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Multiple solutions for a p-Laplacian elliptic problem

Abstract

We consider the following p-Laplacian elliptic equation on W 1 , p ( R N ): Δ p u+b(|x|) | u | p 2 u=f(|x|,u). For certain f(|x|,u), we are interested in the functional on a group invariant subspace, and we obtain the existence of infinitely many radial solutions and non-radial solutions of the equation, which extends the result of (Bartsch and Willem in J. Funct. Anal. 117:447-460, 1993) to the space W 1 , p ( R N ).

1 Introduction

The interesting equation

Δ p u+b ( | x | ) | u | p 2 u=f ( | x | , u ) ,u W 1 , p ( R N ) ,
(1.1)

originates from different problems in physics and mathematical physics. For p=2, problem (1.1) is interpreted as a stationary state of the reaction-diffusion Klein-Gordon equation in chemical dynamics, and Schrödinger equations in finding certain solitary waves.

In the 1980s, people searched for the spherically symmetric solutions of the autonomous equation Δu=g(u) (where g:RR is continuous and odd in u). Berestycki-Lions [1, 2] advocated it for the first time; they obtained the existence of infinitely many radial solutions of the autonomous equation. Then Struwe [3] got similar results. Gidas et al. [4] further demonstrated that any positive solution of the equation with some properties must be radial.

Then Bartsch-Willem [5] found an unbounded sequence of non-radial solutions of (1.1) in H 1 ( R N ) with p=2, under the assumption that b and f satisfy certain growth conditions and f is odd in u.

In recent years, the existence and structure of solutions for the p-Laplacian equation has found considerable interest, and different approaches have been developed. Bartsch-Liu [6] studied the p-Laplacian problem

Δ p u=f(x,u),u W 0 1 , p (Ω),
(1.2)

on a bounded domain Ω R N with smooth boundary Ω, provided that the nonlinearity f is superlinear and subcritical. They proved (1.2) has a pair of a subsolution and a supersolution. In [7] they studied problem (1.2) on a bounded domain Ω R N , with p>1 arbitrary, and proved a nodal solution provided that f:Ω×RR is subcritical and superlinear. Infinitely many nodal solutions are obtained if, in addition, f(x,t)=f(x,t).

Furthermore Jiu-Su [8] applied Morse theory to study the existence of nontrivial solutions of p-Laplacian type Dirichlet boundary value problems. Agarwal-Perera [9] obtained two positive solutions of singular discrete p-Laplacian problems using variational methods. Chabrowski-Fu [10] studied the p-Laplacian problem:

a(x) Δ p u+b(x) | u | p ( x ) 2 u=f(x,u),

on a bounded domain of R N with Dirichlet boundary condition, where 1< p 1 p(x) p 2 <N ( p 1 , p 2 are positive constants). They applied the mountain pass theorem to prove the existence of solutions in W 0 1 , p ( x ) (Ω) for the equation in the superlinear and sublinear cases.

For (1.1), Drábek-Pohozaev [11] proved the existence of multiple positive solutions of quasilinear problems (1.1) of second order by using the fibering method. They considered solutions both in the bounded domain Ω R N and in the whole space R N . Moreover, De Nápoli-Mariani [12] introduced a notion of uniformly convex functional that generalizes the notion of uniformly convex norm. They proved the existence of at least one solution of (1.1), and the existence of infinitely many solutions under further assumptions.

In the present paper, we aim to find the existence of infinitely many radial and non-radial solutions of problem (1.1), and extend the result of [5] to the space W 1 , p ( R N ).

A direct extension to the case p2 is faced with serious difficulties. First the energy functional associated to (1.1) is defined on W 1 , p ( R N ), which is not a Hilbert space for p2. Another difficulty is the lack of a powerful regularity theory. For the Laplace operator there exists a sequence of Banach spaces E 0 E 1 E n with W 1 , 2 E n and E 0 C 1 . But the imbedding W 1 , p ( R N ) L q ( R N ) (p<q< p :=Np/(Np), for 1<p<N) is not compact.

We study the functional on a group invariant subspace {u W 1 , p ( R N )gu(x)=u( g 1 x)=u(x),gO(N)} (where O(N) is the group of orthogonal linear transformations in R N ), then we apply the principle of symmetric criticality [[13], Theorem 5.4] and the fountain theorem to obtain the existence of multiple solutions.

2 The main results and preliminaries

This paper is devoted to the study of infinitely many radial and non-radial solutions for a p-Laplacian equation:

Δ p u+b ( | x | ) | u | p 2 u=f ( | x | , u ) ,u W 1 , p ( R N ) ,

where Δ p u=div( | u | p 2 u) (1<p<N), b(r)C([0,),R) is bounded from below by a positive constant a 0 . The growth condition of f(|x|,u) will be given in the following.

The corresponding functional is

J(u):= R N ( 1 p | u | p + 1 p b ( | x | ) | u | p F ( | x | , u ) ) dx,

where F(|x|,u)= 0 u f(|x|,t)dt, u W 1 , p ( R N ).

We require the following assumptions on the nonlinearity f:

(f1) fC([0,+)×R,R), and f(r,t)=o(|t|) for t0, uniformly on [0,+).

(f2) There exists μ>p such that for all r0, tR, 0<μF(r,t)tf(r,t).

(f3) For 1<p<N, and p<q< p , there exists a constant C>0, such that for any r0, tR, |f(r,t)|C | t | q 1 .

The main result of this paper is as follows.

Theorem 2.1 If N2, the assumptions (f1)-(f3) hold, and f is odd in u, then problem (1.1) possesses infinitely many radial solutions.

Theorem 2.2 Suppose N=4 or N6, if the assumptions (f1)-(f3) hold and f is odd in u, then for problem (1.1) there exist infinitely many non-radial solutions.

Remark 2.3 The assumptions (f1)-(f3) are from [5]. Bartsch-Willem [5] considered the existence of non-radial solutions for the Euclidean scalar field equation Δu+V(|x|)u=f(|x|,u) (u H 1 ( R N )).

(f2) means that the nonlinearity f is superlinear, and (f3) means that f is subcritical. These two conditions enable us to use a variational approach for the study of (1.1).

Condition (f2) corresponds to the standard superlinearity condition of Ambrosetti-Rabinowitz in the case p=2. In the case p=2 without the assumption (f3), the above theorems may not be true. It can be seen from Pohozaev’s identity for p-Laplacian equations that (1.1) has only a trivial solution u=0.

Remark 2.4 If N=5 we fail to define the action of G 3 in the proof of Theorem 2.2.

We shall use the norm

u p := ( R N ( | u | p + b ( | x | ) | u | p ) d x ) 1 p .

We denote E:= W 1 , p ( R N ) is the completion of D( R N ) with the norm p , where D( R N ):={u C ( R N )supp(u) is a compact subset in  R N }. Denote by | | p the usual norm in L p ( R N ).

Let G 1 =O(N) be the group of orthogonal linear transformations in R N , G 2 = Z 2 , and E G 1 :={u W 1 , p ( R N )gu(x)=u( g 1 x)=u(x),gO(N)}.

Throughout this paper, we will use C and C i to represent various positive constants.

Now, we recall some definitions for the action of a topological group and the fountain theorem.

Definition 2.5 ([[14], Definition 1.27])

The action of a topological group G on a normed space Z is a continuous map G×ZZ:[g,z]gz, such that for all g,hG, zZ,

1z=z,(gh)z=g(hz),zgzis linear.

The action is isometric if gz=z.

The set of invariant points is defined by FixG:={zZ;gz=z,gG}. A set AZ is G-invariant if gA=A for every gG. A function φ:ZR is G-invariant if φg=φ for every gG.

A map f:ZZ is G-equivariant if fg=gf, for every gG.

Definition 2.6 ([[15], p.99])

Suppose Z is a G-Banach space, that is, there is a G isometric action on Z. Let A={AZA is closed and gA=A,for any gG} be the set of G-invariant subsets of Z, and Γ={h C 0 (Z,Z);hg=gh,for all gG} be the class of G-equivariant mappings of Z.

Definition 2.7 ([14])

Let Z be a Banach space, I C 1 (Z,R) and cR. The functional I satisfies the ( PS ) c condition if any sequence { z n }Z such that

I( z n )c, I ( z n )0,as n,

has a convergent subsequence.

Theorem 2.8 (Fountain theorem [[14], Theorem 3.6])

The compact group G acts isometrically on the Banach space X= j N X j ¯ , the spaces X j are invariant and there exists a finite-dimensional space V such that for every jN, X j V. The action of G on V is admissible.

Let I C 1 (X,R) be an G-invariant functional. If for every kN, there exists ρ k > r k >0 such that

(A1) a k := max u Y k , u = ρ k I(u)0, where Y k := j = 1 k X j .

(A2) b k := inf u Z k , u = r k I(u), as k, where Z k := j = k X j ¯ .

(A3) I satisfies the ( PS ) c condition, for every c>0.

Then I possesses an unbounded sequence of critical values c k . c k can be characterized as

c k = inf γ Γ k sup u B k I ( γ ( u ) ) ,

where Γ k ={γC( B k ,X)γ is equivariant and γ | B k =id}, B k :={u Y k u ρ k }.

In fact, for each k2, if b k > a k , then there exists a critical value c k > b k .

3 Proof of theorems

Definition 3.1 ([[14], Definition A.3])

On the space L p ( R N ) L q ( R N ), we define the norm

| u | p q = | u | p + | u | q .

On the space L p ( R N )+ L q ( R N ), we define the norm

| u | p q =inf { | v | p + | ω | q v L p ( R N ) , ω L q ( R N ) , u = v + ω } .

Lemma 3.2 ([[14], Theorem A.4])

Assume 1p,q,r,s<, fC([0,+)×R,R) and f(|x|,u)C( | u | p r + | u | q s ), then for every u L p ( R N ) L q ( R N ), f(,u) L p ( R N )+ L q ( R N ), the operator

T 1 : L p ( R N ) L q ( R N ) L p ( R N ) + L q ( R N ) :uf ( | x | , u )

is continuous.

Lemma 3.3 Let T 2 : E G 1 E G 1 be the mapping given by

T 2 u,v= R N ( | u | p 2 u v + b ( | x | ) | u | p 2 u v ) dx,

then T 2 is bounded and continuous.

Proof By the definition of T 2 ,

T 2 u E G 1 = sup v p 1 R N ( | u | p 2 u v + b ( | x | ) | u | p 2 u v ) d x sup v p 1 ( | u | p p 1 | v | p + b ( | x | ) | u | p p 1 | v | p ) | u | p p 1 + b ( | x | ) | u | p p 1 .
(3.1)

Therefore T 2 is bounded. If u n u ˜ in E G 1 , by (3.1), then T 2 u n T 2 u ˜ E G 1 0. Hence T 2 is continuous. □

Lemma 3.4 Suppose the nonlinearity f satisfies (f1)-(f3), then the functional J C 1 ( E G 1 ,R), and

J ( u ) , v = R N | u | p 2 uvdx+ R N b ( | x | ) | u | p 2 uvdx ϕ ( u ) , v ,

where ϕ (u),v= R N f(|x|,u)vdx, here ϕ (u) is compact. In addition, each critical point of J is a weak solution of problem (1.1).

Proof By Lemma 3.3, we only need to prove ϕ (u) is continuous. By Hölder inequality

| ϕ ( u n ) , v ϕ ( u ) , v | R N | f ( | x | , u n ) f ( | x | , u ) | | v | d x | f ( | x | , u n ) f ( | x | , u ) | p q | v | p q ,

where 1/p+1/ p =1, 1/q+1/ q =1. If u n u in E G 1 , then u n u in L p ( R N ) L q ( R N ). It follows from (f3) and Lemma 3.2 that

f ( | x | , u n ) f ( | x | , u ) in  L p ( R N ) + L q ( R N ) .

So

ϕ ( u n ) ϕ ( u ) p = sup v p 1 | ϕ ( u n ) ϕ ( u ) , v |0.
(3.2)

Assume u n u in E G 1 . Since E G 1 L q ( R N ) is compact, then u n u in L q ( R N ). By Lemma 3.2 and (3.2), ϕ (u) is compact. □

Lemma 3.5 ([[16], Lemma 2.1])

There exist constants C 1 and C 2 , such that for all ξ,η R N , N1, we have

( | ξ | p 2 ξ | η | p 2 η ) ( ξ η ) C 1 ( | ξ | + | η | ) p 2 | ξ η | 2 , if  1 < p < 2 , ( | ξ | p 2 ξ | η | p 2 η ) ( ξ η ) C 2 | ξ η | p , if  p 2 .

Lemma 3.6 Let T 2 be defined in Lemma  3.3. If u n u ˜ in E G 1 and T 2 u n T 2 u ˜ , u n u ˜ 0, then u n u ˜ in E G 1 .

Proof If u n u ˜ in E G 1 , then { u n } is bounded in E G 1 .

If p2, by Lemma 3.5,

T 2 u n T 2 u ˜ , u n u ˜ = R N ( | u n | p 2 u n | u ˜ | p 2 u ˜ ) ( u n u ˜ ) d x + R N ( b ( | x | ) | u n | p 2 u n b ( | x | ) | u ˜ | p 2 u ˜ ) ( u n u ˜ ) d x C 3 ( u n p p u ˜ p p )

so u n u ˜ in E G 1 .

If 1<p<2, by Lemma 3.5 and the Hölder inequality,

u n u ˜ p p R N | u n u ˜ | p ( | u n | + | u ˜ | ) p ( p 2 ) 2 ( | u n | + | u ˜ | ) p ( 2 p ) 2 d x + R N b ( | x | ) | u n u ˜ | p ( | u n | + | u ˜ | ) p ( p 2 ) 2 ( | u n | + | u ˜ | ) p ( 2 p ) 2 d x ( R N | u n u ˜ | 2 ( | u n | + | u ˜ | ) p 2 d x ) p 2 ( R N ( | u n | + | u ˜ | ) p d x ) 2 p 2 + ( R N b ( | x | ) | u n u ˜ | 2 ( | u n | + | u ˜ | ) p 2 d x ) p 2 ( R N ( | u n | + | u ˜ | ) p d x ) 2 p 2 ( 1 C 1 R N ( | u n | p 2 u n | u ˜ | p 2 u ˜ ) ( u n u ˜ ) d x ) p 2 C 3 + ( 1 C 1 R N b ( | x | ) ( | u n | p 2 u n | u ˜ | p 2 u ˜ ) ( u n u ˜ ) d x ) p 2 C 3 0 .

 □

Lemma 3.7 Assume that f satisfies (f1)-(f3). Let { u n } E G 1 be a sequence such that

J( u n )c, J ( u n )0,as n,
(3.3)

then { u n } has a subsequence which converges to a critical point of the functional J.

Proof First we show that each sequence { u n } E G 1 satisfying J( u n )c, J ( u n )0, as n, is bounded. By (f2) and (f3),

c + 1 + u n p J ( u n ) μ 1 J ( u n ) , u n = ( 1 p 1 μ ) u n p p + R N 1 μ f ( | x | , u n ) u n F ( | x | , u n ) d x ( 1 p 1 μ ) u n p p ,

where μ>p in the assumption (f2), so { u n } is bounded in E G 1 .

Since E is reflexive, E G 1 is reflexive, then { u n } has a weakly convergent subsequence. Going if necessary to a subsequence, let u n u ¯ . By Lemma 3.4 ϕ ( u n ) ϕ ( u ¯ ), and by the definition of T 2 , T 2 u n ,v= J ( u n ),v+ ϕ ( u n ),v, then T 2 u n converges. So we assume T 2 u n u . Observe that

T 2 u n T 2 u ¯ , u n u ¯ = T 2 u n u , u n u ¯ + u T 2 u ¯ , u n u ¯ 0.
(3.4)

By Lemma 3.6, u n u ¯ .

Next we want to show that u ¯ is a critical point of J, i.e. J ( u ¯ )=0. By Lemma 3.3, T 2 u n T 2 u ¯ ,

J ( u n )= T 2 u n ϕ ( u n ) T 2 u ¯ ϕ ( u ¯ )= J ( u ¯ ).

By (3.3), u ¯ is a critical point of J. □

Now we give the proof of Theorems 2.1 and 2.2 by applying the fountain theorem and the principle of symmetric criticality. First we recall some properties of Banach space.

According to the results in [17], there exists a Schauder basis { e n } n = 1 for E. Let e n = O ( N ) e n (g(x))d μ g , then { e n } n = 1 is a Schauder basis for E G 1 . Since E G 1 is reflexive, there are { e n } n = 1 , which are characterized by the relations e m , e n = δ m , n , forming a basis for E G 1 .

We denote

E G 1 ( n ) =span{ e 1 ,, e n }, E G 1 ( n ) = span { e n + 1 , } ¯ ,

and define a group action of G 2 ={1, τ 1 } Z 2 .

Lemma 3.8 If p<q< p , then

δ k := sup u E G 1 ( k ) , u p = 1 | u | q 0,as k.

Proof It is clear that 0< δ k + 1 δ k , so we assume for δ0, δ k δ, as k. For every k0, there exists u k E G 1 ( k ) such that u k p =1 and | u k | q > δ k 2 . By the definition of E G 1 ( k ) , u k 0 in E G 1 . Since the imbedding E G 1 L q ( R N ) is compact, then u k 0 in L q ( R N ). Thus we get δ=0. □

Proof of Theorem 2.1 Note that J is G 1 -invariant, by the principle of symmetric criticality [[13], Theorem 5.4], any critical point of J | E G 1 is a solution of problem (1.1). J is invariant with respect to the action G 2 .

Now we claim that J | E G 1 satisfies the assumptions of the fountain theorem.

By the assumptions (f1) and (f3), for u E G 1 ,

J ( u ) = 1 p u p p R N F ( | x | , u ) d x 1 p u p p C 2 q R N | u | q d x 1 p u p p C δ k q 2 q a 0 u p q ,

where a 0 >0 is the lower bound of b(|x|). Choose r k = ( 2 C q δ k q ) 1 p q , by Lemma 3.8, for u E G 1 ( k 1 ) , u p = r k , J(u)( 1 p 1 2 q a 0 ) ( 2 C q δ k q ) p p q , and as k,

b k = inf u E G 1 ( k 1 ) , u p = r k J(u).

This proves (A2).

Now we want to show that the condition (A1) is satisfied. By integrating, we obtain from (f2) and (f3) that, there exist two constants C 1 , C 2 >0, such that for any x R N , F(|x|,u) C 1 | u | μ + C 2 | u | q . Hence,

J(u)= 1 p u p p R N F ( | x | , u ) dx 1 p u p p C 2 R N | u | q dx C 1 | u | μ μ .

Since E G 1 ( k ) is finite dimensional, all norms are equivalent on E G 1 ( k ) . Therefore, μ>p and q>p imply that

sup u E k , u p R J(u),as R.

So there exists ρ k > r k >0 such that (A1) is satisfied.

( PS ) c condition is proved above. By Theorem 2.8, we find, for k2, that

c k = inf γ Γ k sup u B k I ( γ ( u ) )

are critical values of the functional J. So we can get an unbounded sequence of solutions of (1.1), and the solutions are radial. □

Proof of Theorem 2.2 In this proof, we will show that it suffices to find the critical points of J restricted to a subspace of invariant functions. The proof is similar to Theorem 1.31 in [14].

Let 2mN/2 be a fixed integer different from (N1)/2. The action of G 3 :=O(m)×O(m)×O(N2m) on E is defined by gu(x):=u( g 1 x). For R N is compatible with G 3 , the embedding E G 3 L p ( R N ) (2<p< 2 ) is compact (or see [18] for details).

Let τ 2 be the involution defined on R N = R m R m R N 2 m by

τ 2 ( x 1 , x 2 , x 3 ):=( x 2 , x 1 , x 3 ).

The action of H:={id, τ 2 } on E G 3 is defined by

hu(x)= { u ( x ) , h = id , u ( h 1 x ) , h = τ 2 .

It is clear that 0 is the only radial function on the set

E G 3 , H :={uu E G 3 ,hu=u,hH}.

Moreover, the embedding E G 3 , H L p ( R N ) is compact. As in the proof of Theorem 2.1, we obtain a sequence of non-radial solutions ± u k of (1.1). □

References

  1. Berestycki H, Lions PL: Nonlinear scalar field equations. I. Existence of a ground state. Arch. Ration. Mech. Anal. 1983, 82: 313-345.

    MathSciNet  Google Scholar 

  2. Berestycki H, Lions PL: Nonlinear scalar field equations. II. Existence of infinitely many solutions. Arch. Ration. Mech. Anal. 1983, 82: 347-375.

    MathSciNet  Google Scholar 

  3. Struwe M: Multiple solutions of differential equations without the Palais-Smale condition. Math. Ann. 1992, 261: 399-412.

    Article  MathSciNet  Google Scholar 

  4. Gidas B, Ni WM, Nirenberg L: Symmetry of positive solutions of nonlinear elliptic equations in R n . Adv. Math. Supp. Stud. 7. Mathematical Analysis and Applications, Part A 1981, 369-402.

    Google Scholar 

  5. Bartsch T, Willem M: Infinitely many nonradial solutions of a Euclidean scalar field equation. J. Funct. Anal. 1993, 117: 447-460. 10.1006/jfan.1993.1133

    Article  MathSciNet  Google Scholar 

  6. Bartsch T, Liu Z: On a superlinear elliptic p -Laplacian equation. J. Differ. Equ. 2004, 198: 149-175. 10.1016/j.jde.2003.08.001

    Article  MathSciNet  Google Scholar 

  7. Bartsch T, Liu Z, Weth T: Nodal solutions of a p -Laplacian equation. Proc. Lond. Math. Soc. 2005, 91: 129-152. 10.1112/S0024611504015187

    Article  MathSciNet  Google Scholar 

  8. Jiu QS, Su JB: Existence and multiplicity results for Dirichlet problems with p -Laplacian. J. Math. Anal. Appl. 2003, 281: 587-601. 10.1016/S0022-247X(03)00165-3

    Article  MathSciNet  Google Scholar 

  9. Agarwal RP, Perera K: Multiple positive solutions of singular discrete p -Laplacian problems via variational methods. Adv. Differ. Equ. 2005, 2005(2):93-99.

    Article  MathSciNet  Google Scholar 

  10. Chabrowski J, Fu Y: Existence of solutions for p(x)-Laplacian problems on a bounded domain. J. Math. Anal. Appl. 2005, 306: 604-618. 10.1016/j.jmaa.2004.10.028

    Article  MathSciNet  Google Scholar 

  11. Drábek P, Pohozaev SI: Positive solutions for the p -Laplacian: application of the fibrering method. Proc. R. Soc. Edinb., Sect. A, Math. 1997, 127: 703-726. 10.1017/S0308210500023787

    Article  Google Scholar 

  12. De Nápoli P, Mariani MC: Mountain pass solutions to equations of p -Laplacian type. Nonlinear Anal. 2003, 54: 1205-1219. 10.1016/S0362-546X(03)00105-6

    Article  MathSciNet  Google Scholar 

  13. Palais RS: The principle of symmetric criticality. Commun. Math. Phys. 1979, 69: 19-30. 10.1007/BF01941322

    Article  MathSciNet  Google Scholar 

  14. Willem M: Minimax Theorems. Birkhäuser, Boston; 1996.

    Book  Google Scholar 

  15. Struwe M: Variational Methods. Springer, Berlin; 2000.

    Book  Google Scholar 

  16. Damascelli L: Comparison theorems for some quasilinear degenerate elliptic operators and applications to symmetry and monotonicity results. Ann. Inst. Henri Poincaré, Anal. Non Linéaire 1998, 15: 493-516. 10.1016/S0294-1449(98)80032-2

    Article  MathSciNet  Google Scholar 

  17. Triebel H: Interpolation Theory, Function Spaces, Differential Operator. North-Holland, Amsterdam; 1978.

    Google Scholar 

  18. Lions PL: Symétrie et compactité dans les espaces de Sobolev. J. Funct. Anal. 1982, 49: 315-334. 10.1016/0022-1236(82)90072-6

    Article  MathSciNet  Google Scholar 

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Acknowledgements

The first author is supported by the project of ‘Min-Hong Kong cooperation postdoctoral training’ funded by Fujian Provincial Civil Service Bureau and the Nonlinear Analysis Innovation Team (IRTL1206) funded by Fujian Normal University. The second author is sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.

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Zeng, J., Cai, S. Multiple solutions for a p-Laplacian elliptic problem. Bound Value Probl 2014, 124 (2014). https://doi.org/10.1186/1687-2770-2014-124

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