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Existence and stability of solitary waves for the generalized Korteweg-de Vries equations

Boundary Value Problems20132013:121

https://doi.org/10.1186/1687-2770-2013-121

  • Received: 31 March 2013
  • Accepted: 16 April 2013
  • Published:

Abstract

In this paper, we consider the fractional Korteweg-de Vries equations with general nonlinearities. By studying constrained minimization problems and applying the method of concentration-compactness, we obtain the existence of solitary waves for the generalized Korteweg-de Vries equations under some assumptions of the nonlinear term. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, in the sense that a solution which starts near the set will remain near it for all time.

Keywords

  • generalized Korteweg-de Vries equations
  • constrained minimization problems
  • concentration-compactness
  • stability

1 Introduction

This paper is devoted to studying the existence and stability of solitary wave solutions of the generalized Korteweg-de Vries equation
u t + ( f ( u ) ) x ( L ( u ) ) x = 0 in  R ,
(1.1)
where f ( u ) satisfies the following assumption:
  1. (A)
    f ( u ) C ( R , R ) , lim u 0 f ( u ) | u | = 0 and lim | u | f ( u ) | u | γ = 0 for some 1 < γ < 1 + 4 α ,
    L ( u ) ˆ ( ξ ) = | ξ | 2 α u ˆ ( ξ ) ,
     

0 < α 1 , the Fourier transform F ψ ( ξ ) = ψ ˆ ( ξ ) = 1 ( 2 π ) 1 2 R u ( x ) e i ξ x d x .

When f ( u ) = 1 2 u 2 and α = 1 , equation (1.1) is the well-known Korteweg-de Vries equation, introduced by Korteweg and de Vries in 1895 (cf. [1]). The existence and stability of solitary waves of the Korteweg-de Vries equation is considered by Benjamin in [2]. Recently, in [3], Pelinovsky obtained a Korteweg-de Vries equation with a forcing term, which is a simple analytical model of tsunami generation by submarine landslides.

Here, we consider the generalized Korteweg-de Vries equation (1.1). Let F ( u ) = 0 u f ( s ) d s . Since the functionals
E ( u ) = + [ 1 2 u L ( u ) F ( u ) ] d x
and
Q ( u ) = 1 2 + u 2 d x
are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to equation (1.1) will be founded as minimizers of
where q > 0 . Denote the set of minimizers of the problem ( I q ) by

Inspired by the methods used in [4, 5], by studying the problem ( I q ), we obtain the existence of solitary waves for equation (1.1) with some special nonlinearities f ( u ) = 1 p u p , where 1 < p < 1 + 4 α , and general nonlinearities satisfying the assumption (A). Moreover, we prove that the set G q of minimizers is a stable set for the initial value problem of equation (1.1) in the sense that a solution which starts near the set will remain near it for all time. In order to obtain those results, we have to overcome one main difficulty: the minimization problem ( I q ) is given in the unbounded domain which results in the loss of compactness. As is done in [4, 6], we overcome the difficulty of loss of compactness by the method of concentration-compactness introduced by Lions in [7, 8] for solving some minimization problems in unbounded domains.

Now we give our main results.

Theorem 1.1 Suppose that α = 1 and f ( u ) satisfies condition (A) and I q 0 < 0 for some q 0 > 0 . Then there exists 0 < q q 0 such that G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exist a sequence { y n } R and g G q such that { u n ( + y n ) } contains a subsequence converging strongly in H 1 ( R ) to g, and
lim n + inf g G q u n g = 0 ,

where is the norm of H 1 ( R ) .

Theorem 1.2 Under the assumptions of Theorem  1.1, the set G q is H 1 ( R ) -stable with respect to equation (1.1), i.e., for any ε > 0 , there exists δ > 0 such that if
inf g G q u 0 g < δ ,
then the solution u ( x , t ) to equation (1.1) with initial data u 0 satisfies
inf g G q u ( t , ) g < ε

for any t [ 0 , T ) .

Theorem 1.3 Suppose that 0 < α < 1 and f ( u ) satisfies condition (A) and I q 0 < 0 for some q 0 > 0 . Then there exists 0 < q q 0 such that G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exists a sequence { y n } R and g G q such that { u n ( + y n ) } contains a subsequence converging strongly in H α ( R ) to g, and
lim n + inf g G q u n g α , 2 = 0 ,

where α , 2 is the norm of H α ( R ) given in Section  5.

Theorem 1.4 Under the assumptions of Theorem  1.1, the set G q is H α ( R ) -stable with respect to equation (1.1), i.e., for any ε > 0 , there exists δ > 0 such that if
inf g G q u 0 g α , 2 < δ ,
then the solution u ( x , t ) to equation (1.1) with initial data u 0 satisfies
inf g G q u ( t , ) g α , 2 < ε

for any t [ 0 , T ) .

The paper is organized as follows. In Section 2, we give some preliminaries. In Section 3, we study the existence and stability of solitary waves of equation (1.1) with some special nonlinearities f ( u ) = 1 p u p . Section 4 is devoted to studying equation (1.1) with general nonlinearities f ( u ) satisfying the assumption (A). We shall consider the existence and stability of solitary waves of equation (1.1) with 0 < α < 1 in Section 5.

2 Some preliminaries

At first, we give some notations. The set of all integers and the set of all real numbers are written as and , respectively. And all the integrals will be taken over unless specified. L p ( R ) denotes the usual Lebesgue space with the norm | | p given by
| | p = ( | u | p d x ) 1 p for  1 p < + .
The Sobolev space H 1 ( R ) is defined by
H 1 ( R ) : = { u : u L 2 ( R )  and  u x L 2 ( R ) } ,
whose norm is given by
= ( ( | u x | 2 + | u | 2 ) d x ) 1 2 .

Now, we give Lemma 2.1 and Lemma 2.2 which will be used to study the behavior of the minimizing sequence for the problem ( I q ). Lemma 2.2 is due to Lions [7, 8].

Lemma 2.1 Suppose that B > 0 and δ > 0 are given. Then there exists η = η ( B , δ ) such that if u H 1 ( R ) with u B and | u | p + 1 δ , then
sup y R y 1 2 y + 1 2 | u | p + 1 d x η .
Proof We have
j Z j 1 2 j + 1 2 [ ( u x ) 2 + u 2 ] d x = u 2 B 2 | u | p + 1 p + 1 | u | p + 1 p + 1 = j Z B 2 | u | p + 1 p + 1 j 1 2 j + 1 2 | u | p + 1 d x .
Therefore, there exists some j 0 Z such that
j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] d x B 2 | u | p + 1 p + 1 j 0 1 2 j 0 + 1 2 | u | p + 1 d x .
Applying the Sobolev embedding theorem [9], there exists a constant A such that
( j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] d x ) 1 p + 1 A ( j 0 1 2 j 0 + 1 2 [ ( u x ) 2 + u 2 ] d x ) 1 2 A B | u | p + 1 p + 1 2 ( j 0 1 2 j 0 + 1 2 | u | p + 1 d x ) 1 2 .
Thus, we obtain
j 0 1 2 j 0 + 1 2 | u | p + 1 d x ( | u | p + 1 p + 1 2 A B ) 2 ( p + 1 ) p 1 δ ( p + 1 ) 2 p 1 ( A B ) 2 ( p + 1 ) p 1 .
Taking η = δ ( p + 1 ) 2 p 1 ( A B ) 2 ( p + 1 ) p 1 , it follows that
sup y R y 1 2 y + 1 2 | u | p + 1 d x j 0 1 2 j 0 + 1 2 | u | p + 1 d x η .

 □

Lemma 2.2 Let { u n } be a bounded sequence in H 1 ( R ) such that
sup y R y 1 y + 1 | u n | 2 d x 0 as n +

for some r > 0 . Then u n 0 in L s ( R ) for 2 < s < .

Proof Let 2 < s < . Without loss of generality, we may assume r = 1 . It follows from the interpolation inequalities that
| u | L s ( B ( y , 1 ) ) A | u | L 2 ( B ( y , 1 ) ) s + 2 2 u H 1 ( B ( y , 1 ) ) s 2 2 ,

where A > 0 is a constant independent of u.

Covering by a family of intervals ( y i 1 , y i + 1 ) such that each point of is contained in at most two such intervals and summing this inequality over this family of intervals, we get
| u | L s s 2 A ( sup y R y 1 y + 1 | u | 2 d x ) s + 2 4 u s 2 2 .

Since { u n } is bounded in H 1 ( R ) and sup y R y 1 y + 1 | u n | 2 d x 0 as n + , applying the above inequality, we know that u n 0 in L s ( R ) for 2 < s < . □

Next, we establish a convergence result that will be used in the proof of Theorem 1.3.

Lemma 2.3 Let f C ( R , R ) and suppose that
| f ( t ) | C ( | t | + | t | p 1 ) for all t R ,
(2.1)
where 1 < p 1 < . If u n u 0 in H 1 ( R ) and u n u 0 a.e. on , then
lim n [ + F ( u n ) d x + F ( u 0 ) d x + F ( u n u 0 ) d x ] = 0 ,

where F ( u ) = 0 u f ( s ) d s .

Proof Let R > 0 . Applying the mean value theorem, we have
+ F ( u n ) d x = | x | < R F ( u n ) d x + | x | R F ( u 0 + ( u n u 0 ) ) d x = | x | < R F ( u n ) d x + | x | R [ F ( u n u 0 ) d x + f ( u n u 0 + θ u 0 ) u 0 ] d x ,
where 0 < θ < 1 is dependent on x and R. Now we write
| + F ( u n ) d x + F ( u 0 ) d x + F ( u n u 0 ) d x | | | x | < R [ F ( u n ) F ( u 0 ) ] d x | + | | x | R F ( u 0 ) d x | + | | x | < R F ( u n u 0 ) d x | + | | x | R f ( u n u 0 + θ u 0 ) u 0 d x | .
(2.2)
It follows from (2.1), the mean value theorem and the Hölder inequality that
| | x | < R [ F ( u n ) F ( u 0 ) ] d x | = | | x | < R f ( u 0 + θ ( u n u 0 ) ) ( u n u 0 ) d x | C | x | < R | u 0 + θ ( u n u 0 ) | | u n u 0 | d x + C | x | < R | u 0 + θ ( u n u 0 ) | p 1 | u n u 0 | d x C | x | < R | u 0 | | u n u 0 | d x + C | x | < R | u n u 0 | 2 d x + ε | x | < R | u 0 | p 1 | u n u 0 | d x + C ε | x | < R | u n u 0 | p 1 + 1 d x C ( | x | < R | u 0 | 2 d x ) 1 2 ( | x | < R | u n u 0 | 2 d x ) 1 2 + C | x | < R | u n u 0 | 2 d x + ε ( | x | < R | u 0 | 2 p 1 d x ) 1 2 ( | x | < R | u n u 0 | 2 d x ) 1 2 + C ε | x | < R | u n u 0 | p 1 + 1 d x
(2.3)
and
| | x | < R F ( u n u 0 ) d x | C | x | < R | u n u 0 | 2 d x + C | x | < R | u n u 0 | p 1 + 1 d x .
(2.4)
Since the embedding H 1 ( R ) L loc s ( R ) ( 2 s < ) is compact, the inequalities (2.3) and (2.4) imply that
| | x | < R [ F ( u n ) F ( u 0 ) ] d x | 0 as  n ,
(2.5)
| | x | < R F ( u n u 0 ) d x | 0 as  n .
(2.6)
Similarly, by the Hölder inequality, the Sobolev embedding theorem and (2.1), we get
| | x | R f ( u n u 0 + θ u 0 ) u 0 d x | C | x | R | u n u 0 + θ u 0 | | u 0 | d x + C | x | R | u n u 0 + θ u 0 | p 1 | u 0 | d x C ( | x | R | u n | 2 d x ) 1 2 ( | x | R | u 0 | 2 d x ) 1 2 + C | x | R | u 0 | 2 d x + ε ( | x | R | u n | 2 p 1 d x ) 1 2 ( | x | R | u 0 | 2 d x ) 1 2 + C ε | x | R | u 0 | p 1 + 1 d x C u n ( | x | R | u 0 | 2 d x ) 1 2 + C | x | R | u 0 | 2 d x + ε u n p 1 ( | x | R | u 0 | 2 d x ) 1 2 + C ε | x | R | u 0 | p 1 + 1 d x .
Since { u n } is bounded in H 1 ( R ) , we see that
| | x | R f ( u n u 0 + θ u 0 ) u 0 d x | 0 as  R .
(2.7)
Hence, combining (2.2), (2.5), (2.6) and (2.7), we obtain
+ F ( u n ) d x + F ( u 0 ) d x + F ( u n u 0 ) d x 0 as  n .

 □

3 The case of special nonlinearity

In this section, we only consider the case of α = 1 and f ( u ) = 1 p u p , where 1 < p < 5 . Correspondingly,
E ( u ) = + [ 1 2 ( u x ) 2 1 p 1 p + 1 u p + 1 ] d x .

At first, we commence by studying some properties of the functional I q : ( 0 , + ) R and the behavior of the minimizing sequences for the problem ( I q ).

Lemma 3.1 For any q > 0 ,
  1. (i)

    < I q < 0 ;

     
  2. (ii)

    If { u n } is a minimizing sequence for the problem ( I q ), there exists a constant B > 0 such that u n B for all n;

     
  3. (iii)
    If { u n } is a minimizing sequence for the problem ( I q ), there exist a positive constant δ and a sequence { y n } of real numbers such that
    y n 1 2 y n + 1 2 | u n | p + 1 d x δ
     

for sufficiently large n.

Proof (i) Choose any function u H 1 ( R ) such that Q ( u ) = q and u p + 1 d x 0 . For any θ > 0 , define u θ ( x ) = θ u ( θ x ) . Then we have
Q ( u θ ) = q
and
E ( u θ ) = 1 2 θ 2 ( u x ) 2 d x 1 p ( p + 1 ) θ p 1 2 u p + 1 d x .

For 1 < p < 5 , by taking θ > 0 sufficiently small, we get I q E ( u θ ) < 0 .

Next we prove I q > . Let u H 1 ( R ) such that Q ( u ) = q . By the Sobolev embedding theorems and interpolation inequalities, we get
| u p + 1 d x | | u | p + 1 p + 1 A | u | P 1 2 ( P + 1 ) P + 1 A u p 1 2 | u | 2 p + 3 2 ,
(3.1)
where A denotes various constants which are independent of u. Using the Young inequality, we derive from (3.1)
| u p + 1 d x | ε u 2 + A ε | u | 2 2 p + 6 5 p ε u 2 + A ε , q ,
where ε > 0 is arbitrary and A ε , q depends on ε and q, but not on u. Therefore,
E ( u ) = E ( u ) + Q ( u ) Q ( u ) = 1 2 u 2 1 p ( p + 1 ) u p + 1 d x 1 2 u 2 d x 1 2 u 2 ε p ( p + 1 ) u 2 1 p ( p + 1 ) A ε , q q .
Choosing ε < p ( p + 1 ) 2 , we obtain the lower bound of the functional E
E ( u ) 1 p ( p + 1 ) A ε , q q ,
which implies I q 1 p ( p + 1 ) A ε , q q > .
  1. (ii)
    Let { u n } be a minimizing sequence for the problem ( I q ). Then, by (3.1), we have
    1 2 u n 2 = E ( u n ) + Q ( u n ) + 1 p ( p + 1 ) u n p + 1 d x sup n E ( u n ) + q + 1 p ( p + 1 ) | u n | p + 1 p + 1 A + q + A u p 1 2 | u | 2 p + 3 2 A ( 1 + u p 1 2 ) ,
     
where A denotes various constants which are independent of n. Since 1 < p < 5 , the existence of the desired bound B follows.
  1. (iii)
    Let { u n } be a minimizing sequence for the problem ( I q ). Then we claim: there exists a constant η > 0 such that | u n | p + 1 η for all sufficiently large n. We argue by contradiction: if no such η > 0 exists, then lim inf n | u n | p + 1 d x 0 . Hence
    I q = lim n E ( u n ) lim inf n | u n | p + 1 d x 0 ,
     

which contradicts (i). So, the claim is achieved.

Combining (ii) and Lemma 2.1, there exist a positive constant δ and a sequence { y n } of real numbers such that
y n 1 2 y n + 1 2 | u n | p + 1 d x δ

for sufficiently large n. The proof of Lemma 3.1 is completed. □

The next lemma will establish a subadditivity inequality which will be a crucial step in the proof of the existence minimizer for the problem ( I q ).

Lemma 3.2 For all q 1 , q 2 > 0 , I q 1 + q 2 < I q 1 + I q 2 .

Proof For given u H 1 ( R ) , | u | 2 2 = q 1 , θ > 0 , let u θ ( x ) = θ 2 p + 1 u ( θ p 1 p + 1 x ) , where θ = ( q 2 q 1 ) p + 1 5 p . Then it follows that
Q ( u θ ) = q 2 q 1 Q ( u ) = q 2 ,
and
E ( u θ ) = ( q 2 q 1 ) p + 3 5 p E ( u ) .
Hence we get
I q 2 = inf { ( q 2 q 1 ) p + 3 5 p E ( u ) : Q ( u ) = q 1 } = ( q 2 q 1 ) p + 3 5 p I q 1 .
(3.2)
Now, from (3.2) and Lemma 3.1, we obtain for all q 1 , q 2 > 0 ,
I q 1 + q 2 = ( q 2 + q 1 ) p + 3 5 p I 1 < ( q 2 p + 3 5 p + q 1 p + 3 5 p ) I 1 = I q 1 + I q 2 .

 □

Now we formulate the following two theorems, which are special cases corresponding to Theorem 1.1 and Theorem 1.2, and give their proof with the aim of Lemma 3.1 and Lemma 3.2.

Theorem 3.1 Let α = 1 and f ( u ) = 1 p u p , where 1 < p < 5 . For any q > 0 , the set G q is not empty. Moreover, if { u n } is a minimizing sequence for the problem ( I q ), then there exist a sequence { y n } R and g G q such that { u n ( + y n ) } contains a subsequence converging strongly in H 1 ( R ) to g, and
lim n + inf g G q u n g = 0 .
Theorem 3.2 Let α = 1 and f ( u ) = 1 p u p , where 1 < p < 5 . For any q > 0 , the set G q is H 1 ( R ) -stable with respect to equation (1.1), i.e., for any ε > 0 , there exists δ > 0 such that if
inf g G q u 0 g < δ ,
then the solution u ( x , t ) to equation (1.1) with initial data u 0 satisfies
inf g G q u ( t , ) g < ε

for any t [ 0 , T ) .

Proof of Theorem 3.1 From (3.2), it is easy to check that I q is continuous on ( 0 , ) . Let { u n } be a minimizing sequence for the problem ( I q ). By Lemma 3.1, there exist a positive constant δ and a sequence { y n } of real numbers such that
y n 1 2 y n + 1 2 | u n | p + 1 d x δ

for sufficiently large n.

Let us define v n = u n ( x + y n ) . Hence Q ( V n ) = Q ( u n ) = q , E ( v n ) = E ( u n ) I q , as n , and
1 2 1 2 | v n | p + 1 d x = y n 1 2 y n + 1 2 | u n | p + 1 d x δ > 0 .
(3.3)
Since { v n } is bounded in H 1 ( R ) , by Lemma 3.1, we may assume going, if necessary, to a subsequence
v n g in  H 1 ( R ) , v n g in  L loc p + 1 ( R ) , v n g a.e. on  R .
(3.4)
Hence, by (3.3), we get g 0 . And applying the Brezis-Lieb lemma [10], we have
| v n | 2 2 = | v n g | 2 2 + | g | 2 2 ,
(3.5)
| v n | p + 1 p + 1 = | v n g | p + 1 p + 1 + | g | p + 1 p + 1 .
(3.6)
Now we show that Q ( g ) = 1 2 g 2 d x = q . In the contrary case, 0 < Q ( g ) = λ < q . By (3.5), we obtain lim n q n = lim n Q ( v n g ) = q λ . Then it follows from (3.5) and (3.6) that
I q = E ( v n ) + o ( 1 ) 1 2 v n 2 1 2 | v n | 2 2 1 p ( p + 1 ) | v n | p + 1 p + 1 + o ( 1 ) = E ( v n g ) + E ( g ) + o ( 1 ) I q n + I λ + o ( 1 ) .
(3.7)
Since I q is continuous on ( 0 , ) , letting n , we get I q I q λ + I λ , which contradicts Lemma 3.2. Therefore Q ( g ) = q . It then follows from (3.5) that
v n g in  L 2 ( R ) .
(3.8)
Applying the interpolation inequality, (3.4) and (3.8), we get
v n g in  L p + 1 ( R ) .
(3.9)
Using the weak low semi-continuity of the norm in H 1 ( R ) , we know that
I q 1 2 v n 2 1 2 | v n | 2 2 1 p ( p + 1 ) | v n | p + 1 p + 1 + o ( 1 ) 1 2 g 2 1 2 | v n g | 2 2 1 2 | g | 2 2 1 p ( p + 1 ) | v n g | p + 1 p + 1 1 p ( p + 1 ) | g | p + 1 p + 1 + o ( 1 ) = E ( g ) 1 2 | v n g | 2 2 1 p ( p + 1 ) | v n g | p + 1 p + 1 + o ( 1 ) .
Letting n , by (3.8) and (3.9), we obtain E ( g ) I q . On the other hand, it follows from Q ( g ) = q that E ( g ) I q . Therefore E ( g ) = I q , which implies that g is a minimizer of the problem ( I q ) (i.e., G q ). Then it follows from (3.7), (3.8) and (3.9) that
v n = u n ( + y n ) g in  H 1 ( R ) .
We prove lim n + inf g G q u n g = 0 with an argument by contradiction. Assume that there exist ε 0 > 0 and a subsequence { u n k } of { u n } such that
inf g G q u n k g ε 0 > 0
(3.10)
for all n k . With the result of the above proof, we obtain that there exist a subsequence of { u n k } , denoted again by { u n k } , { y n k } R and g G q such that
u n k ( + y n k ) g in  H 1 ( R ) .
Since g ( y n k ) G q ,
u n k g ( y n k ) = u n k ( + y n k ) g 0 as  n k + ,

which contradicts (3.10). □

An immediate consequence of Theorem 3.1 is that G q forms a stable set for the initial-value problem for equation (1.1).

Proof of Theorem 3.2 We prove Theorem 3.2 with an argument by contradiction. Assume that the set G q is not H 1 ( R ) -stable. Then there exist ε 0 > 0 , { ψ n } H 1 ( R ) and a sequence of times { t n } such that
inf g G q ψ n g < 1 n ,
(3.11)
and
inf g G q u n ( , t n ) g ε 0
(3.12)

for all n, where u n ( x , t ) solves equation (1.1) with u n ( x , 0 ) = ψ n .

Equation (3.11) implies that
E ( ψ n ) I q , Q ( ψ n ) q .
Choose { μ n } R such that Q ( μ n ψ n ) = q for all n. Thus μ n 1 as n + . Hence the sequence v n = μ n u n ( , t n ) satisfies Q ( v n ) = q and
lim n E ( v n ) = lim n E ( u n ( , t n ) ) = lim n E ( ψ n ) = I q .
Therefore { v n } is a minimizing sequence for the problem ( I q ). By Theorem 1.1, there exists { g n k } G q such that
v n k g n k < ε 0 2
(3.13)
for sufficiently large n k . Since μ n 1 and u n ( , t n ) is bounded, we derive from (3.12) and (3.13)
ε 0 u n k ( , t n k ) g n k u n k ( , t n k ) μ n k u n k ( , t n k ) + μ n k u n k ( , t n k ) g n k ( | μ n k 1 | ) u n k ( , t n k ) + ε 0 2 3 4 ε 0
(3.14)

for sufficiently large n k . (3.14) is a contradiction. Therefore, the set G q is H 1 ( R ) -stable with respect to equation (1.1). □

4 The case for more general nonlinearities

In this section, we consider (1.1) with α = 1 and more general nonlinearities f satisfying condition (A). At first, we study the properties of the functional I q : ( 0 , ) R and the minimizing sequence of the problem ( I q ).

Lemma 4.1
  1. (i)

    For any q > 0 , I q is finite and continuous on ( 0 , ) . Moreover, each minimizing sequence for ( I q ) is bounded;

     
  2. (ii)

    I q 0 for any q > 0 .

     
Proof (i) According to assumption (A), we observe that for each ε > 0 there exists C ε > 0 such that
| F ( u ) | ε | u | 2 + ε | u | γ + 1 + C ε | u | α ,
(4.1)
where 2 < α < γ + 1 . By the Sobolev embedding theorems and interpolation inequalities, we obtain
+ | u | α d x A u α 2 2 | u | 2 α + 2 2
(4.2)
and
+ | u | γ + 1 d x A u γ 1 2 | u | 2 γ + 3 2 ,
(4.3)
where A > 0 is independent of u. Then using the Young inequality, we can derive from (4.2) and (4.3) that for all η > 0 , there exists C η > 0 such that
+ | u | α d x η u 2 + C η | u | 2 2 ( α + 2 ) 6 α
(4.4)
and
+ | u | γ + 1 d x η u 2 + C η | u | 2 2 γ + 6 5 γ .
(4.5)
Let u H 1 ( R ) such that Q ( u ) = q . It follows from (4.1), (4.4) and (4.5) that
E ( u ) = 1 2 u 2 1 2 + u 2 d x + F ( u ) d x 1 2 u 2 1 2 | u | 2 2 ε | u | 2 2 ε η u 2 ε C η | u | 2 2 γ + 6 5 γ C ε η u 2 C ε C η | u | 2 2 ( α + 2 ) 6 α ( 1 2 ε η C ε η ) u 2 C ε , η , q ,
(4.6)

where C ε , η , q is a positive constant dependent only on ε and η for given q > 0 . Choosing ε > 0 and η > 0 such that 1 2 ε η C ε η > 0 , we see that I q > .

Since | θ u | 2 2 = θ 2 | u | 2 2 for θ > 0 , it is easy to check that I q is continuous on ( 0 , ) .

Let { u n } be a minimizing sequence for the problem ( I q ). From (4.6) and the fact that I q is finite, we know that { u n } is bounded in H 1 ( R ) .
  1. (ii)
    For given u H 1 ( R ) such that Q ( u ) = q , let u θ ( x ) = 1 θ u ( x θ ) for θ > 0 . We obtain that
    | u θ | 2 2 = | u | 2 2 = q
    (4.7)
     
and
E ( u θ ) = 1 2 θ 2 + ( u x ) 2 d x θ + F ( 1 θ u ) d x .
(4.8)
Combining (4.1), (4.7) and (4.8), we obtain
I q E ( u θ ) 0 as  θ + .

 □

Lemma 4.2 Suppose that I q 0 < 0 for some q 0 > 0 . Then the following two properties hold:
  1. (i)

    I q q is non-increasing on ( 0 , + ) and lim q 0 + I q q = 0 ;

     
  2. (ii)
    there exists q 1 q 0 such that
    I q q > I q 0 q 0 for q ( 0 , q 1 ) .
     
Proof First we observe that if σ > 0 and β > 0 with Q ( u ) = β and u σ ( x ) = u ( 1 σ x ) , then Q ( u σ ) = σ β and
E ( u σ ) = 1 2 σ + ( u x ) 2 d x σ + F ( u ) d x .
Consequently, for q 1 > 0 and q 2 > 0 , we have
I q 2 = inf { q 1 2 q 2 + ( u x ) 2 d x q 2 q 1 + F ( u ) d x , Q ( u ) = q 1 } .
If q 1 > q 2 > 0 , then for each ε > 0 , there exists u H 1 ( R ) with Q ( u ) = q 1 such that
I q 2 + ε > 1 2 q 1 q 2 + ( u x ) 2 d x q 2 q 1 + F ( u ) d x > q 2 q 1 E ( u ) q 2 q 1 I q 1 .
(4.9)

This inequality yields I q 2 q 2 I q 1 q 1 for 0 < q 2 < q 1 .

Since I q 0 for all q > 0 , we see that
lim q 0 + I q q = A 0 .
We claim that A = 0 . Letting ε = q 2 , 0 < q q 0 , from (4.9), there exists u ( q ) H 1 ( R ) with Q ( u ( q ) ) = q 0 such that
I q + q 2 1 2 q 0 q + ( u x ( q ) ) 2 d x q q 0 + F ( u ( q ) ) d x q q 0 [ 1 2 + ( u x ( q ) ) 2 d x + F ( u ( q ) ) d x ] .
(4.10)
It follows from (4.6) and (4.10) that
I q + q 2 q q 0 ( C 1 ( q 0 ) + ( u x ( q ) ) 2 d x C 2 ( q 0 ) ) ,
where C 1 ( q 0 ) > 0 and C 2 ( q 0 ) > 0 are constants independent of q 0 . Hence we obtain q 0 2 C 1 ( q 0 ) + ( u ( q ) ) x 2 d x C 2 ( q 0 ) , which implies
+ ( u x ( q ) ) 2 d x C 3 ( q 0 ) ,
(4.11)
where C 3 ( q 0 ) is dependent only on q 0 . Combining (4.1), (4.4), (4.5) and (4.11), we also get
| + F ( u ( q ) ) d x | C 4 ( q 0 ) ,
(4.12)

where C 4 ( q 0 ) is dependent only on q 0 .

We claim that
+ ( u x ( q ) ) 2 d x 0 as  q 0 + .
(4.13)
Indeed, if there exist ε 0 > 0 and q n 0 + such that + ( u x ( q n ) ) 2 d x ε 0 , then by (4.11) and (4.12), we obtain
I q n q n + q n q 0 ε 0 2 1 q n 2 1 q 0 C 4 ( q 0 ) + as  q n 0 + ,
which contradicts lim q 0 + I q q = A 0 . Therefore (4.13) is achieved and this implies that lim q 0 + + F ( u ( q ) ) d x = 0 and, consequently,
I q q + q 1 q 0 + F ( u ( q ) ) d x 0 as  q 0 + .
This shows that lim q 0 + I q q = 0 .
  1. (2)

    We observe that lim q 0 + I q q = 0 > I q 0 q 0 , which implies (ii). □

     

Then we establish a subadditivity inequality similar to Lemma 3.2 with the aim of Lemma 4.2.

Lemma 4.3 Suppose that I q 0 < 0 for some q 0 > 0 . Then there exists 0 < q q 0 such that I q < I q q + I q for 0 < q < q .

Proof According to Lemma 4.2, the set
{ q 1 | q 1 q 0  and  I q q > I q 0 q 0  or each  q ( 0 , q 1 ) }
is nonempty. We define
q = sup { q 1 | q 1 q 0  and  I q q > I q 0 q 0  for each  q ( 0 , q 1 ) } .
(4.14)
It follows from the continuity of I q and lim q 0 + I q q = 0 that 0 < q q 0 ,
I q = q q 0 I q 0 < 0 ,
(4.15)
I q > q q 0 I q 0 for all  q ( 0 , q ) .
(4.16)
Therefore,
I q = q q 0 I q 0 = q q q 0 I q 0 + q q 0 I q 0 < I q q + I q ,

for all q ( 0 , q ) . □

Now we give the proof of Theorem 1.1 with the aim of Lemma 4.1, Lemma 4.2 and Lemma 4.3. Since the proof of Theorem 1.1 is similar to that of Theorem 3.1, we only give the sketch of the proof.

Proof of Theorem 1.1 Let { u n } be a minimizing sequence of I q , where q is defined in (4.14). Since { u n } is bounded, we may assume
u n u in  H 1 ( R ) , u n u a.e. on  R .
First, we consider the case u = 0 . In this case, by Lemma 2.2, either
  1. (a)

    u n 0 in L s ( R ) for 2 < s < , or

     
  2. (b)
    there exists a sequence { y n } R such that
    υ n ( x ) = u n ( x + y n ) g 0 in  H 1 ( R ) .
     
In the case (a), combining Lemma 2.2 and condition (A), we obtain
lim n + F ( u n ) d x = 0
and, consequently,
I q = lim n E ( u n ) = lim n [ 1 2 + ( u n x ) 2 d x + F ( u n ) d x ] 0 ,

which contradicts Lemma 4.1. Hence (b) holds. Then it follows from Lemma 2.3 and Lemma 4.3 that g is the minimizer for the problem ( I q ) (i.e., G q ) and the result of Theorem 1.1 holds. The proof is similar to that of Theorem 3.1, we omit the details.

If u 0 , we repeat the previous argument in the case (b) to obtain the result of Theorem 1.1. □

Proof of Theorem 1.2 Theorem 1.2 is an immediate result of Theorem 1.1. We can prove it with an argument similar to that of Theorem 3.2. Here, we omit the details of the proof. □

5 The case for 0 < α < 1

In this section, we only consider the case of 0 < α < 1 , i.e., we consider the existence and stability of solitary waves for the fractional Korteweg-de Vries equations with general nonlinearities. At first, we give the definition of H α ( R ) . The fractional order Sobolev space H α ( R ) is defined by
H α ( R ) : = { u ; u : R C , u L 2 ( R )  and  F 1 [ ( 1 + | ξ | 2 ) α 2 F u ] L 2 ( R ) } ,
whose norm is given by
α , 2 = | F 1 [ ( 1 + | ξ | 2 ) α 2 F ] | 2 .
Since the functionals
E ( u ) = + [ 1 2 | ( ) α 2 u | 2 F ( u ) ] d x
and
Q ( u ) = 1 2 + u 2 d x
are two conserved quantities with (1.1), for studying the existence of solitary wave solutions to (1.1), by the variational methods, the solitary wave solutions to the equation (1.1) will be founded as minimizers of
where q > 0 . Denote the set of minimizers of the problem ( I q ) by

Similar to Lemma 4.1 and Lemma 4.3, we obtain the following two lemmas.

Lemma 5.1
  1. (i)

    I q and I q are finite and continuous on ( 0 , + ) ; moreover, for any q > 0 , each minimizing sequence for the problem ( I q ) or ( I q ) is bounded in H α ( R ) ;

     
  2. (ii)

    I q 0 for any q > 0 .

     
Lemma 5.2 If I q 0 < 0 for some q 0 > 0 , then there exists q 1 , 0 < q 1 q 0 , such that
I q 1 < I q 1 q + I q for 0 < q < q 1 .

Applying the above two lemmas and commutator estimates [[5], Lemma 2.5], we prove Theorem 1.3 and Theorem 1.4 by similar steps to those given in Section 4. Here we omit the details of Theorem 1.3 and Theorem 1.4.

Declarations

Acknowledgements

The work was supported in part by Special Fund of Fundamental Scientific Research Business Expense for Higher School of Central Government (projects for young teachers, No. ZY20110226) and the National Natural Science Foundation of China (No. 41276020).

Authors’ Affiliations

(1)
Institute of Disaster Prevention, Sanhe Hebei, 065201, China

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