Open Access

Positivity of the infimum eigenvalue for equations of p ( x ) -Laplace type in R N

Boundary Value Problems20132013:214

DOI: 10.1186/1687-2770-2013-214

Received: 29 June 2013

Accepted: 21 August 2013

Published: 2 October 2013

Abstract

We study the following elliptic equations with variable exponents

div ( ϕ ( x , | u | ) u ) = λ f ( x , u ) in  R N .

Under suitable conditions on ϕ and f, we show the existence of positivity of the infimum of all eigenvalues for the problem above, and then give an example to demonstrate our main result.

MSC:35D30, 35J60, 35J92, 35P30, 47J10.

Keywords

p ( x ) -Laplacian variable exponent Lebesgue-Sobolev spaces weak solution eigenvalue

1 Introduction

The variable exponent problems appear in a lot of applications, for example, elastic mechanics, electro-rheological fluid dynamics and image processing, etc. The study of variable mathematical problems involving p ( x ) -growth conditions has attracted interest and attention in recent years. We refer the readers to [14] and references therein.

In this paper, we are concerned with the eigenvalue problem of a class of equations of p ( x ) -Laplacian type
div ( ϕ ( x , | u | ) u ) = λ f ( x , u ) in  R N ,
(E)

where the function ϕ ( x , t ) is of type | t | p ( x ) 2 with continuous nonconstant function p : R N ( 1 , ) and f : R N × R R satisfies a Carathéodory condition. Recently, the authors in [5] obtained the positivity of the infimum of all eigenvalues for the p ( x ) -Laplacian type subject to the Dirichlet boundary condition. As far as the authors know, there are no results concerned with the eigenvalue problem for a more general p ( x ) -Laplacian type problem in the whole space R N .

When ϕ ( x , t ) = | t | p ( x ) 2 , the operator involved in (E) is called the p ( x ) -Laplacian, i.e., Δ p ( x ) u : = div ( | u | p ( x ) 2 u ) . The studies for p ( x ) -Laplacian problems have been extensively performed by many researchers in various ways; see [511]. In particular, by using the Ljusternik-Schnirelmann critical point theory, Fan et al. [8] established the existence of the sequence of eigenvalues of the p ( x ) -Laplacian Dirichlet problem; see [12] for Neumann problems. Mihăilescu and Rădulescu in [13] obtained the existence of a continuous family of eigenvalues in a neighborhood of the origin under suitable conditions.

The p ( x ) -Laplacian is a natural generalization of the p-Laplacian, where p > 1 is a constant. There are a bunch of papers, for instance, [1418] and references therein. But the p ( x ) -Laplace operator possesses more complicated nonlinearities than the p-Laplace operator, for example, it is nonhomogeneous, so a more complicated analysis has to be carefully carried out. Some properties of the p-Laplacian eigenvalue problems may not hold for a general p ( x ) -Laplacian. For example, under some conditions, the infimum of all eigenvalues for the p ( x ) -Laplacian might be zero; see [8]. The purpose of this paper is to give suitable conditions on ϕ and f to satisfy the positivity of the infimum of all eigenvalues for (E) still. This result generalizes Benouhiba’s recent result in [6] in some sense.

This paper is organized as follows. In Section 2, we state some basic results for the variable exponent Lebesgue-Sobolev spaces, which are given in [19, 20]. In Section 3, we give sufficient conditions on ϕ and f to obtain the positivity of the infimum eigenvalue for the problem (E) above. Also, we present an example to illustrate our main result.

2 Preliminaries

In this section, we state some elementary properties for the variable exponent Lebesgue-Sobolev spaces, which will be used in the next section. The basic properties of the variable exponent Lebesgue-Sobolev spaces can be found from [19, 20].

To make a self-contained paper, we first recall some definitions and basic properties of the variable exponent Lebesgue spaces L p ( x ) ( R N ) and the variable exponent Lebesgue-Sobolev spaces W 1 , p ( x ) ( R N ) .

Set
C + ( R N ) = { h C ( R N ) : inf x R N h ( x ) > 1 } .
For any h C + ( R N ) , we define
h + = sup x R N h ( x ) and h = inf x R N h ( x ) .
For any p C + ( R N ) , we introduce the variable exponent Lebesgue space
L p ( x ) ( R N ) : = { u : u  is a measurable real-valued function,  R N | u ( x ) | p ( x ) d x < } ,
endowed with the Luxemburg norm
u L p ( x ) ( R N ) = inf { λ > 0 : R N | u ( x ) λ | p ( x ) d x 1 } .

The dual space of L p ( x ) ( R N ) is L p ( x ) ( R N ) , where 1 / p ( x ) + 1 / p ( x ) = 1 . The variable exponent Lebesgue spaces are a special case of Orlicz-Musielak spaces treated by Musielak in [21].

The variable exponent Sobolev space W 1 , p ( x ) ( R N ) is defined by
W 1 , p ( x ) ( R N ) = { u L p ( x ) ( R N ) : | u | L p ( x ) ( R N ) } ,
where the norm is
u W 1 , p ( x ) ( R N ) = u L p ( x ) ( R N ) + u L p ( x ) ( R N ) .
(2.1)
Definition 2.1 The exponent p ( ) is said to be log-Hölder continuous if there is a constant C such that
| p ( x ) p ( y ) | C log | x y |
(2.2)

for every x , y R N with | x y | 1 / 2 .

Smooth functions are not dense in the variable exponent Sobolev spaces, without additional assumptions on the exponent p ( x ) . Zhikov [22] gave some examples of Lavrentiev’s phenomenon for the problems with variable exponents. These examples show that smooth functions are not dense in variable exponent Sobolev spaces. However, when p ( x ) satisfies the log-Hölder continuity condition, smooth functions are dense in variable exponent Sobolev spaces, and there is no confusion in defining the Sobolev space with zero boundary values, W 0 1 , p ( x ) ( R N ) , as the completion of C 0 ( R N ) with respect to the norm u W 1 , p ( x ) ( R N ) (see [23, 24]).

Lemma 2.2 [19, 20]

The space L p ( x ) ( R N ) is a separable, uniformly convex Banach space, and its conjugate space is L p ( x ) ( R N ) , where 1 / p ( x ) + 1 / p ( x ) = 1 . For any u L p ( x ) ( R N ) and v L p ( x ) ( R N ) , we have
| R N u v d x | ( 1 p + 1 p ) u L p ( x ) ( R N ) v L p ( x ) ( R N ) 2 u L p ( x ) ( R N ) v L p ( x ) ( R N ) .

Lemma 2.3 [19]

Denote
ρ ( u ) = R N | u | p ( x ) d x for all  u L p ( x ) ( R N ) .
Then
  1. (1)

    ρ ( u ) > 1 (=1; <1) if and only if u L p ( x ) ( R N ) > 1 (=1; <1), respectively;

     
  2. (2)

    if u L p ( x ) ( R N ) > 1 , then u L p ( x ) ( R N ) p ρ ( u ) u L p ( x ) ( R N ) p + ;

     
  3. (3)

    if u L p ( x ) ( R N ) < 1 , then u L p ( x ) ( R N ) p + ρ ( u ) u L p ( x ) ( R N ) p .

     

Lemma 2.4 [11]

Let q L ( R N ) be such that 1 p ( x ) q ( x ) for almost all x R N . If u L q ( x ) ( R N ) with u 0 , then
  1. (1)

    if u L p ( x ) q ( x ) ( R N ) > 1 , then u L p ( x ) q ( x ) ( R N ) q | u | q ( x ) L p ( x ) ( R N ) u L p ( x ) q ( x ) ( R N ) q + ;

     
  2. (2)

    if u L p ( x ) q ( x ) ( R N ) < 1 , then u L p ( x ) q ( x ) ( R N ) q + | u | q ( x ) L p ( x ) ( R N ) u L p ( x ) q ( x ) ( R N ) q .

     

Lemma 2.5 [23]

Let Ω R N be an open, bounded set with Lipschitz boundary, and let p C + ( Ω ¯ ) with 1 < p p + < N satisfy the log-Hölder continuity condition (2.2). If q L ( Ω ) with q > 1 satisfies
q ( x ) p ( x ) : = N p ( x ) N p ( x ) for all  x Ω ,
then we have
W 1 , p ( x ) ( Ω ) L q ( x ) ( Ω ) ,

and the imbedding is compact if inf x Ω ( p ( x ) q ( x ) ) > 0 .

Lemma 2.6 [25]

Suppose that p : R N R N is a Lipschitz function with 1 < p p + < N . Let q L ( R N ) and p ( x ) q ( x ) p ( x ) for almost all x R N . Then there is a continuous embedding W 1 , p ( x ) ( R N ) L q ( x ) ( R N ) .

3 Main result

In this section, we shall give the proof of the existence of the positive eigenvalue for the problem (E), by applying the basic properties of the spaces L p ( x ) ( R N ) and W 1 , p ( x ) ( R N ) , which were given in the previous section.

Throughout this paper, let p C + ( R N ) satisfy the log-Hölder continuity condition (2.2) and X : = W 0 1 , p ( x ) ( R N ) with the norm
u X = inf { λ > 0 : R N | u ( x ) λ | p ( x ) d x 1 } ,

which is equivalent to norm (2.1).

Definition 3.1 We say that u X is a weak solution of the problem (E) if
R N ϕ ( x , | u | ) u ( x ) φ ( x ) d x = λ R N f ( x , u ) φ ( x ) d x

for all φ X .

Denote
Ω 1 = { x R N : 1 < p ( x ) < 2 } , Ω 2 = { x R N : p ( x ) 2 }

(we allow the case that one of these sets is empty). Then it is obvious that R N = Ω 1 Ω 2 . We assume that:

(H1) p , q C + ( R N ) , p ( x ) < N , and 1 < p p + < q q + < p ( x ) .

  1. (HJ1)

    ϕ : R N × [ 0 , ) [ 0 , ) satisfies the following conditions: ϕ ( , η ) is measurable on R N for all η 0 and ϕ ( x , ) is locally absolutely continuous on [ 0 , ) for almost all x R N .

     
  2. (HJ2)
    There are a function a L p ( x ) ( R N ) and a nonnegative constant b such that
    | ϕ ( x , | v | ) v | a ( x ) + b | v | p ( x ) 1

    for almost all x R N and for all v R N .

     
  3. (HJ3)
    There exists a positive constant c such that the following conditions are satisfied for almost all x R N :
    ϕ ( x , η ) c η p ( x ) 2 and η ϕ η ( x , η ) + ϕ ( x , η ) c η p ( x ) 2
    (3.1)
    for almost all η ( 0 , 1 ) . In case x Ω 2 , assume that condition (3.1) holds for almost all η ( 1 , ) , and in case x Ω 1 , assume that for almost all η ( 1 , ) instead
    ϕ ( x , η ) c and η ϕ η ( x , η ) + ϕ ( x , η ) c .
    (3.2)
     
  4. (HJ4)
    For all x R N and all ξ R N , the estimate holds
    0 a ( x , ξ ) ξ p + Φ 0 ( x , | ξ | ) ,

    where a ( x , ξ ) = ϕ ( x , | ξ | ) ξ .

     
Let us put
Φ 0 ( x , t ) = 0 t ϕ ( x , η ) η d η
and define the functional Φ : X R by
Φ ( u ) = R N Φ 0 ( x , | u ( x ) | ) d x .
Then Φ C 1 ( X , R ) [5], and its Gateaux derivative is
Φ ( u ) , φ : = R N ϕ ( x , | u ( x ) | ) u ( x ) φ ( x ) d x .
(3.3)
Let f : R N × R R be a real-valued function. We assume that the function f satisfies the Carathéodory condition in the sense that f ( , t ) is measurable for all t R and f ( x , ) is continuous for almost all x R N . Denote
γ ( x ) = r ( x ) r ( x ) q ( x ) for almost all  x R N ,

where q is given in (H1) and q ( x ) < r ( x ) < p ( x ) . We assume that

(F1) For all ( x , t ) R N × R , f ( x , t ) t 0 , and there is a nonnegative measurable function m with m L γ ( x ) ( R N ) such that
| f ( x , t ) | m ( x ) | t | q ( x ) 1 .
Denoting F ( x , t ) = 0 t f ( x , s ) d s , it follows from (F1) that
(F1 ) 0 F ( x , t ) m ( x ) q ( x ) | t | q ( x ) for all  ( x , t ) R N × R .
Define the functional Ψ , I λ : X R by
Ψ ( u ) = R N F ( x , u ) d x and I λ ( u ) = Φ ( u ) λ Ψ ( u ) .
Then it is easy to check that Ψ C 1 ( X , R ) , and its Gateaux derivative is
Ψ ( u ) , φ = R N f ( x , u ) φ ( x ) d x
(3.4)
for any u , φ X . Let us consider the following quantity:
λ = inf u X { 0 } R N Φ 0 ( x , | u | ) d x R N F ( x , u ) d x .
(3.5)

For the case of ϕ ( x , | t | ) = | t | p ( x ) 2 and f ( x , t ) = m ( x ) | t | q ( x ) 2 t , where m ( x ) satisfies a suitable condition, Benouhiba [6] proved that λ > 0 . In this section, we shall generalize the conditions on f and ϕ to satisfy λ > 0 still.

The following lemma plays a key role in obtaining the main result in this section.

Lemma 3.2 Assume that assumptions (HJ3)-(HJ4), (H1), and (F1) hold and satisfy
(H2) q + 1 2 p < q ,
then the functionals Φ and Ψ satisfy the following relations:
lim u X 0 Φ ( u ) Ψ ( u ) =
(3.6)
and
lim u X Φ ( u ) Ψ ( u ) = .
(3.7)
Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
| Ψ ( u ) | = | R N F ( x , u ) d x | R N | m ( x ) q ( x ) | u | q ( x ) | d x 2 q m L γ ( x ) ( R N ) | u | q ( x ) L r ( x ) q ( x ) ( R N ) 2 q m L γ ( x ) ( R N ) ( u L r ( x ) ( R N ) q + + u L r ( x ) ( R N ) q ) 2 C q m L γ ( x ) ( R N ) ( u X q + + u X q )
(3.8)
for some positive constant C. Let u in X with u X 1 . Then it follows from (HJ3), (HJ4), (3.8) and Lemma 2.3(3) that
| Φ ( u ) Ψ ( u ) | R N Φ 0 ( x , | u | ) d x 4 C q m L γ ( x ) ( R N ) u X q c p + u X p + 4 C q m L γ ( x ) ( R N ) u X q .
(3.9)
Since q > p + , we conclude that
Φ ( u ) Ψ ( u ) as  u X 0 .
Next, we show that relation (3.7) holds. From (H2), there exists a positive constant δ such that q + ( 1 / 2 ) p < δ < q , and thus we have
p > 2 ( q + δ ) > 2 ( q δ ) .
(3.10)
Let ( x ) be a measurable function such that
max { p ( x ) γ ( x ) p ( x ) + δ γ ( x ) , p ( x ) p ( x ) + δ q ( x ) } ( x ) min { p ( x ) γ ( x ) p ( x ) + δ γ ( x ) , p ( x ) p ( x ) + δ q ( x ) }
(3.11)
holds for almost all x R N and
δ ( + + 1 ) < q .
(3.12)
Then we have L ( R N ) and 1 < ( x ) < γ ( x ) . Let u X with u X > 1 . Then it follows from (F1′) and Lemma 2.2 that
| Ψ ( u ) | 1 q R N m ( x ) | u | δ | u | q ( x ) δ d x 2 q m | u | δ L ( x ) ( R N ) | u | q ( x ) δ L ( x ) ( R N ) .
Therefore, without loss of generality, we may suppose that m | u | δ L ( x ) ( R N ) > 1 . From the inequality above, by using Lemma 2.3, Lemma 2.2 and Lemma 2.4 in order, we have
| Ψ ( u ) | 2 q ( R N m ( x ) | u | δ ( x ) ) 1 | u | q ( x ) δ L ( x ) ( R N ) 4 q m ( x ) L γ ( x ) ( x ) ( R N ) 1 | u | δ ( x ) L ( γ ( x ) ( x ) ) ( R N ) 1 | u | q ( x ) δ L ( x ) ( R N ) 4 q m L γ ( x ) ( R N ) α ( u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) δ + + u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) δ ) × ( u L ( q ( x ) δ ) ( x ) ( R N ) q + δ + u L ( q ( x ) δ ) ( x ) ( R N ) q δ ) ,

where α = { + / if  m L γ ( x ) ( R N ) > 1 , 1 if  m L γ ( x ) ( R N ) 1 .

By Young’s inequality, we get
| Ψ ( u ) | 4 q m L γ ( x ) ( R N ) α ( u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) 2 δ + + u L δ ( x ) ( γ ( x ) ( x ) ) ( R N ) 2 δ + u L ( q ( x ) δ ) ( x ) ( R N ) 2 ( q + δ ) + u L ( q ( x ) δ ) ( x ) ( R N ) 2 ( q δ ) ) .
Using (3.11), we get that
p ( x ) < δ ( x ) ( γ ( x ) ( x ) ) p ( x ) , p ( x ) < ( q ( x ) δ ) ( x ) p ( x )
holds for almost all x R N . Hence it follows from Lemma 2.6 that
| Ψ ( u ) | 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) )
(3.13)
for some positive constant C. Therefore, we obtain that
| Φ ( u ) Ψ ( u ) | c p + u X p 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) ) .

From (3.10), with the inequality above, we conclude that relation (3.7) holds. □

Lemma 3.3 Assume that (HJ1)-(HJ3) and (H1) hold. Then Φ is weakly lower semi-continuous, i.e., u n u in X implies that Φ ( u ) lim inf n Φ ( u n ) .

Proof Suppose that u n u in X as n . Since (HJ3) implies that Φ is strictly monotone on X, we have that Φ is convex, and so,
Φ ( u n ) Φ ( u ) + Φ ( u ) , u n u
for any n. Then we get that
lim inf n Φ ( u n ) Φ ( u ) + lim inf n Φ ( u ) , u n u = Φ ( u ) .

The proof is complete. □

Lemma 3.4 Assume that (H1) and (F1) hold. For any K [ 0 , ) and all u X , the following estimate holds:
| x | K F ( x , u ) d x 2 C q ( | x | K m ( x ) d x ) 1 γ 1 ( u X q + + u X q ) ,
(3.14)

where γ 1 is either γ + or γ .

Proof Applying Lemmas 2.2, 2.4 and 2.6, we get
| x | K F ( x , u ) d x | x | K m ( x ) q ( x ) | u | q ( x ) d x 2 q m L γ ( x ) ( { x R N : | x | K } ) | u | q ( x ) L r ( x ) q ( x ) ( { x R N : | x | K } ) 2 q ( | x | K m ( x ) d x ) 1 γ 1 ( u L r ( x ) ( R N ) q + + u L r ( x ) ( R N ) q ) 2 C q ( | x | K m ( x ) d x ) 1 γ 1 ( u X q + + u X q )

for some positive constant C. □

Lemma 3.5 Assume that (H1) and (F1) hold. For almost all x R N and all t R , the following estimate holds:
F ( x , t ) 1 q ( m ( x ) γ ( x ) γ + | t | r ( x ) ( γ + ) ) .
(3.15)

Proof Since q ( x ) ( γ ( x ) ) = r ( x ) , estimate (3.15) is obtained from (F1′) and Young’s inequality. □

Lemma 3.6 Assume that (H1) and (F1) hold. Then Ψ is weakly-strongly continuous, i.e., u n u in X implies that Ψ ( u n ) Ψ ( u ) .

Proof Let { u n } be a sequence in X such that u n u in X. Then { u n } is bounded in X. By Lemma 3.4, for each ε > 0 , there is a positive constant K ε such that
| x | K ε F ( x , u n ) d x ε and | x | K ε F ( x , u ) d x ε
(3.16)
holds for each n N . It follows from Lemma 3.5 that the Nemytskij operator
u F ( x , u ( x ) )
is continuous from L r ( x ) ( B K ε ( 0 ) ) into L 1 ( B K ε ( 0 ) ) ; see Theorem 1.1 in [26]. This together with Lemma 2.5 yields that
| x | < K ε F ( x , u n ) d x | x | < K ε F ( x , u ) d x .
(3.17)

Using (3.16) and (3.17), we deduce that Ψ ( u n ) Ψ ( u ) as n . The proof is complete. □

We are in a position to state the main result about the existence of the positive eigenvalue for the problem (E).

Theorem 3.7 Assume that (HJ1)-(HJ4), (H1), (H2), and (F1) hold. Then λ is a positive eigenvalue of the problem (E). Moreover, the problem (E) has a nontrivial weak solution for any λ λ .

Proof It is trivial by (3.5) that λ 0 . Suppose to the contrary that λ = 0 . Let { u n } be a sequence in X { 0 } such that
lim n Φ ( u n ) Ψ ( u n ) = 0 .
As in (3.9), we have
| Φ ( u n ) Ψ ( u n ) | C u n X p + q
for some positive constant C. Since p + < q , we obtain that u n X as n . Hence it follows from Lemma 3.2 that
lim n Φ ( u n ) Ψ ( u n ) = ,

which contradicts with the hypothesis. Hence we get λ > 0 . The analogous argument as that in the proof of Theorem 4.5 in [5] proves that λ is an eigenvalue of the problem (E); see also Theorem 3.1 in [6].

Finally, we show that the problem (E) has a nontrivial weak solution for any λ λ . Notice that u is a weak solution of (E) if and only if u is a critical point of I λ . Assume that λ > λ is fixed. Let u X with u X > 1 . With the help of (HJ3) and (HJ4), it follows from proceeding as in the proof of relation (3.13) in Lemma 3.2 that
I λ ( u ) c p + u X p λ 4 C q m L γ ( x ) ( R N ) α ( u X 2 δ + + u X 2 ( q + δ ) ) .
Since p > 2 ( q + δ ) > 2 δ ( + / ) , the inequality above implies that I λ ( u ) as u X for λ > λ , that is, I λ is coercive. Also since the functional I λ is weakly lower semi-continuous by Lemmas 3.3 and 3.6, we deduce that there exists a global minimizer u 0 of I λ in X. Since λ > λ , we verify by definition (3.5) that there is an element ω in X { 0 } such that Φ ( ω ) / Ψ ( ω ) < λ . Then I λ ( ω ) < 0 . So we obtain that
I λ ( u 0 ) = inf v X { 0 } I λ ( v ) < 0 .

Consequently, we conclude that u 0 0 . This completes the proof. □

Now, we consider an example to demonstrate our main result in this section.

Example 3.8 Let p C ( R N ) with 2 p ( x ) < N satisfy the log-Hölder continuity condition (2.2). Suppose that a L 2 p ( x ) ( R N ) , and there is a positive constant a 0 such that a ( x ) a 0 for almost all x R N . Let us consider
https://static-content.springer.com/image/art%3A10.1186%2F1687-2770-2013-214/MediaObjects/13661_2013_Article_460_Equal_HTML.gif
In this case, put
ϕ ( x , | v | ) = ( a ( x ) + | v | 2 ) p ( x ) 2 2 and Φ 0 ( x , | v | ) = 1 p ( x ) ( a ( x ) + | v | 2 ) p ( x ) 2
for all v R N . Denote the quantities
λ = inf u X { 0 } R N 1 p ( x ) ( a ( x ) + | u | 2 ) p ( x ) 2 d x R N m ( x ) p ( x ) | u | q ( x ) d x and λ = inf u X { 0 } R N ( a ( x ) + | u | 2 ) p ( x ) 2 d x R N m ( x ) | u | q ( x ) d x .
If conditions (H1)-(H2) hold, then we have
  1. (i)

    0 < λ λ ,

     
  2. (ii)

    λ is a positive eigenvalue of the problem (E0),

     
  3. (iii)

    the problem (E0) has a nontrivial weak solution for any λ λ ,

     
  4. (iv)

    λ is not an eigenvalue of (E0) for λ < λ .

     
Proof It is clear that conditions (HJ1)-(HJ4) and (F1) hold. From the definitions of λ and λ , we know that
q p + λ λ q + p λ ,
and thus λ λ . Also, from the same argument as that in Theorem 3.7, we have λ > 0 , and thus λ > 0 . Applying Theorem 3.7, the conclusions (ii) and (iii) hold. Let λ < λ . Suppose that λ is an eigenvalue of the problem (E0). Then there is an element v X { 0 } such that
R N ( a ( x ) + | v | 2 ) p ( x ) 2 d x λ R N m ( x ) | v | q ( x ) d x = 0 .
By the definition of λ , we get that
λ R N m ( x ) | v | q ( x ) d x R N ( a ( x ) + | v | 2 ) p ( x ) 2 d x = λ R N m ( x ) | v | q ( x ) d x < λ R N m ( x ) | v | q ( x ) d x ,

a contradiction. □

Declarations

Acknowledgements

The first author was supported by the Incheon National University Research Grant in 2012, and the second author was supported by the National Research Foundation of Korea Grant funded by the Korean Government (NRF-2012R1A1A2042187) and a 2013 Research Grant from Sangmyung University.

Authors’ Affiliations

(1)
Department of Mathematics, Incheon National University
(2)
Department of Mathematics Education, Sangmyung University

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