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Contents lists available at SciVerse ScienceDirect

Journal of Differential Equations

www.elsevier.com/locate/jde

Elliptic problems involving the fractional Laplacian in R N

Giuseppina Autuori a , Patrizia Pucci b ,

aDipartimento di Ingegneria Industriale e Scienze Matematiche, Università Politecnica delle Marche, Via Brecce Bianche, 60131 Ancona, Italy

bDipartimento di Matematica e Informatica, Università degli Studi di Perugia, Via Vanvitelli 1, 06123 Perugia, Italy

a r t i c l e i n f o a b s t r a c t

Article history:

Received 18 October 2012 Revised 11 June 2013 Available online 8 July 2013

MSC:

primary 35J60, 35R11 secondary 35S30, 35J20

Keywords:

Existence

Fractional Sobolev spaces Variational methods

We study the existence and multiplicity of solutions for elliptic equations in

R

N, driven by a non-local integro-differential operator, which main prototype is the fractional Laplacian. The model under consideration, denoted by

( P

λ

)

, depends on a real parameter

λ

and involves two superlinear nonlinearities, one of which could be critical or even supercritical. The main theorem of the paper establishes the existence of three critical values of

λ

which divide the real line in different intervals, where

( P

λ

)

admits no solutions, at least one nontrivial non-negative entire solution and two nontrivial non-negative entire solutions.

©

2013 Elsevier Inc. All rights reserved.

1. Introduction

In this paper we prove the existence and multiplicity of solutions for non-local integro-differential equations in R

N

, whose prototype is given by

(−)

s

u + a ( x ) u = λ w ( x )| u |

q2

uh ( x )| u |

r2

u in R

N

, ( P

λ

)

where λ ∈ R , 0 < s < 1, 2s < N and ( −)

s

is the fractional Laplacian operator. Up to normalization factors, ( −)

s

u is defined pointwise for x in R

N

by

*

Corresponding author.

E-mail addresses:autuori@dipmat.univpm.it(G. Autuori),pucci@dmi.unipg.it(P. Pucci).

0022-0396/$ – see front matter

©

2013 Elsevier Inc. All rights reserved.

http://dx.doi.org/10.1016/j.jde.2013.06.016

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( −)

s

u ( x ) = − 1 2



RN

u ( x + y ) + u ( xy )2u ( x )

| y |

N+2s

dy ,

along any rapidly decaying function u of class C

( R

N

) , see Lemma 3.5 of [18].

The nonlinear terms in (P

λ

) are related to the main elliptic part by the request that 2 < q < min 

r , 2



, (1.1)

where 2

= 2N /( N2s ) is the critical Sobolev exponent for H

s

( R

N

) . The coefficient a is supposed to be in L

loc

( R

N

) and to satisfy for a.a. x ∈ R

N

ν ( x ) = max  a ( x ), 

1 + | x | 

2s



, a ( x )  κν ( x ), (1.2) for some constant κ ∈ ( 0 , 1 ] . The weight w verifies

wL

 R

N



L σ

loc

 R

N



, with = 2

/  2

q 

, σ > ℘, (1.3)

while h is a positive weight of class L

1loc

( R

N

) . Finally, h and w are related by the condition



RN

 w ( x )

r

h ( x )

q



1/(rq)

dx = H ∈ R

+

. (1.4)

The main result of the paper is

Theorem 1.1. Under the above assumptions there exist λ

, λ

∗∗

and λ , with 0 < λ

 λ

∗∗

 λ such that Eq. ( P

λ

) admits

(i) only the trivial solution if λ < λ

;

(ii) a nontrivial non-negative entire solution if and only if λ  λ

∗∗

; (iii) at least two nontrivial non-negative entire solutions if λ > λ .

The definition of entire solution for (P

λ

) , as well as the proof of Theorem 1.1(i), are given in Section 2, after the introduction of the main solution space X . Some preliminary results for existence are presented in Section 3 and in Appendix A. The proof of Theorem 1.1(ii) is discussed in Section 4, while Theorem 1.1(iii) is proved in Section 5.

For standing wave solutions of fractional Schrödinger equations in R

N

we refer to [20,22,32,13, 28], [19, Section 5] and to the references therein. Models governed by unbounded potentials V are investigated in [14] and in its recent extension [27]. All these papers, however, deal with problems which are not directly comparable to (P

λ

) . The present work is more related to the results on general quasilinear elliptic problems given in [4]. Indeed, in [4], as a corollary of the main theorems, we proved under (1.4) that there exists λ

> 0 such that

div 

|∇ u |

p2

u 

+ a ( x )| u |

p2

u = λ w ( x )| u |

q2

uh ( x )| u |

r2

u in R

N

, 1 < p < N , max { 2 , p } < q < min 

r , p



, p

= Np

Np , ( E

λ

) admits at least a nontrivial non-negative entire solution if and only if λ  λ

. Theorem 1.1(ii) extends Theorem A of [4] to non-local integro-differential equations. It would be interesting to understand if λ

= λ

∗∗

in Theorem 1.1. This possible gap does not rise in [4]. Indeed, if u is a solution of ( E

λ

) also

| u | is. The situation is more delicate for (P

λ

) , since the fractional Laplacian itself does not guarantee

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the same property. Hence, it remains an open problem to establish whether λ

= λ

∗∗

in the non-local setting.

The extension of Theorem A of [4] to ( P

λ

) is not trivial and requires to overcome several diffi- culties which arise in the new context. In particular, the proof of the main preliminary Theorem 4.2 needs a special care.

Furthermore, Theorem 1.1(iii) is a complete extension of Theorem B of [4] to the non-local equation ( P

λ

) and its proof is based on a new strategy.

For previous related results in the local setting and in bounded domains we refer to [3,2,15,23]

for the semilinear case and to [16] for the quasilinear case. We also refer to [24] for the semilinear case in R

N

. Actually, for semilinear elliptic equations assumption (1.4) first appears in the existence Theorem 1.1 of [2] for Dirichlet problems in bounded domains Ω , see also [26] for quasilinear equa- tions in R

N

. In the existence Theorem 1.2 of [2] Alama and Tarantello use the weaker assumption that w ( w / h )

(q2)/(rq)

is in L

N/2

(Ω) . It is still an open problem to produce nontrivial solutions of (P

λ

) when w ( w / h )

(q2)/(rq)

L

N/2s

(R

N

) replaces (1.4) and of (E

λ

) when w ( w / h )

(qp)/(rq)

L

N/p

(R

N

) .

In the last years a great attention has been devoted to the study of fractional and non-local prob- lems. For example, some of the most recent contributions on the existence of positive solutions for critical fractional Laplacian elliptic Dirichlet problems in bounded domains are given in [5], where the effects of lower order perturbations are considered. Comparison and regularity results and a priori estimates on the solutions of special fractional Laplacian elliptic boundary value problems in bounded domains are presented in [17], via symmetrization techniques. For the existence, non-existence, mul- tiplicity and bifurcation of solutions for square root Laplacian Dirichlet problems in bounded domains with sign-changing weights we refer to [33]. A mountain pass theorem and applications to Dirichlet problems in bounded domains involving non-local integro-differential operators of fractional Laplacian type are given in [29]. Existence of positive solutions of concave–convex Dirichlet fractional Laplacian problems in bounded domains is proved in [8].

However, the interest in non-local integro-differential problems goes beyond the mathematical curiosity. Indeed, they have impressive applications in different fields, as the thin obstacle problem, optimization, finance, phase transitions, stratified materials, anomalous diffusion, crystal dislocation, deblurring and denoising of images, and so on. For further details we refer to [10,11,13,14,18,22,27, 30–32] and the references therein.

Theorem 1.1 continues to hold when (−)

s

u in (P

λ

) is replaced by any non-local integro- differential operator L

K

u, defined pointwise by

L

K

u ( x ) = − 1 2



RN

u ( x + y ) + u ( xy )2u ( x )

K ( y ) dy ,

along any rapidly decaying function u of class C

( R

N

) , where the positive weight K : R

N

\ { 0 } → R

+

satisfies the main properties

( k

1

) g KL

1

( R

N

) , where g ( x ) = min { 1 , | x |

2

} ;

( k

2

) there exists γ > 0 such that K ( x )  γ | x |

−(N+2s)

for all x ∈ R

N

\ { 0 } ; ( k

3

) K ( x ) = K (x ) for all x ∈ R

N

\ { 0 } .

Few details of the main changes, in passing from (−)

s

u to L

K

u in (P

λ

) , are given in Appendix B.

Of course L

K

u reduces to the fractional Laplace operator ( −)

s

u when K ( x ) = | x |

−(N+2s)

. 2. Preliminaries and non-existence

Let D

s

( R

N

) denote the completion of C

0

( R

N

) with respect to the Gagliardo norm

[ u ]

s

=  

R2N

| u ( x )u ( y ) |

2

| xy |

N+2s

dx dy

1/2

.

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The embedding D

s

( R

N

) L

2

( R

N

) is continuous, that is

 u 

2

 C

2

[ u ]

s

for all uD

s

 R

N



, (2.1)

where C

22

= c ( N )

(sN(12ss))

by Theorem 1 of [25], see also Theorem 1 of [7]. The space E denotes the completion of C

0

( R

N

) with respect to the norm

 u 

E

=

[ u ]

2s

+



RN

ν ( x ) | u |

2

dx

1/2

.

Clearly,  · 

E

is a Hilbertian norm induced by the inner product

u , v

E

=

 

R2N

[ u ( x )u ( y )] · [ v ( x )v ( y )]

| xy |

N+2s

dx dy +



RN

ν ( x ) u ( x ) v ( x ) dx

= u , v

s

+ u , v ν .

Finally, X is the completion of C

0

( R

N

) with respect to the norm

 u  = 

 u 

2E

+  u 

2r,h



1/2

, where  u 

rr,h

=



RN

h ( x ) | u |

r

dx .

From now on B

R

will denote the ball in R

N

of center zero and radius R > 0.

Lemma 2.1. The embeddings X E D

s

(R

N

) L

2

(R

N

) are continuous, with [ u ]

s

  u 

E

for all uE and  u 

E

  u  for all uX .

Moreover, for any R > 0 and p ∈ [ 1 , 2

) the embeddings E →→ L

p

( B

R

) and X →→ L

p

( B

R

) are com- pact.

Proof. The first two embeddings of the chain X E D

s

(R

N

) L

2

(R

N

) are obviously con- tinuous, with [ u ]

s

  u 

E

for all uE and  u 

E

  u  for all uX . The continuity of the third embedding follows from (2.1), as recalled above.

Fix R > 0. By the first part of the lemma the embedding E H

s

( B

R

) is continuous, since 0 < k

1

 ν ( x )  k

2

for a.a. xB

R

and for some positive numbers k

1

and k

2

depending only on R, being aL

loc

( R

N

) by (1.2). The embedding H

s

( B

R

) →→ L

p

( B

R

) is compact for all p ∈ [ 1 , 2

) by Corollary 7.2 of [18], and so the embeddings E →→ L

p

( B

R

) and X →→ L

p

( B

R

) are compact. 2

From the structural assumptions (1.2)–(1.4) all the coefficients a, w, h in (P

λ

) are weights in R

N

. We indicate with L

2

( R

N

, a ) = ( L

2

( R

N

, a ),  · 

2,a

) , L

q

( R

N

, w ) = ( L

q

( R

N

, w ),  · 

q,w

) and L

r

( R

N

, h ) = ( L

r

( R

N

, h ),  · 

r,h

) the corresponding weighted Lebesgue spaces, which are uniformly convex Banach spaces by Proposition A.6 of [4].

Lemma 2.2. The embedding D

s

( R

N

) L

q

( R

N

, w ) is continuous, with

 u 

q,w

 C

w

[ u ]

s

for all uD

s

 R

N



, (2.2)

and C

w

= C

2

 w 

1/q

> 0. The embeddings

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E →→ L

q

 R

N

, w 

and X →→ L

q

 R

N

, w 

are compact. Furthermore, for all uE

[ u ]

2s

+  u 

22,a

 κ  u 

2E

, (2.3) where κ is given in (1.2).

Proof. By (1.3), (2.1) and Hölder’s inequality, for all uD

s

(R

N

) ,

 u 

q,w

 

RN

w ( x )

dx

1/℘q

· 

RN

| u |

2

dx

1/2

 C

2

 w 

1/q

[ u ]

s

,

that is, (2.2) holds.

In order to prove the last part of the lemma it is enough to show that E →→ L

q

(R

N

, w ) , that is, that  u

n

u 

q,w

0 as n → ∞ whenever u

n

u in E. By Hölder’s inequality,



RN\BR

w ( x ) | u

n

u |

q

dx  M 

RN\BR

w ( x )

dx

1/℘

= o ( 1 )

as R → ∞ , being wL

(R

N

) by (1.3) and M = sup

n

 u

n

u 

q2

<. For all ε > 0 there exists > 0 so large that sup

n

RN\BRε

w ( x ) | u

n

u |

q

dx < ε / 2. Moreover, by (1.3), Hölder’s inequality and Lemma 2.1 as n → ∞



BRε

w ( x )| u

n

u |

q

dx   w 

Lσ(BRε)

 u

n

u 

q q(BRε)

= o ( 1 ),

since σ

q < 2

. Hence, there exists Nε > 0 such that

BRε

w ( x ) | u

n

u |

q

dx < ε / 2 for all n  Nε. In conclusion, for all n 

 u

n

u 

qq,w

=



RN\BRε

w ( x )| u

n

u |

q

dx +



BRε

w ( x )| u

n

u |

q

dx < ε ,

as required. Now, from (1.2) we directly get (2.3), being κ ∈ ( 0 , 1 ] . 2 We say that uX is a (weak) entire solution of (P

λ

) if

u , ϕ

s

+



RN

a ( x ) u ϕ dx = λ



RN

w ( x ) | u |

q2

u ϕ dx



RN

h ( x ) | u |

r2

u ϕ dx (2.4)

for all ϕ X .

Hence the entire solutions of (P

λ

) correspond to the critical points of the C

1

energy functional Φ

λ

: X → R , defined by

Φ

λ

( u ) = 1

2 [ u ]

2s

+ 1

2  u 

22,a

λ

q  u 

qq,w

+ 1

r  u 

rr,h

,

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see the next Lemma 3.4. Similarly, non-negative entire solutions of ( P

λ

) are the critical points of the C

1

functional

Ψ

λ

( u ) = 1

2 [ u ]

2s

+ 1

2  u 

22,a

λ q  u

+



q

q,w

+ 1 r  u 

rr,h

,

well-defined for all uX , see the next Lemma 3.4. Indeed, both u

+

and u

X for all uX , being

| u

+

( x )u

+

( y ) |  | u ( x )u ( y ) | and | u

( x )u

( y ) |  | u ( x )u ( y ) | for all x, y ∈ R

N

. Furthermore, for all uX

 u , u



s

=

 

R2N

u

( x )

2

+ u

( y )

2

2u

( x ) u ( y )

| xy |

N+2s

dx dy ,

since

R2Nu(x)u(y)

|xy|N+2s

dx dy =

R2Nu

(x)u(y)

|xy|N+2s

dx dy. Therefore, if uX is a critical point of Ψ

λ

, then by (1.2)

0 =  u , u



s

+



RN

a ( x )  u



2

dx +



RN

h ( x )  u



r

dx



 

R2N

[ u

( x )u

( y )]

2

+ 2u

( x ) u

+

( y )

| xy |

N+2s

dx dy + κ  u



2

2,

ν

 κ  u



2

E

+ 2

 

R2N

u

( x ) u

+

( y )

| xy |

N+2s

dx dy  0 ,

in other words u

= 0 in E, that is, the critical point u of Ψ

λ

is non-negative in R

N

. Lemma 2.3. If uX \ { 0 } and λ ∈ R satisfy

[ u ]

2s

+  u 

22,a

+  u 

rr,h

= λ u 

qq,w

, (2.5) then λ > 0 and

c

1

λ

1/(2q)

  u 

q,w

 c

2

λ

r/2(rq)

, (2.6) where

c

1

= 

κ /C

2w



1/(q2)

and c

2

=

( rq )C

2w

H / r κ

1/2

.

Proof. Let uX \ { 0 } and λ ∈ R satisfy (2.5). Then 0 < κ  u 

2E

 λ u 

qq,w

by (2.3), so that λ > 0 and moreover

 u 

q2,w

 C

2w

 u 

2E

 λC

2w

κ  u 

q

q,w

(2.7)

by (2.2). Using Young’s inequality

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t τ  t α α +

τ

β

β ,

with t = h ( x )

q/r

| u |

q

 0, τ = λ w ( x ) h ( x )

q/r

 0, α = r / q > 1 and β = r /( rq ) > 1, we find

λ w ( x ) | u |

q

 q

r h ( x ) | u |

r

+ rq r

λ w ( x ) h ( x )

q/r

r/(rq)

.

Integration over R

N

gives

λ u 

qq,w

 q

r  u 

rr,h

+ rq

r H λ

r/(rq)

. (2.8)

Thus, by (2.5) we obtain

[ u ]

2s

+  u 

22,a

 qr

r  u 

rr,h

+ rq

r H λ

r/(rq)

 rq

r H λ

r/(rq)

,

being q < r. Hence, since u ≡ 0 by assumption, the last inequality and (2.7) give (2.6), with c

1

and c

2

as stated. 2

If (P

λ

) admits a nontrivial entire solution uX , then λ  λ

0

by (2.6), where λ

0

= ( c

1

/ c

2

)

2(rq)(q2)/q(r2)

> 0. Define

λ

= sup 

λ > 0: ( P μ ) admits only the trivial solution for all μ < λ  .

Theorem 1.1(i) follows directly by the definition of λ

. Similarly, put

λ

Ψ

λ

= sup 

λ > 0: ( P μ ) admits no nontrivial non-negative solution for all μ < λ  .

Clearly λ

Ψλ

 λ

 λ

0

> 0.

3. Preliminary results for existence

By the results of Section 2 from now on we consider only the case λ > 0.

Lemma 3.1. The functionals Φ

λ

and Ψ

λ

are coercive in X . In particular, any sequence ( u

n

)

n

in X such that either

λ

( u

n

))

n

or

λ

( u

n

))

n

is bounded admits a weakly convergent subsequence in X .

Proof. Let us consider the following elementary inequality: for every k

1

, k

2

> 0 and 0 < α < β

k

1

| t | αk

2

| t |

β

 C α

β

k

1

k

1

k

2

α

/(β−

α

)

for all t ∈ R, (3.1)

where Cα

β

> 0 is a constant depending only on α and β .

Taking k

1

= λ w ( x )/ q, k

2

= h ( x )/ 2r, α = q, β = r and t = u ( x ) in (3.1), for all x ∈ R

N

we have λ

q w ( x ) u ( x )

q

h ( x )

2r  u ( x )

r

 C λ

r/(rq)

 w ( x )

r

h ( x )

q



1/(rq)

,

where C = C

qr

[ 2r / q ]

q/(rq)

/ q. Integrating the above inequality over R

N

, we get by (1.4)

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λ

q  u 

qq,w

1

2r  u 

rr,h

 C

λ

,

where C

λ

= C H λ

r/(rq)

> 0.

Therefore, by (2.3) for all uX Φ

λ

( u ) = 1

2 [ u ]

2s

+ 1

2  u 

22,a

 λ

q  u 

qq,w

1 2r  u 

rr,h



1

2r  u 

rr,h

+ 1 r  u 

rr,h

 κ

2  u 

2E

+ 1

2r  u 

rr,h

C

λ

 κ

2  u 

2E

+ 1 2r

  u 

2r,h

1 

C

λ

 min { κ , r

1

}

2  u 

2

C

λ

1 2r .

Hence, Φ

λ

is coercive in X . This implies at once that also Ψ

λ

is coercive in X , being Ψ

λ

( u )  Φ

λ

( u ) for all uX .

The last part of the claim follows at once by the coercivity of Φ

λ

and Ψ

λ

and the reflexivity of the space X , see Proposition A.1. 2

Lemma 3.2. The functional Ψ : X → R , Ψ ( u ) =

12

[ u ]

2s

, is convex and of class C

1

. In particular, Ψ is weakly lower semicontinuous in X .

Proof. The convexity is trivial. Now, let ( u

n

)

n

, uX be such that u

n

u in X . Then clearly [ u

n

u ]

s

0 as n → ∞ . Consider the following elementary inequality || t |

2

− | τ |

2

|  2 (| tτ |

2

+ | τ | · | tτ |) which is valid for all t, τ ∈ R . Applying this relation and Hölder’s inequality, we have

Ψ ( u

n

) − Ψ ( u )   1 2

 

R2N

|| u

n

( x )u

n

( y ) |

2

− | u ( x )u ( y ) |

2

|

| xy |

N+2s

dx dy



 

R2N

| u

n

( x )u

n

( y )u ( x ) + u ( y ) |

2

| xy |

N+2s

dx dy +

 

R2N

| u ( x )u ( y )| · | u

n

( x )u

n

( y )u ( x ) + u ( y )|

| xy |

N+2s

dx dy

 [ u

n

u ]

2s

+ [ u ]

s

[ u

n

u ]

s

= o ( 1 )

as n → ∞ . This shows the continuity of Ψ .

Moreover, Ψ is Gâteaux-differentiable in X and for all u, ϕX

 Ψ

( u ), ϕ  =

 

R2N

[ u ( x )u ( y ) ] · [ ϕ ( x )ϕ ( y ) ]

| xy |

N+2s

dx dy = u , ϕ

s

.

Now, let ( u

n

)

n

, uX be such that u

n

u in X as n → ∞ . Of course

 Ψ

( u

n

) − Ψ

( u ) 

X

= sup

ϕ

X



ϕ

=1

 u

n

u , ϕ

s

 [ u

n

u ]

s

  u

n

u ,

that is, Ψ is of class C

1

, as claimed. Finally, Ψ is weakly lower semicontinuous in X by Corollary 3.9

of [9]. 2

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For any ( x , u ) ∈ R

N

× R put

f ( x , u ) = λ w ( x )| u |

q2

uh ( x )| u |

r2

u , (3.2) so that

F ( x , u ) =



u

0

f ( x , v ) dv = λ

q w ( x )| u |

q

h ( x ) | u |

r

r . (3.3)

Lemma 3.3. For any fixed uX the functional F

u

: X → R , defined by F

u

( v ) =



RN

f 

x , u ( x )  v ( x ) dx ,

is in X

. In particular, if v

n

v in X then F

u

( v

n

)F

u

( v ) .

Proof. Take uX . Clearly F

u

is linear. Moreover, using (2.2), we get for all v ∈ X

 F

u

( v )   λ



RN

w ( x )| u |

q1

| v | dx +



RN

h ( x )| u |

r1

| v | dx

 λ u 

qq,w1

 v 

q,w

+  u 

rr,h1

 v 

r,h

 

λC

w

 u 

qq,w1

+  u 

rr,h1



 v ,

and so F

u

is continuous in X . 2

In the next result we strongly use the assumption q > 2. An interesting open question occurs when 1 < q < 2 < r, cf. Theorem 2.1 of [3] for homogeneous Dirichlet problems in bounded domains of R

N

. Lemma 3.4. The functionals Φ

λ

and Ψ

λ

are of class C

1

( X ) and Φ

λ

is sequentially weakly lower semicontinu- ous in X , that is, if u

n

u in X , then

Φ

λ

( u )  lim inf

n→∞

Φ

λ

( u

n

). (3.4)

Proof. Lemmas 3.2 and A.3–A.5 imply that Φ

λ

and Ψ

λ

are of class C

1

( X ) . Let ( u

n

)

n

, uX be such that u

n

u in X . The definition of Φ

λ

and (3.2) give

Φ

λ

( u ) − Φ

λ

( u

n

) = 1 2

 [ u ]

2s

− [ u

n

]

2s

+  u 

22,a

−  u

n



22,a

 +



RN

F ( x , u

n

)F ( x , u )

dx . (3.5)

Since u

n

u in X , Lemmas 3.2 and A.3 imply that

[ u ]

2s

 lim inf

n→∞

[ u

n

]

2s

and  u 

22,a

 lim inf

n→∞

 u

n



22,a

.

(10)

Hence, by (3.5)

lim sup

n→∞

Φ

λ

( u ) − Φ

λ

( u

n

)

 lim sup

n→∞



RN

F ( x , u

n

)F ( x , u )

dx . (3.6)

By (3.2) and (3.3), for all t ∈ [ 0 , 1 ] ,

F

u



x , u + t ( u

n

u ) 

= f 

x , u + t ( u

n

u ) 

= f ( x , u ) + ( u

n

u )



t

0

f

u



x , u + τ ( u

n

u ) 

d τ , (3.7)

where clearly

f

u

( x , z ) = λ( q1 ) w ( x )| z |

q2

h ( x )( r1 )| z |

r2

.

Multiplying (3.7) by u

n

u and integrating over [ 0 , 1 ] , we obtain

F ( x , u

n

)F ( x , u ) = f ( x , u )( u

n

u ) + ( u

n

u )

2



1

0

 

t 0

f

u

 x , u + τ ( u

n

u )  d τ



dt . (3.8)

By (3.1), with t = z, k

1

= λ w ( x )( q1 ) , k

2

= h ( x )( r1 ) , α = q2 > 0 and β = r2 > 0, we get f

u

( x , z )  2C

1

w ( x )

2/q

 w ( x )

r/q

h ( x )



(q2)/(rq)

,

where C

1

is a positive constant, depending only on q, r and λ . Consequently, (3.8) yields



RN

F ( x , u

n

)F ( x , u ) dx 



RN

f ( x , u )( u

n

u ) dx

+ C

1



RN

w ( x )

2/q

( u

n

u )

2

 w ( x )

r/q

h ( x )



(q2)/(rq)

dx





RN

f ( x , u )( u

n

u ) dx + C

1

H

(q2)/q

 u

n

u 

2q,w

, (3.9)

by Hölder’s inequality and (1.4). Now, Lemma 3.3 gives

n

lim

→∞



RN

f ( x , u )( u

n

u ) dx = 0 , (3.10)

(11)

and Lemma 2.2 implies

n

lim

→∞

 u

n

u 

q,w

= 0 . (3.11) Combining (3.9)–(3.11) with (3.6) we get the claim (3.4). 2

4. Existence if λ is large Define

λ = inf

uX

uq,w=1

 q

2 [ u ]

2s

+ q

2  u 

22,a

+ q r  u 

rr,h

 .

Note that λ > 0. Indeed, for any uX with  u 

q,w

= 1, by Hölder’s inequality and (1.4), we have 1 =  u 

qq,w

=



RN

w ( x )

h ( x )

q/r

h ( x )

q/r

| u |

q

dx  H

(rq)/r

 u 

qr,h

.

Consequently, using also (2.3) and Lemma 2.1, we get

q

2 [ u ]

2s

+ q

2  u 

22,a

+ q

r  u 

rr,h

 κ q

2  u 

2E

+ q

r H

(qr)/q

 κ q 2 C

2w

+ q

r H

(qr)/q

,

where C

w

> 0 is given in (2.2). In other words,

λ  κ q 2 C

2w

+ q

r H

(qr)/q

> 0 .

Lemma 4.1. For all λ > λ there exists a global nontrivial non-negative minimizer eX of Φ

λ

with negative energy, that is Φ

λ

( e ) < 0. Furthermore, e is also a critical point of Ψ

λ

and Ψ

λ

( e ) = Φ

λ

( e ) < 0.

Proof. For all λ > 0 the functional Φ

λ

is sequentially weakly lower semicontinuous, bounded below and coercive in the reflexive Banach space X by Lemmas 3.1, 3.4 and Proposition A.1. Hence, Theo- rem 6.1.1 of [6] implies that for all λ > 0 there exists a global minimizer eX of Φ

λ

, that is

Φ

λ

( e ) = inf

vX

Φ

λ

( v ).

Clearly e is a solution of (P

λ

) . We prove that e0 whenever λ > λ , showing that inf

vX

Φ

λ

( v ) < 0.

Let λ > λ . Then there exists a function ϕX , with  ϕ 

q,w

= 1, such that

λ ϕ 

qq,w

= λ > q 2

 [ ϕ ]

2s

+  ϕ 

22,a

 + q

r  ϕ 

rr,h

.

This can be rewritten as

Φ

λ

( ϕ ) = 1 2

 [ ϕ ]

2s

+  ϕ 

22,a



λ

q  ϕ 

qq,w

+ 1

r  ϕ 

rr,h

< 0

and consequently Φ

λ

( e ) = inf

vX

Φ

λ

( v )  Φ

λ

( ϕ ) < 0.

(12)

Hence, for any λ > λ Eq. ( P

λ

) has a nontrivial entire solution eX such that Φ

λ

( e ) < 0. Finally, we may assume e  0 in R

N

. Indeed, | e | ∈ X and Φ

λ

( | e |)  Φ

λ

( e ) , being [| u |]

s

 [ u ]

s

. This gives Φ

λ

( e ) = Φ

λ

(| e |) , due to the minimality of e.

The second part of the lemma is almost trivial, being Φ

λ

( u ) = Ψ

λ

( u ) for all uX , with u  0 in R

N

, so that e is also a nontrivial global minimizer of Ψ

λ

in X . 2

Define

λ

∗∗

= inf 

λ > 0: ( P

λ

) admits a nontrivial non-negative entire solution  .

Lemma 4.1 assures that this definition is meaningful and that λ  λ

∗∗

.

Theorem 4.2. For any λ > λ

∗∗

Eq. (P

λ

) admits a nontrivial non-negative entire solution u

λ

X .

Proof. Fix λ > λ

∗∗

. By definition of λ

∗∗

there exists μ ∈ (λ

∗∗

, λ) such that Φ μ has a nontrivial critical point uμX , with uμ  0 in R

N

. Of course, uμ is a subsolution for (P

λ

) . Consider the following minimization problem

v

inf

∈M

Φ

λ

( v ), M = { vX: v  u μ }.

First note that M is closed and convex, and in turn also weakly closed. Moreover, as shown in the proof of Lemma 4.1, Theorem 6.1.1 of [6] can be applied in X and so in the weakly closed set M . Hence, Φ

λ

attains its infimum in M , i.e. there exists u

λ

 uμ such that Φ

λ

( u

λ

) = inf

v∈M

Φ

λ

( v ) .

We claim that u

λ

is a solution of ( P

λ

) , which is clearly non-negative. Indeed, take ϕ C

0

( R

N

) and ε > 0. Put

ϕ ε = max { 0 , u μu

λ

εϕ }  0 and v ε = u

λ

+ εϕ + ϕ ε ,

so that vεM . Of course 0  

Φ

λ

( u

λ

), v εu

λ

 = ε  Φ

λ

( u

λ

), ϕ  +  Φ

λ

( u

λ

), ϕ ε  ,

and in turn

 Φ

λ

( u

λ

), ϕ   − 1 ε

 Φ

λ

( u

λ

), ϕ ε  . (4.1)

Since uμ is a subsolution of ( P

λ

) and ϕ ε  0 we get that Φ

λ

( ), ϕ ε  0. In particular,

 Φ

λ

( u

λ

), ϕ ε  = 

Φ

λ

( u μ ), ϕ ε  + 

Φ

λ

( u

λ

) − Φ

λ

( u μ ), ϕ ε   

Φ

λ

( u

λ

) − Φ

λ

( u μ ), ϕ ε  .

Define Ω ε = { x ∈ R

N

: u

λ

( x ) + εϕ ( x )  ( x ) < u

λ

( x ) } . Clearly Ω ε is a subset of supp ϕ . Put u = u

λ

uμ and

U ε ( x , y ) = [ u ( x )u ( y ) ] · [ ϕ ε ( x )ϕ ε ( y ) ]

| xy |

N+2s

,

so that

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