GLOBAL ATTRACTOR FOR A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATION ON R N

Abstract. The aim of this paper is to prove the existence of the global attractor for a semilinear strongly degenerate parabolic equation on RN with the locally Lipschitz nonlinearity satisfying a subcritical growth condition

GLOBAL ATTRACTOR FOR A SEMILINEAR STRONGLY

CUNG THE ANH

Abstract The aim of this paper is to prove the existence of the global

at-tractor for a semilinear strongly degenerate parabolic equation on R N with

the locally Lipschitz nonlinearity satisfying a subcritical growth condition.

1 Introduction The understanding of asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics One way to attack the problem for a dissipative dynamical system is to consider its global attractor The existence of global attractors has been proved for a large class of nondegenerate partial differential equations (see e.g [5, 20, 23] and references therein) In the last few years, a number of papers are devoted to the study of long-time behavior of solutions to degenerate parabolic equations

One of the classes of degenerate equations that has been studied widely in recent years is the class of equations involving an operator of Grushin type

Gαu = ∆xu + |x|2α∆yu, α ≥ 0

This operator was studied by Grushin in [9] when α is an integer, and by Franchi

& Lanconelli in [6, 7, 8] when α is not an integer Note that G0 = ∆ is the Laplacian operator, and Gα, when α > 0, is not elliptic in domains intersecting the surface x = 0 The long-time behavior of solutions to semilinear parabolic equations involving this operator has been studied recently in [1, 2] We also refer the reader

to some recent results on the generalized Grushin operators [10, 13, 15, 16, 17] Recently, Thuy and Tri [21] considered a strongly degenerate operator

Pα,βu = ∆xu + ∆yu + |x|2α|y|2β∆zu, α, β ≥ 0, which is degenerate on two intersecting surfaces x = 0 and y = 0, and established some compact embedding theorems for weighted Sobolev spaces associated to this operator in bounded domains This operator falls into the class of ∆λ operators [12], or more general, the class of X-elliptic operators [14]

In this paper we study the existence and long-time behavior of solutions to the following semilinear strongly degenerate parabolic equation







∂u

∂t − Pα,βu + λu + f (X, u) = g(X), X ∈ RN, t > 0,

(1.1)

2010 Mathematics Subject Classification 35D35, 35K65, 35B41.

Key words and phrases strongly degenerate; unbounded domains; mild solution; global at-tractor; tail-estimates method; sectorial operator; Lyapunov function.

where X = (x, y, z) ∈ RN1 × RN2

× RN3

= RN, λ > 0, u0 ∈ S1

(RN) (see the definition of this function space below) are given, the nonlinearity f and the external force g satisfy the following conditions:

(F) f : RN × R → R is a function satisfying

|fu0(X, u)| ≤ C

 a(X) + |u|ρ

 , 0 ≤ ρ < 4

Nα,β− 2, (1.3)

f (X, u)u ≥ −µu2− C1(X), (1.5)

F (X, u) ≥ −µ

2u

2

where Nα,β = N1+ N2+ (α + β + 1)N3, C, ` are positive constants, a ∈

LNα,β(RN)∩LNα,β2 (RN) is a nonnegative function, F (X, s) =R0sf (X, τ )dτ ,

µ < λ, C1, C2∈ L1

(RN) are two nonnegative functions;

(G) g ∈ L2

(RN)

Under similar above conditions, following the approach used in [1], Thuy and Tri [22] recently proved the existence and uniqueness of a global mild solution to problem (1.1) in a bounded domain Ω with the homogeneous Dirichlet boundary condition, and they also proved the existence of a compact global attractor for the semigroup generated by this problem (see also [13] for a more general situation) The aim of the present paper is to extend these results to the case of unbounded domains, the more complicated case due to the lack of compactness of the embed-ding theorems For other results related to problem (1.1) with the nonlinearity of polynomial type, we refer the reader to some very recent works [3, 4]

In order to study problem (1.1), we use some weighted Sobolev spaces We use the space S1(RN) defined as the completion of C0∞(RN) in the norm

kuk2

S 1 (R N )=

Z

RN



|u|2+ |∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2

 dX

Then S1

(RN) is a Hilbert space with the inner product

(u, v)S1 (R N )=

Z

RN



uv + ∇xu∇xv + ∇yu∇yv + |x|2α|y|2β∇zu∇zv

 dX

We also use the space S2(RN) defined as the completion of C0∞(RN) in the norm

kuk2

S 2 (R N )=

Z

RN



|u|2+ |Pα,βu|2

 dX

In Lemma 2.1 below, we will establish some embedding results related to these function spaces

Denote A = −Pα,β+ λu, the positive and self-adjoint unbounded linear operator with domain of the definition

D(A) = {u ∈ S1(RN) : Au ∈ L2(RN)} = S2(RN), and define the corresponding Nemytski map f by

ˆ

f (u)(X) = f (X, u(X)), u ∈ S1(RN)

Then, (1.1) can be formulated as an abstract evolutionary equation







du

dt + Au + ˆf (u) = g,

The aim of this paper is to study the existence of a global mild solution and of a global attractor for the semigroup generated by problem (1.1)

One basic feature of (1.1) is that it admits the following (strict) natural Lyapunov function

Φ(u) =1

2

Z

RN



|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2



dX + λ

Z

RN

u2dX +

Z

RN

(F (X, u) − gu)dX

(1.7)

One can check that

d

dtΦ(u(t)) = −kutk2L2 (R N )

for any solution u of problem (1.1) This Lyapunov function plays an essential role

in proving the global existence of solutions and it implies the structure of the global attractor obtained later

The structure of the paper is as follows In Section 2, we prove the existence of

a global mild solution to problem (1.1) by using the existence theorem for abstract parabolic equations and the Lyapunov function Φ Then we can construct the semi-group S(t) generated by problem (1.1) In Section 3, we prove the existence of a compact global attractor for S(t) by showing the existence of a bounded absorb-ing set and the asymptotic compactness of the semigroup S(t) The existence of bounded absorbing sets can be quite easily deduced by using a priori estimates of solutions, while the proof of the asymptotic compactness is much more involved To prove the asymptotic compactness of S(t) in L2(RN) we exploit the tail-estimates method introduced by B Wang [24] to overcome the difficulty arising due to the lack of compactness of the embedding theorems, while the asymptotic compactness

of S(t) in S1

(RN) is proved by combining the asymptotic compactness in L2

(RN), the singular Gronwall inequality and the arguments introduced by Prizzi and Ry-bokowski in [18, 19] It is noticed that the results obtained in the paper are also true for problem (1.1) in an arbitrary (bounded or unbounded) domain Ω in RN, not necessary the whole space RN, with the homogeneous Dirichlet boundary con-dition Then, instead of S1

(RN) and S2

(RN), we use the spaces S1(Ω) and S2(Ω), defined as the completions of C0∞(Ω) in the corresponding norms

2 Existence and uniqueness of a global mild solution

First, we prove the following embedding results

Lemma 2.1 The following embeddings are continuous:

i) S1

(RN) ,→ Lp

(RN) for all 2 ≤ p ≤ 2∗α,β := 2Nα,β

Nα,β− 2, where Nα,β =

N1+ N2+ (α + β + 1)N3;

ii) S2

(RN) ,→ S1

(RN)

Proof i) We follow the ideas in the case of bounded domains (see the proofs of Theorem 3.3 and Proposition 3.2 in [12]) More precisely, we first embed S1

(RN)

into an anisotropic Sobolev-type space, and then use an embedding theorem for classical anisotropic Sobolev-type spaces of fractional orders Because the proof is very similar to the case of bounded domains [12], so we omit it here

ii) For all u ∈ C∞

0 (RN), we have kuk2

S 1 (R N )=

Z

RN

(|u|2+ |∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2)dX

=

Z

RN

u(u − Pα,βu)dX ≤ C

Z

RN

|u|2dX

1/2Z

RN

(|u|2+ |Pα,βu|2)dX

1/2

= CkukL2 (R N )kukS2 (R N )

Noting that kukL2 (R N ) ≤ kukS1 (R N ), we get kukS1 (R N ) ≤ CkukS2 (R N ), that is, the embedding S2

(RN) ,→ S1

Next, we discuss on the operator A = −Pα,β+ λu First, one can check that A

is a positive definite sectorial operator on the Hilbert space X = L2(RN) Thus,

we can define fractional powers of A on X = L2(RN) as follows (see [11, Chapter 1]):

For θ > 0, we define, as usual, the operator A−θ: X → X as

A−θu = 1

Γ(θ)

Z ∞ 0

tθ−1e−Atudt, u ∈ X

Then A−θ is injective and we define Xθ to be the range of A−θ, and Aθ: Xθ→ X

to be the inverse of A−θ We also set X0= X and A0= IdX

We know that Xθ= D(Aθ) is a separable Hilbert space endowed with the inner scalar product

(u, v)Xθ = (Aθu, Aθv), kukXθ = kAθukX For any θ, η > 0 and θ > η, Xθis continuously embedded into Xη

From the above definition, one can see that

X1= {u ∈ S1(RN) : Au ∈ L2(RN)} = S2(RN),

X1/2= S1(RN)

We have the following basic estimate (see e.g [11, Theorem 1.4.3, p 26]):

kAθe−Atk 6 Cθt−θe−δt for all t > 0, (2.1) Now we deal with the nonlinear term f (u)

Lemma 2.2 Suppose (F) − (G) hold Then, there exists 0 ≤ γ < 1

2 such that the map ˆf : X1/2 = S1

(RN) → X−γ is Lipschitzian continuous on every bounded subset of S1

(RN)

Proof We consider two cases:

Case 1: 0 < ρ ≤ N 2

α,β −2 In this case it is easy to check that the map ˆf :

S1

(RN) → L2

(RN) is Lipschitzian continuous on every bounded subset of S1

(RN)

by using H¨older inequality and the embedding S1(RN) ,→ L2(ρ+1)(RN)

Case 2: N 2

α,β −2 < ρ < N 4

α,β −2 We will show that ˆf : S1

(RN) → Lq

(RN), where

q := 2

∗

α,β

ρ + 1, is Lipschitzian continuous on every bounded subset of S

1

(RN)

By (1.3) we have |f (X, u)| ≤ C(|a||u| + |u|ρ+1) Hence for u ∈ S1(RN), by H¨older’s inequality, we have

Z

RN

|f (X, u)|qdX ≤ C

Z

RN

(|a|q|u|q+ |u|q(ρ+1))dX

≤ Ckakq

L2∗α,β/ρ(R N )kukq

L2∗α,β (R N )+ kuk2

∗ α,β

L2∗α,β (R N )



< +∞

since S1(RN) is continuously embedded into L2∗α,β(RN) and the assumption of a This shows that ˆf is a map from S1

(RN) to Lq

(RN)

Let u, v ∈ S1

(RN) and kukS1 (R N )≤ R, kvkS1 (R N )≤ R, we have from (1.3) Z

RN

|f (X, u) − f (X, v)|qdX ≤ C

Z

RN

|u − v|q(|a|q+ |u|qρ+ |v|qρ)dX

≤ C

Z

RN

|a|q|u − v|qdX + C

Z

RN

|u|qρ|u − v|qdX + C

Z

RN

|v|qρ|u − v|qdX Applying H¨older’s inequality, we have

Z

RN

|a|q|u − v|qdX ≤ kakqρ

L2∗α,β/ρ(R N )ku − vkq

L2∗α,β (R N ), Z

RN

|u|qρ|u − v|qdX ≤ kukqρ

L2∗α,β (R N )ku − vkq

L2∗α,β (R N ), Z

RN

|v|qρ|u − v|qdX ≤ kvkqρ

L2∗α,β (R N )ku − vkq

L2∗α,β (R N ) Since S1(RN) is continuously embedded into L2∗α,β(RN) and 1 < q < 2∗α,β, there exists a positive number M (R) such that

k ˆf (u) − ˆf (v)kLq (R N )≤ M (R)ku − vkS1 (R N ) Since N 2

α,β −2 < ρ < N 4

α,β −2, we have 1 < q < 2 Thus,

p = q

q − 1 =

2∗α,β

2∗α,β− 1 ∈ (2, 2

∗ α,β)

Repeating the arguments used in the proof of Lemma 3.1 in [1], just replacing N (k)

by Nα,β, one can show that there exists γ ∈ (0, 1/2) such that Xγ is continuously embedded in Lp

(RN) Hence Lq

(RN) = (Lp

(RN))0 is continuously embedded in

X−γ

Since both L2

(RN) and Lq

(RN) are continuously embedded in X−γ, we get the

We are now ready to prove the main result in this section

Theorem 2.1 Suppose (F) − (G) hold Then for any u0∈ S1

(RN) given, problem (1.1) has a unique global mild solution u ∈ C([0, ∞); S1

(RN))

Proof Because ˆf : X1/2 → X−γ is Lipschitzian continuous on every bounded subset of X1/2, by Theorem 3.3.3 in [11, p 54], we get the local existence of a mild solution u

Suppose that the solution u is defined on the maximal interval [0, tmax) We now show that tmax= +∞ by using the Lyapunov function (1.7) and the dissipativeness condition (1.6) Using (1.6) and the Cauchy inequality we get

Φ(u(t)) ≥ 1

2

Z

RN



|u(t)|2+ |∇xu(t)|2+ |∇yu(t)|2+ |x|2α|y|2β|∇zu(t)|2

 dX

+λ

2ku(t)k2

L 2 (R N )−µ

2ku(t)k2

L 2 (R N )− kC2kL1 (R N )− εku(t)k2

L 2 (R N )− 1

4εkgk2

L 2 (R N ) Choosing ε small enough such that µ + 2 < λ we obtain

Φ(u(0)) ≥Φ(u(t))

≥1

2

Z

RN



|u(t)|2+ |∇xu(t)|2+ |∇yu(t)|2+ |x|2α|y|2β|∇zu(t)|2

 dX

+λ − µ − 2ε

2 ku(t)k2

L 2 (R N )− 1

4εkgk2

L 2 (R N )− kC2kL1 (R N ) Hence

ku(t)kS1 (R N )≤ M ∀t ∈ [0, tmax)

This implies that tmax= +∞ Indeed, let tmax< ∞ and lim supt→t−

maxku(t)kS1 (R N )< +∞ Then there exist a sequence (tn)n≥1and a constant K such that tn→ t−

maxas

n → +∞ and ku(tn)kS1 (R N )< K, n = 1, 2, As we have already shown above, for each n ∈ N there exists a unique mild solution of problem (1.1) with the initial da-tum u(tn) on [tn, tn+ T∗], where T∗> 0 depending on K and independent of n ∈

N Thus, we can get tmax< tn+ T∗, for n ∈ N large enough This contradicts the

3 Existence of a global attractor in S1

(RN)

By Theorem 2.1, we can define a continuous semigroup S(t) : S1(RN) → S1(RN)

as follows

S(t)u0:= u(t), where u(t) is the unique global mild solution of (1.1) subject to u0as initial datum 3.1 Existence of bounded absorbing sets For the sake of brevity, in the following lemmas, we give some formal calculations, the rigorous proof is done by use of Galerkin approximations and Lemma 11.2 in [20]

We first prove the existence of a bounded absorbing set for S(t) in S1

(RN) Lemma 3.1 Suppose (F) − (G) hold Then the semigroup S(t) generated by (1.1) has a bounded absorbing set in S1

(RN), that is, there exists a positive constant ρ, such that for every bounded subset B in S1

(RN), there is a number T = T (B) > 0, such that for all t ≥ T , u0∈ B, we have

ku(t)kS1 (R N )≤ ρ

Proof Taking the inner product of (1.1) with u in L2(RN) we get

1

2

d

dtkuk2

L 2 (R N )+

Z

RN

(|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2)dX + λkuk2L2 (R N )+

Z

N

f (X, u)udX = (g, u)L2 (R)

(3.1)

Using (1.5), we have

Z

RN

f (X, u)udX ≥ −µ

Z

RN

|u|2dX −

Z

RN

C1(X)dX (3.2)

By the Cauchy inequality, the right-hand side of (3.1) is estimated as follows

|(g, u)L2 (R)| ≤ kgkL2 (R N )kukL2 (R N )≤λ − µ

2 kuk2

L 2 (R N )+ 1

2(λ − µ)kgk2

L 2 (R N ) (3.3)

It follows from (3.1)-(3.3) that

d

dtkuk2L2 (R N )+ 2

Z

RN

(|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2)dX + (λ − µ)kuk2L2 (R N )

≤ 2kC1kL1 (R N )+ 1

λ − µkgk2

L 2 (R N )

(3.4) Hence, in particular, we have

d

dtku(t)k2

L 2 (R N )+ (λ − µ)ku(t)k2≤ 2kC1kL1 (R N )+ 1

λ − µkgk2

L 2 (R N )

Using the Gronwall inequality, we obtain

ku(t)k2

L 2 (R N )≤ e−(λ−µ)tku0k2

L 2 (R N )+2kC1kL1 (R N )

1 (λ − µ)2kgk2

L 2 (R N )



Hence we deduce the existence of a bounded absorbing set in L2

(RN): There are a constant R and a time t0(ku0kL2 (R N )) such that for the solution u(t) = S(t)u0,

ku(t)kL 2 (R N )≤ R for all t ≥ t0(ku0kL 2 (R N )) (3.5)

Integrating (3.4) on (t, t + 1), t ≥ t0(ku0kL2 (R N )), and using (1.6), we find that

Z t+1

t

Z

RN

|∇xu(s)|2+ |∇yu(s)|2+ |x|2α|y|2β|∇zu(s)|2
dXds

+(λ − µ)

Z t+1 t

ku(s)k2

L 2 (R N )ds ≤ Cku(t)k2

L 2 (R N )+ 1 + kgk2L2 (R N )



≤ CR2+ 1 + kgk2L2 (R N )



(3.6)

On the other hand, multiplying (1.1) by −Pα,βu and integrating over RN, then integrating by parts and using (1.4), we obtain

d

dt

Z

RN

|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2
dX+

Z

RN

|Pα,βu|2dX + λ

Z

RN

|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2
dX

=

Z

RN

f (X, u)Pα,βudX −

Z

RN

gPα,βudX

= −

Z

RN

fu0(X, u) |∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2
dX −

Z

RN

gPα,βudX

≤`

Z

RN

|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2
dX +

Z

RN

|Pα,βu|2dX

+1

4

Z

RN

|g|2dX

(3.7) Hence

d

ds

Z

RN

(|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2)dX

≤ `

Z

RN

|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2
dX +1

4kgk2L2 (R N )

(3.8)

Combining (3.6), (3.8), and applying the uniform Gronwall inequality, we have Z

RN

(|∇xu(t)|2+ |∇yu(t)|2+ |x|2α|y|2β|∇zu(t)|2)dX ≤ CR2+ 1 + kgk2L2 (R N )



(3.9) for all t ≥ t0 Using (3.5), we finish the proof 

We now derive uniform estimates of the derivative of solutions in time

Lemma 3.2 Suppose (F)−(G) hold Then for every bounded subset B in S1(RN), there exists a constant T = T (B) > 0 such that

kut(s)k2L2 (R N )≤ ρ1 for all u0∈ B, and s ≥ T, where ut(s) = dtd(S(t)u0)|t=s and ρ1 is a positive constant independent of B Proof By differentiating (1.1) in time and denoting v = ut, we get

∂v

∂t − Pα,βv + λv + ∂f

∂u(X, u)v = 0.

Taking the inner product of the above equality with v in L2

(RN), we obtain 1

2

d

dtkvk2

L 2 (R N )+

Z

RN



|∇xv|2+ |∇yv|2+ |x|2α|y|2β|∇zv|2dX +λkvk2L2 (R N )+

Z

RN

∂f

∂u(X, u)|v|

2dX = 0

By (1.4), it follows that

d

dtkvk2

L 2 (R N )≤ 2`kvk2

On the other hand, integrating (3.7) from t to t + 1 and using (3.9), we obtain

Z t+1 t

kut(s)k2L2 (R N )ds ≤ C(ρ, kgk2L2 (R N )) (3.11)

as t large enough Combining (3.10) with (3.11), and using the uniform Gronwall inequality, we have

kut(s)k2L2 (R N )≤ C(ρ, kgk2

L 2 (R N ))

We now show the existence of a bounded absorbing set in S2

(RN)

Lemma 3.3 Let assumptions (F) − (G) hold Then there exists a positive number

R such that for any solution u of problem (1.1) we have

kPα,βuk2L2 (R N )+ kuk2L2 (R N )≤ R2 (3.12) Proof Taking the L2-inner product of (1.1) with −Pα,βu + u, we have

kPα,βuk2L2 (R N )+ λkuk2L2 (R N )+

Z

RN

f (X, u)udX

≤ −

Z

RN

ut − Pα,βu + u
dX + (λ + 1)

Z

RN

uPα,βudX −

Z

RN

f (X, u)Pα,βudX +

Z

RN

g − Pα,βu + u
dX

Using (1.5) and integrating by parts, we get

kPα,βuk2L2 (R N )+ (λ − µ)kuk2L2 (R N )

≤ −

Z

RN

ut − Pα,βu + u
dX − (λ + 1)

Z

RN

(|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu|2)dX

−

Z

RN

fu0(X, u)(|∇xu|2+ |∇yu|2+ |x|2α|y|2β|∇zu)|2)dX +

Z

RN

g − Pα,βu + u
dX + kC1kL1 (R N )

Using (1.4) and the Cauchy inequality, we have

kPα,βuk2L2 (R N )+kuk2L2 (R N )≤ Ckutk2L2 (R N )+kuk2S1 (R N )+kgk2L2 (R N )+kC1kL 1 (R N )



3.2 Asymptotic compactness of the semigroup S(t) In order to prove the existence of a global attractor, we mainly have to prove the asymptotic compactness

of S(t) Firstly, in Lemma 3.4 below, we prove a tail-estimate of solutions Then, using this estimate, we prove the asymptotic compactness of S(t) in L2

(RN) Fi-nally, using the singular Gronwall inequality and properties of analytic semigroup generated by sectorial operators, we obtain the asymptotic compactness of the semigroup S(t) in S1

(RN)

Lemma 3.4 Let ¯ϑ ∈ C1([0, ∞)) be a function such that ¯ϑ ∈ [0, 1] and

¯ ϑ(s) = 0 for s ∈ [0, 1] and ¯ϑ(s) = 1 for s ∈ [2, ∞)

Setting ϑ = ¯ϑ2 and the functions ¯ϑk: RN −→ R and ϑk: RN −→ R be defined by

¯

ϑk(X) = ¯ϑ |X|2)

k2

! and ϑk(X) = ϑ |X|2

k2

! for any k ∈ N

Then whenever u : [0, ∞) −→ S1(RN) is a solution of (1.1), for every η > 0 there exist two constants K0> 0 and T0> 0 we have

Z

RN

ϑk(X)|u(t, X)|2dX < η for all t ≥ T0, k ≥ K0 (3.13)

Proof We follow the arguments in the proof of Theorem 6.5 in [19] Since ∇ϑk(X) = 2

k2ϑ0 |X|2

k2



X and ∇ ¯ϑk(X) = 2

k2ϑ¯0 |X|2

k2



X, we have

sup

X∈R N

|∇ϑk(X)| ≤ Cϑ

k and X∈RsupN

|∇ ¯ϑk(X)| ≤ Cϑ¯

k . Let Vk: S1

(RN

) −→ R+ be a function defined by

Vk(u) = 1

2 Z

RN

ϑk(X)|u(X)|2dX, u ∈ S1(RN)

It is easy to check that Vk is Fr´echet differentiable and

(Vk(u))0(v) =

Z

RN

ϑk(X)u(X)v(X)dX = (ϑku, v)L2 (R N ), u, v ∈ S1(RN)

Therefore, if u : [0, ∞) −→ S1(RN) is a positive semi-trajectory of the semigroup S(t), we have

(Vk◦ u)0(t) =

Z

RN

ϑk(X)u(t)ut(t)dX = (ut(t), ϑku(t))L2 (R N ) Let q be a positive number such that q + µ < λ We have

(Vk◦ u)0(t) + 2q(Vk◦ u)(t) = (ut, ϑku(t))L2 (R N )+ 2q(Vk◦ u)(t)

=(Pα,βu, ϑk(X)u)L2 (R N )− λ

Z

RN

ϑk(X)|u|2dX −

Z

RN

f (X, u)ϑk(X)udX

−

Z

RN

gϑk(X)udX + q

Z

RN

ϑk(X)|u|2dX

≤(Pα,βu, ϑku) + (q + µ − λ)

Z

RN

ϑk(X)|u|2dX

−

Z

RN

g(X)ϑk(X)udX +

Z

RN

C1(X)ϑk(X)dX, where we have used (1.5) Since

(Pα,βu, ϑku) ≤

Z

N

ϑk(X) |u|2+ |Pα,βu|2
dX ≤

Z

|X|≥k

|u|2+ |Pα,βu|2
dX,