

p q p q

W p q W q p q p p p q p p p p q q p q

p p _p

p p

p p q

p

q q

q

f g u v

p _{u v} p

q S q

p q _S

q

p

f

p q _S

p q

S f g

q g q                               _ _           _{ }                                                   _        _      _  _             p q p  _       _

which shows that

 

, 0,

 

, ,

I u v _d _ u v _N

since









0, 1 ,

p

p p _{p q}

p q p q q

f g p               _ _{ } _       

where d0d0



 , , , , , , , , ,q N S S f g 



is a positive

constant.

For all

 

u v, W such that G u v

 

, 0 ,set



  



  

1 max , 0. , p _p W

p q u v

t

q G u v

       _           

Lemma 2.5. Suppose that











0, 1



p p

p q p q

f g

   

    

and

 

u v, W is a function satisfying G u v

 

, 0. (i) If R u v

 

, 0



, then there exists a unique t1tmax

such that 1 1



N





0

, sup_t

,

t u t v  and



1 1



,



I t u t v  _ I tu tv .

(ii) If R u v

 

, 









2 max 3 2 2 3 3

0_{ }t t _t t u t v, , _Nand t u t v, _N_. Furthermore,







max

2 , 2 ₀inf_{t t} ,



,

I t u t v I tu tv

 





3 , 3



supt 0



,



I t u t v   I tu tv .

Proof. Fix

 

u v, W with G u v

 

, 0. For all

,let

0

t

 

p q

 

, p q

 

, W

t t  u v t   G u v

   ,

then it is obvious that Ψ(0) = 0,Ψ(t) →−∞as t →+∞,

 

t 0

  as small enough. So we can deduce that Ψ′(t) = 0 at

0

t

 

max, 0

t t  t  for t



0,tmax



,

 

t 0



  for t



tmax,



. Then Ψ(t) that achieves

its maximum at is increasing for and decreasing for max

t t



0,tmax





max,



.

t t  Moreover,

 



  



  

 



  



  

 













 

max – , , – , – , , – , – – – – , , , p q p _p p W W q p p W

p q q

p p p q q W W p q t

p q u v

u v q G u v

p q u v

G u v q G u v

p q p q

q q

u v

u v G u v

p q _S

q                                            _{ }                             _ _ _ _                  _               

 

, 0

p q p

q W

p _{u v} q   _{ }      _{ }     .

(i) If R u v

 

, 0,then there exists a unique

1 max0 such that

t t 

 

t1 R u v

 

,  0, 

 

t1 0.

Note that



 



 





 





1 1 1 1

1 1 1

1 1 , , , , , , , , , 0, p p q W p

q p q q

W q

I t u t v t u t v

t u v t R u v t G u v

t t u v t G u v R u v

t t R u v

                   ,

thus we get



t u t v1 ,1



N.

From



 





  







 

1 1 1 1

1 1 1 1 1 , , , , , 0, p p W q

t u t v t u t v

p q t u v q t G u v

t t                 



0 , then there exist and such

that 2

(5)

we have



t u t v1 ,1



N . For all it follows that 

 t t _max,



 





  







 

1

, , ,

, ,

0,

p p

W q

tu tv tu tv

p q t u v q t G u v

t t

 

  



 

    



  







2 2

d

, 0

dt I tu tv  ,





1



 



1

d

, , 0

d

q

I tu tv t t R u v t t

t



    for  .

So we get that



1 1







0

, sup ,

t



 .

(ii) If R u v

 

, 0, for











0,Λ1



,

p p

p q p q

f g

   

   

 









 

1

max

0 0 ,

,

–

,

– –

,

p q

q _p _p _p

q

p p q p q

W

q p q

q

p p

W p q

p

q p

W

R u v

S f g u v

S u v

p q p

S

q q

t

   

 

 

   

 

 

 

  _{ }

  

 

 _  _

 

 

  _{ }

  

 

 

 

 

u v

3 . t



, then there exist t2and t3 such that

 

2 2 max 3 2 3

0 t t u v, t  t , t R u v,   t and

 

t2 0

 



    By the similar argument in (i), we

get



t u t v2 , 2



N ,



t u t v3 , 3



N and

 

 





2 3

d

, 0 for or .

dtI tu tv  t t t t





2 2

d ,

dt I tu tv 0 for t



0,tmax







2 2

d , dt I tu tv 0



, for t



tmax,



.

Then it follows that







max

2 ,2 ₀inf_{t t} ,

 





3 3







0

, sup ,

t



 .

The proof of this Lemma is completed.

For each

 

u v, Wwith R u v

 

, 0 ,we write



  



  

1

max

,

0. ,

p q

p W

q R u v t

p u v

   



 _{ } 

 

 

 _{ } 

 

(2.9)









0, 1

p

p p _{p q}

p q p q q

f g

p

    

 

 

 

 _ _{ }  _

 

 

 

and

 

u v, W is a function satisfying R u v

 

, 0.

(i) If G u v

 

, 0, then there exists a unique t1tmax such that



t u t v1 ,1



N



 and









max

1 ,1 sup ,

t t



 .

(ii) If G u v

 

, 0 , then there exist and such

that 2

t t3





2 max 3 2 2

0 t t t t u t v, , N and



t u t3 ,3v



N



 .

Furthermore,









max

2 , 2 ₀inf_{t t} ,

 





3 3







0

, sup ,

t





Proof. Fix

 

u v, W with R u v

 

, 0. For all

let 0,

t

 

p

 

, p q



,

W

t t    u v t   R u v

  



(2.10)

then it is obvious that 

 

t  ast 0 . So we can deduce that 

 

t 0 at t t _max,

 

t 0 fort



0,tmax



,



   

 

t 0 for t



tmax,



.

Then 

 

t that achieves its maximum at tmax is in-

creasing for t



0,tmax



and decreasing for



max,



t t  . Using the similar argument in Lemma 2.5, we can obtain the result of Lemma 2.6.

3. Proof of Theorem 1.1











0, 1



,

p p

p q p q

f g

   

    

then the functional has a minimizer I

 

u v, _N _{and it}

satisfies

(i) I u v

 

, .

(ii)

 

u v, is a nontrivial solution of (1.1).

Proof.Let





u vn, n





N



 be a minimizing sequence such that



_n, _n



_n

 

1 ,



_n, _n



_n

 

I u v __o I u v_ _o _{1 .}_(3.1)

Since I is coercive on N, we get that





u v_n, _n





is bounded on W . Passing to a subsequence (still denoted by





u vn, n





), there exists



u v,



such that



u vn, n

  

 u v, v

v



weakly in W, ,

n n

u u v  a.e. in Ω, (3.2) ,

n n

u u v  strongly in _Lq



_ and in _L 

 

_ _. This implies

(6)

Since





u v_n, _n





_N_,_{we get}



_n, _n



_

_{  }

, q

_

_{ }

_n _n



W

p q _,

I u v u v R u v

p q

   

   

 

 

By Lemma 2.4 (i) we get I u v



n, n



 and



0

then R u v

 

,  . Now we prove that u v_n, _n

  

 u v, strongly in W. Suppose otherwise, then either

1,p lim inf_n n 1,por 1,p lim inf_n n1,p.

u u v

 

  v (3.3)

Fix

 

u v, W with R u v

 

, 0. Let

 

,

E t   t G u v ,where 

 

t is as in (2.10). Clearly, E t

 

  as t 0 ,and

 

G u v

 

,

E t  

 

, as t . Since

E t   t by an argument similar to the one in the proof of Lemma 2.6, we have that the function

 

E t achieves its maximum at tmax, is increasing for



0,tmax



t and decreasing for t



tmax,



, where max

t is as in (2.9). Since R u v

 

, 0, by Lemma 2.6, there is unique 0 t0 tmax

, such that













max

0 ,0 , 0 ,0 ₀inf_{t t} ,

t u t v N I t u t v I tu tv

 

 

Then

 











0 0

0 0 0 0 0 0 0

,

, ,

0.

p W

E t t G u v

t   t u t v R t u t v G t u t v

  

 

 _   _

 





, (3.4)

By (3.3) and (3.4), we obtain E t

 

₀ 0 for n sufficiently large for the sequence





u v_n, _n





. Since



u vn, n



N



 , we have tmax



u vn, n



1. Moreover,

 

1



_n, _n



p



_n, _n





_n, _n



W

E  u v R u v G u v 0,

and E t

 

is increasing for t



0,tmax



u vn, n



. This

implies E t

 

0 for all t



0,1



and n sufficiently large. We obtain 1 t0 tmax

 

u v . , But



t u t v0 ,0



N

and







max

0 , 0 ₀inf_{t t} ,



,

 

 this implies



0 ,0



I u v

 

, _nlim



n n



,

I t u t v I u v



  ,

which is a contradiction. Hence



u v_n, _n

  

 u v

 

,

I u v

, strongly in W. This implies I u v



n, n







 

I N.

as .

Thus is a minimizer for on Since

n 



u v,

 

,v I





,



I u u v and



_{u v}_,



_N_{, by Lemma 2.2 we} may assume that is a nontrivial nonnegative so-lution of Equation (1.1).



,

u v



Next we prove Arguing by contradiction, without loss of generality, we may assume that v ≡ 0. Then as u is a nonzero solution of

0, 0.

u v

 

2

2 2

0, ,

, ,

p p

p q

u m x u u x

u

u f x u u x



 

    



 _

  





 n 

(3.5)

we have

 

,0 p

 

qd 0.

W

u 

_

_f x u s (3.6) Choose _{z W}_ 1,p

   

__﹨ ₀ _{such that}

 

0, p

 

qd 0,

W

z 

_

_g x z s (3.7) then

 

qd

 

qd

f x u s g x z s



_

_ 

_

_ 0.

By Lemma 2.6, there is a unique 0 t t_max such that



tu tz,



_N_._{Moreover, from (3.6) and (3.7), it} follows that



  



  

1

max

1

, ,

1,

p q

p W

p q

q R u z t

p u z

q p

   



 _{ } 

 



 

_ _ 





  

 

    

and









max

0

, inf ,

t t .

I tu tz I tu tz

 



This implies



,

    

,

I tu tv I u v I u,0 ,

which contradict with that (u,0) is the minimizer and hence u0,v0. So

 

u v, is a nontrivial nonnegative solution of Equation (1.1).









0, 1 .

p

p p _{p q}

p q p q q

f g

p

    

 

 

 

 _ _{ } _

 

 

 



Then the functional has a minimizer I



U V,



_N

and it satisfies

(i) I U V



,



_.

(ii)



U V,



is a nontrivial solution of (1.1).

Proof.Let





u vn, n





N



 be a minimizing sequence such that



n, n



n

 

1 ,



n, n



n

 

I u v __o I u v_ _o _{1 .}_(3.8)

Since I is coercive on N, we get that





u v_n, _n





is bounded on W . Passing to a subsequence (still denoted by





u vn, n





), there exists



U V,



such that



u vn, n

 

 U V,



weakly in W ,

,

n n

(7)

V



,

n n

u U v  _{strongly in} _Lq



_ and in _L 

 

_ _.

REFERENCES

This implies _[1] _{J. Garcia-Azorero, I. Peral and J. D. Rossi, “A Convex-} Concave Problem with a Nonlinear Boundary Condition,” Journal of Differential Equations, Vol. 198, No. 1, 2004, pp. 91-128. doi:10.1016/S0022-0396(03)00068-8.



_n, _n





,



,



_n, _n







R u v R U V G u v G U V, . Moreover, by (2.3) we obtain

[2] T.-F. Wu, “A Semilinear Elliptic Problem Involving Non- linear Boundary Condition and Sign-Changing Potential,” Electronic Journal of Differential Equations, Vol. 2006, No. 131, 2006, pp. 1-15.



_n, _n





_n, _n



p W

p q

G u v u v

q

 

 

  ,

then G U V



,



 C 0. Now we prove that



,

 

 U V,



[3] T.-F. Wu, “Multiplicity of Positive Solution of p-Lapla- cian Problems with Sign-Changing Weight Functions,” International Journal of Mathematical Analysis, Vol. 1, No. 9-12, 2007, pp. 557-563.

n n strongly in W. Suppose otherwise,

then either

u v

1, 1,

or

lim inf

lim inf .

n

p _n p

n

p _n p

U u

V v





 (3.10) [4] T.-F. Wu, “Multiple Positive Solutions for Semilinear El- _{liptic Systems with Nonlinear Boundary Condition,”}_Ap-

plied Mathematics and Computation, Vol. 189, No. 2, 2007, pp. 1712-1722. doi:10.1080/028418501127346846. By Lemma 2.6, there is unique to such that



t U t V0 ,0



N





 ,



Since



u vn, n



N I u v,



n, n

 

I tu tvn, n for all , we

have

0

t



0 ,0



lim



0 n,0 n



lim



n, n



,

n n

I t U t V I t u t v I u v 

 

  

[5] T.-S. Hsu, “Multiple Positive Solutions for a Critical Quasilinear Elliptic System with Oncave-Convex Nonlin- earitiesc,” Nonlinear Analysis, Vol. 71, No. 7-8, 2009, pp. 2688-2698. doi:10.1016/ j.na.2009.01.110.

and this is a contradiction. Hence



u v_n, _n

 

 U V,





,



I U V



,



strongly in W. This implies I u vn n  

[6] T.-F. Wu, “On Semilinear Elliptic Equations Involving Concave-Convex Nonlinearities and Sign-Changing Weight Function,” Article Journal of Mathematical Analysis and Applications, Vol. 318, No. 1, 2006, pp. 253-270. doi:10.1016/j.jmaa. 2005.05.057.

as n . Thus



U V,



is a minimizer for I on N. _[7] _{T.-S. Hsu and H.-L. Lin, “Multiple Positive Solutions for} Singular Elliptic Equations with Concave-Convex Non- linearities and Sign-Changing Weights,” Boundary Value Problems, Vol. 2009, 2009, Article ID: 584203.

Since I U



,V



 I U



,V



and



U V,



_N_{, by} Lemma 2.4 and the similar argument as that in Lemma 3.1 we can get



U V,



is also a nontrivial nonnegative

solution of Equation (1.1). [8] L. Wang, Q. Wei and D. Kang, “Existence and Multiplic- _{ity of Positive Solutions to Elliptic Systems Involving} Critical Exponents,” Journal of Mathematical Analysis and Applications, Vol. 383, No. 2, 2011, pp. 541-552. doi:10.1016/j.jmaa. 2011.05.053.

Proof of Theorem 1.1. From Lemma 3.1 and Lemma 3.2, we obtain that Equation (1.1) has two nontrivial nonnegative solutions



u v,



and



U V,



satisfy

 

u v, N and . It remains to show that the solutions found in Lemma 3.1 and Lemma 3.2 are distinct. Since this implies that



U V,



N

,

N_N_{ }

 

_{u v}_,

and



U V,



are distinct. This concludes the proof.

[9] K. Brown and Y. Zhang, “The Nehari Manifold for a Semilinear Elliptic Equation with a Sign-Changing Weight Function,” Journal of Differential Equations, Vol. 193, No. 2, 2003, pp. 481-499.

doi:10.1016/S0022-0396(03)00121-9.

4. Acknowledgements

[10] G. Tarantello, “On Nonhomogeneous Elliptic Equations

Involving Critical Sobolev Exponent,” Annales De L In- stitut Henri Poincare-Analyse Non Lineaire, Vol. 9, No. 3, 1992, pp. 281-304.