Nonlinear Analysis 71 (2009) 697–707

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Nonlinear Analysis

journal homepage: www.elsevier.com/locate/na

Standing waves and global existence for the nonlinear wave equationwith potential and damping termsI

Yi Jiang, Zaihui Gan ∗, Yiran HeCollege of Mathematics and Software Science, Sichuan Normal University, Chengdu 610068, PR China

a r t i c l e i n f o

Article history:Received 15 July 2008Accepted 4 November 2008

MSC:35A1535L1535Q55

Keywords:Nonlinear wave equationStanding waveInstabilityGround statePotential and damping terms

a b s t r a c t

This paper is concerned with the Cauchy problem of the nonlinear wave equationwith potential and damping terms. Firstly, by using intricate variational arguments, theexistence of a standing wave with the ground state is obtained. Then the instability ofthe standing wave is shown by applying potential well arguments and concavity methods.Finally, we answer the question of how small the initial data are for the global solutions toexist.

© 2008 Elsevier Ltd. All rights reserved.

Contents

1. Introduction............................................................................................................................................................................................. 6972. Preliminaries ........................................................................................................................................................................................... 6983. Existence of a standing wave with ground state................................................................................................................................... 6994. Instability of a standing wave ................................................................................................................................................................ 7035. Global existence ...................................................................................................................................................................................... 706

References................................................................................................................................................................................................ 707

1. Introduction

Consider the Cauchy problem for the nonlinear wave equation with potential and damping terms

φtt −1φ + V (x)φ + φt |φt |m−1 = φ|φ|p−1, t ≥ 0, x ∈ RN , (1.1)with initial data

φ(0, x) = φ0(x), φt(0, x) = φ1, x ∈ RN , (1.2)where p > 1,m ≥ 1 and φ(t, x) is a complex-valued function of (t, x) ∈ R+ × RN . With the absence of the dampingterm φt |φt |m−1, Eq. (1.1) can be viewed as an interaction between one or more discrete oscillators and a field or continuousmedium [10].

I This work is supported by the National Natural Science Foundation of PR China (10801102, 10771151 and 10726034), Sichuan Youth Sciences andTechnology Foundation (07ZQ026-009).∗ Corresponding author.E-mail address: ganzaihui2008cn@yahoo.com.cn (Z. Gan).

0362-546X/$ – see front matter© 2008 Elsevier Ltd. All rights reserved.doi:10.1016/j.na.2008.11.019

698 Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707

For the case of linear damping (m = 1) and nonlinear potential, Levine [1] showed that the solutions to (1.1) withnegative initial energy blow up for the abstract version. Zhou [2] proved that the solution to (1.1) blows up in finite timeeven for vanishing initial energy if the initial data (φ0, φ1) satisfy

∫RN φ0φ1 ≥ 0 with V (x) = 0. For the nonlinear damping

and potential terms, the abstract version was studied by Levine and Serrin [3] as well as Levine, Pucci and Serrin [4]. Severalauthors (Ikehata [5], Ohta [6], Vitillaro [7]) obtained some blow-up results for the case of bounded domain and V (x) = 0.Todorova [8] proved that the solutions to the Cauchy problem (1.1) and (1.2) blow up in finite time if the initial energy isnegative, Np/(N + p+ 1) < m < p and V (x) = 0.But to our knowledge, little work on the existence and instability of the standing wave for (1.1) has been carried out. In

this paper, we first study the existence of a standing wave with ground state, which is the minimal action solution of thefollowing elliptic equation

1u− V (x)u+ u|u|p−1 = 0. (1.3)

Next, in terms of the characterization of the ground state and the local well-posedness theory [9], we are interested instudying the instability of the standing wave for the Cauchy problem (1.1) and (1.2). Finally, we derive a sufficient conditionof global existence of solutions to the Cauchy problem (1.1) and (1.2) by using the relation between initial data and theground state solution of (1.3). It should be pointed out that these results in the present paper are unknown to Eq. (1.1) before.For simplicity, throughout this paper we denote

∫RN · dx by

∫· dx and arbitrary positive constants by C .

2. Preliminaries

We define the energy space H in the course of nature as

H :={ϕ ∈ H1(RN),

∫V (x)|ϕ|2 dx <∞

}. (2.1)

By the definition of H , H becomes a Hilbert space, continuously embedded in H1(RN), when endowed with the innerproduct as follows:

〈ϕ, φ〉H :=

∫(∇ϕ∇φ + V (x)ϕφ) dx, (2.2)

whose associated norm we denote by ‖ · ‖H . If ϕ ∈ H , then ‖ϕ‖H =∫|∇ϕ|2 dx+

∫V (x)|ϕ|2 dx.

Throughout this paper, we make the following assumptions on V (x):

(A)

infx∈RNV (x) = V > 0,

V (x) is positive and a C1 bounded measurable function on RN ,limx→∞

V (x) = ∞.

FromTodorova [9], we can get the localwell-posedness of the solution to the Cauchy problem (1.1) and (1.2) in the energyspace H , namely:

Proposition 2.1. Let m ≥ 1 and

1 < p ≤ N/N − 2 if N > 2 and 1 < p <∞ if N ≤ 2. (2.3)

Then, for any compactly supported data,

φ0 ∈ H, φ1 ∈ L2(RN), (2.4)

the Cauchy problem (1.1) and (1.2) has a unique solution such that

φ(t, x) ∈ C([0, T );H),φt(t, x) ∈ C([0, T ); L2(RN)) ∩ Lm+1([0, T )× RN)

provided T is small enough.

Furthermore, we have that ∀t ∈ [0, T ), φ(t, x) satisfies the conservation of energy:

E(t)+∫ t

0‖φt(s)‖m+1Lm+1

ds = E(0), (2.5)

where E(t) is defined as follows:

E(t) =12

∫|φt |

2 dx+12

∫|∇φ|2 dx+

12

∫|φ0|

2 dx−1p+ 1

∫|φ|p+1 dx. (2.6)

Moreover, we give a result in [8].

Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707 699

Proposition 2.2. Assume p ≤ m and let conditions (2.3) and (2.4) be fulfilled. Then the Cauchy problem (1.1) and (1.2) has aunique global solution such that

φ(t, x) ∈ C([0, T );H),φt(t, x) ∈ C([0, T ); L2(RN)) ∩ Lm+1([0, T )× RN)

for any T > 0.

Remark 2.1. FromProposition 2.2, it follows thatm = p is the critical case, namely for p ≤ m, aweak solution exists globallyin time for any compactly supported initial data; while form < p, blow-up of the solution to the Cauchy problem (1.1) and(1.2) occurs.

At the end of this section, we state the main results of the present paper.

Theorem 2.1. There exists D ∈ M such that:(1) J(D) = infD∈M J(u) = d;(2) D is a ground state solution of (1.3).Here, J(u) and M will be defined in Section 3.

Theorem 2.2. Let D be a ground state solution of (1.3). Then for any ε > 0, there exists φ0 ∈ H such that

‖φ0 − D‖H < ε.

Moreover, the following property holds: the solution φ(t, x) of the Cauchy problem for (1.1) corresponding to the compactlysupported initial data

φ(0, x) = φ0(x), φt(0, x) = 0 (2.7)

is defined for 0 < T <∞ such that

φ(t, x) ∈ C([0, T );H),φt(t, x) ∈ C([0, T ); L2(RN)) ∩ Lm+1([0, T )× RN),1 < p ≤ N/(N − 2), 1 ≤ m < p,

and

limt→T‖φ‖H = ∞. (2.8)

Theorem 2.3. Let 1 < p ≤ N/(N − 2) and 1 ≤ m < p. If (φ0, φ1) ∈ H × L2(RN) and satisfies∫|φ1|

2 dx+∫|∇φ0|

2 dx+∫V (x)|φ0|2 dx <

p− 1p+ 1

∫|D|p+1 dx, (2.9)

then the solution φ(t, x) of the Cauchy problem (1.1) and (1.2) globally exists. Furthermore, for all t ∈ [0,∞) one has

‖φt(t)‖2L2(RN ) +p− 1p+ 1

‖φ(t)‖2H <p− 1p+ 1

‖D‖2H . (2.10)

Remark 2.2. Theorem 2.2 shows the instability of the standing wave for (1.1) with minimal action. In fact, this theoremshows that for any neighborhood in H , the solution φ(t, x) of (1.1) with initial data (2.7) goes away from the orbit of thestanding wave associated with the ground state D to infinity in a finite time.

3. Existence of a standing wave with ground state

In this section, in order to prove Theorem 2.1, we will use the following definitions and lemmas.For u ∈ H \ {0}, 1 < p ≤ N/(N − 2) and 1 ≤ m < p. We define two functionals J(u) and K(u) as

J(u) :=12

∫|∇u|2 dx+

12

∫V (x)|u|2 dx−

1p+ 1

∫|u|p+1 dx, (3.1)

K(u) :=∫|∇u|2 dx+

∫V (x)|u|2 dx−

∫|u|p+1 dx, (3.2)

and define a manifoldM as

M := {u ∈ H \ {0}, K(u) = 0}. (3.3)

700 Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707

Then we consider the constrained variational problemd := inf{sup

λ≥0J(λu) : K(u) < 0, u ∈ H \ {0}}, (3.4)

and obtain the following result, namely.

Lemma 3.1. The constrained variational problem

d := inf{supλ≥0J(λu) : K(u) < 0, u ∈ H \ {0}}

is equivalent to

d1 := inf{supλ≥0J(λu) : K(u) = 0, u ∈ H \ {0}}

= infu∈MJ(u), (3.5)

and d > 0 provided 1 < p ≤ N/(N − 2) as well as 1 ≤ m < p.Proof. Let u ∈ H \ {0}. Since

J(λu) =λ2

2

∫|∇u|2 dx+

λ2

2

∫V (x)|u|2 dx−

λp+1

p+ 1

∫|u|p+1 dx, (3.6)

we have

ddλJ(λu) = λ

∫|∇u|2 dx+ λ

∫V (x)|u|2 dx− λp

∫|u|p+1 dx. (3.7)

Thus by 1 ≤ m < p, one gets that there exists some λ1 ≥ 0 such that

supλ≥0J(λu) = J(λ1u) = λ21

(12

∫|∇u|2 dx+

12

∫V (x)|u|2 dx−

λp−11

p+ 1

∫|u|p+1 dx

), (3.8)

where λ1 uniquely depends on u and satisfies

− λp−11

∫|u|p+1 dx+

∫|∇u|2 dx+

∫V (x)|u|2 dx = 0. (3.9)

Sinced2

dλ2J(λu) =

∫|∇u|2 dx+

∫V (x)|u|2 dx− pλp−1

∫|u|p+1 dx,

which together with p > 1 and (3.9) implies that d2

dλ2J(λu)|λ=λ1 < 0, then

supλ≥0J(λu) =

p− 12(p+ 1)

(∫|∇u|2 dx+

∫V (x)|u|2 dx)

p+1p−1

(∫|u|p+1 dx)

2p−1

. (3.10)

Therefore, the above estimates lead to

d = inf{supλ≥0J(λu) : K(u) < 0, u ∈ H \ {0}}

= inf

p− 12(p+ 1)

(∫|∇u|2 dx+

∫V (x)|u|2 dx)

p+1p−1

(∫|u|p+1 dx)

2p−1

: K(u) < 0

. (3.11)

It is easy to see that

d1 = inf{supλ≥0J(λu) : k(u) = 0, u ∈ H \ {0}}

= infu∈M

{p− 12(p+ 1)

∫|∇u|2 dx+

∫V (x)|u|2 dx

}. (3.12)

From (3.1)–(3.3), onM one has

J(u) =p− 12(p+ 1)

∫|∇u|2 dx+

∫V (x)|u|2 dx. (3.13)

It follows that d1 = infu∈M J(u) and J(u) > 0 onM .

Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707 701

Now we establish the equivalence of the two minimization problems (3.4) and (3.5).For any u0 ∈ H and K(u0) < 0, let uβ(x) = βu0. Thus there exists a β0 ∈ (0, 1) such that K(uβ0) = 0 and from (3.10) we

get

supλ≥0J(λuβ0) =

p− 12(p+ 1)

(∫|∇uβ0 |

2 dx+∫V (x)|uβ0 |

2 dx)p+1p−1

(∫|uβ0 |p+1 dx)

2p−1

=p− 12(p+ 1)

β2(p+1)p−10 (

∫|∇u0|2 dx+

∫V (x)|u0|2 dx)

p+1p−1

β2(p+1)p−10 (

∫|u0|p+1 dx)

2p−1

=p− 12(p+ 1)

(∫|∇u0|2 dx+

∫V (x)|u0|2 dx)

p+1p−1

(∫|u0|p+1 dx)

2p−1

. (3.14)

Consequently the two minimization problems (3.11) and (3.12) are equivalent, that is, (3.4) and (3.5) are equivalent.Finally, we prove d > 0 by showing d1 > 0 in terms of the above equivalence. Since 1 < p ≤ N/(N − 2), the Sobolev

embedding inequality yields that∫|u|p+1 dx ≤ C

(∫|∇u|2 dx+

∫V (x)|u|2 dx

) p+12

. (3.15)

From K(u) = 0, it follows that∫|∇u|2 dx+

∫V (x)|u|2 dx =

∫|u|p+1 dx ≤ C

(∫|∇u|2 dx+

∫V (x)|u|2 dx

) p+12

, (3.16)

which together with p > 1 implies∫|∇u|2 dx+

∫V (x)|u|2 dx ≥ C > 0. (3.17)

Therefore, from (3.13), we get

J(u) ≥ C > 0, for u ∈ M.

Thus the equivalence of the two minimization problems (3.4) and (3.5) concludes that d > 0 for 1 < p ≤ N/(N − 2).This completes the proof of Lemma 3.1. �

Lemma 3.2. Let u ∈ H \ {0} and uλ(x) = λu(x) for λ > 0. Then there exists a unique µ > 0 (depending on u) such thatK(uµ) = 0. Moreover,

K(uλ) > 0 for λ ∈ (0, µ);K(uλ) < 0 for λ ∈ (µ,∞);∀λ > 0, J(uµ) ≥ J(uλ).

Proof. By (3.1) and (3.2), we have

J(λu) =λ2

2

∫|∇u|2 dx+

λ2

2

∫V (x)|u|2 dx−

λp+1

p+ 1

∫|u|p+1 dx,

K(λu) = λ2∫|∇u|2 dx+ λ2

∫V (x)|u|2 dx− λp+1

∫|u|p+1 dx.

From the definition of M (M is not empty), there must exist a unique µ > 0 such that K(uµ) = 0. Moreover, from theexpression of K(uµ) and p > 1, we obtain

K(uλ) > 0 for λ ∈ (0, µ);K(uλ) < 0 for λ ∈ (µ,∞).

SinceddλJ(uλ) = λ−1K(uλ),

702 Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707

noting that K(uµ) = 0, we get

J(uµ) ≥ J(uλ), ∀λ > 0.

This completes the proof of Lemma 3.2. �

In the following, we begin to prove Theorem 2.1.

Proof of Theorem 2.1. From Lemma 3.1, we may let {un : n ∈ N} ⊂ M be a minimizing sequence for (3.5), that is

J(un)→ infu∈MJ(u) = d as n→∞,

K(un) = 0.(3.18)

By (3.12) and (3.18), we know that ‖un‖H is bounded for all n ∈ N. Then there exists a subsequence {unk : k ∈ N} ⊂ {un :n ∈ N} such that

unk ⇀ ue weakly in H.

For simplicity, we still denote {unk : k ∈ N} by {un : n ∈ N}. Now we need to use a compactness lemma (see [11]), that is:If 1 ≤ q < N+2

N−2 (1 ≤ q < ∞ for N = 1, 2) and limx→∞ V (x) = ∞, then the embedding H ↪→ Lq+1(RN) is compact. So wehave

un → ue strongly in L2(RN) as n→∞,

un → ue strongly in Lp+1(RN) as n→∞.(3.19)

Now we assert that ue 6= 0, which will be proved by contradiction.If ue = 0, then

un → 0 strongly in L2(RN) as n→∞,un → 0 strongly in Lp+1(RN) as n→∞.

Since un ∈ M , K(un) = 0 and assumption (A) imply that∫|∇un|2 dx+

∫V (x)|un|2 dx→ 0, as n→∞. (3.20)

On the other hand, by the Sobolev embedding inequality and (3.20), we get∫|un|p+1 dx ≤ C

(∫|∇un|2 dx+

∫V (x)|un|2 dx

) p+12

.

Thus K(un) = 0 concludes∫|∇un|2 dx+

∫V (x)|un|2 dx ≥

(1C

) 2p−1

,

which is contradictory to (3.20), so ue 6= 0.Now we take D = (ue)µ with µ > 0, which is uniquely determined by the condition K(D) = K((ue)µ) = 0. It is easy to

see that

lunµ → D strongly in L2(RN) as n→∞,

unµ → D strongly in Lp+1(RN) as n→∞,unµ ⇀ Dweakly in H as n→∞.

From K(un) = 0 and Lemma 3.2, we get

J((un)µ) ≤ J(un).

Thus

J(D) ≤ limn→∞

J((un)µ) ≤ limn→∞

J(un) = infu∈MJ(u). (3.21)

As D 6= 0 and K(D) = 0, one has D ∈ M . Therefore, by (3.21), D solves the minimization problem

J(D) = minu∈MJ(u) =

p− 12(p+ 1)

‖D‖2H . (3.22)

Thus we proved (1) of Theorem 2.1. In the following, we prove (2) of Theorem 2.1.

Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707 703

Since D is a solution of problem (3.22), there exists a Lagrange multiplierΛ such that

δD[J +ΛK ] = 0, (3.23)

where δuG(u) denotes the variation of G(u). By the formula

δuG(u) =∂

∂ηG(u+ ηδu)

∣∣∣∣η=0,

we get

δu[J +ΛK ] = (1+ 2Λ)∫(−1u · δu+ V (x)uδu) dx− [1+ (p+ 1)Λ]

∫u|u|p−1δu dx,

where δu denotes the variation of u. By (3.23), one has

(1+ 2Λ)∫(|∇D|2 + V (x)|D|2) dx− [1+ (p+ 1)Λ]

∫|D|p+1 dx = 0. (3.24)

From K(D) = 0 and (3.24), we getΛ = 0. Thus (3.24) implies that

1D− V (x)D+ D|D|p−1 = 0.

Therefore D is a solution of (1.3). Noting (3.22), D is a ground state solution of (1.3). �

4. Instability of a standing wave

In this section, we prove Theorem 2.2. Firstly, we give several lemmas which are crucial for the proof of Theorem 2.2.

Lemma 4.1. Let E(0) < J(D). Define

R1 := {u ∈ H, K(u) < 0, J(u) < J(D)},R2 := {u ∈ H, K(u) > 0, J(u) < J(D)},

then R1 and R2 are invariant manifolds under the flow generated by the Cauchy problem (1.1) and (1.2).

Proof. Let φ0 ∈ R1 and φ(t, x) is a solution of the Cauchy problem (1.1) and (1.2) on t ∈ [0, T ). From (2.5), (2.6) and (3.1),we have

J(φ) ≤ E(t) = E(0) < J(D). (∗)

Now we show that K(φ) < 0 for any t ∈ [0, T ). Otherwise, by continuity, there is a t0 ∈ [0, T ) such that K(φ(t0, x)) = 0and φ(t0, x) 6= 0. Thus (3.5), (∗) and Theorem 2.1 imply that

inf{supλ≥0J(λφ(t0, x)) : K(φ(t0, x)) = 0, φ(t0, x) ∈ H \ {0}}

= infφ(t0,x)∈M

J(φ(t0, x)) < J(D) = d,

which contradicts the definition of d. Therefore, for all t ∈ [0, T ), K(φ(t, x)) < 0. Thus R1 is invariant.Similarly, we can prove that R2 is also invariant.Now the proof of Lemma 4.1 is completed. �

From Lemma 4.1, we can get the following result.

Lemma 4.2. If φ0 ∈ R1, then for t ∈ [0, T ), the following inequality holds:∫|∇φ|2 dx+

∫V (x)|φ|2 dx >

2(p+ 1)J(D)p− 1

, (4.1)

where φ(t, x) is a solution of the Cauchy problem (1.1) and (1.2) on t ∈ [0, T ).

Proof. According to Lemma 4.1 and (3.2), we have∫|∇φ|2 dx+

∫V (x)|φ|2 dx <

∫|φ|p+1 dx,

which together with Lemma 3.1 yields that

J(D) <p− 12(p+ 1)

(∫|∇φ|2 dx+

∫V (x)|φ|2 dx

).

Thus we conclude the proof of Lemma 4.2. �

704 Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707

Now we prove Theorem 2.2.

Proof of Theorem 2.2. From (2.5)–(2.7), we get

E(0) = J(φ0). (4.2)

Now we take φ0(x) = λD(x), λ > 1. For any ε > 0, one can always take a λwith λ > 1 such that

‖φ0 − D‖H = (λ− 1)‖D‖H < ε.

Since λ > 1, Lemma 3.2 yields thatK(φ0) < K(D) = 0,

J(φ0) < J(D) =p− 12(p+ 1)

(∫|∇D|2 dx+

∫V (x)|D|2 dx

).

(4.3)

From (4.2), it follows that

E(0) < J(D) =p− 12(p+ 1)

(∫|∇D|2 dx+

∫V (x)|D|2 dx

). (4.4)

Therefore (4.3), (4.4) and Lemma 4.1 imply that K(φ) < 0.From [9], we know that for some t1 ≥ 0, E(t1) becomes nonnegative and the solution φ(t, x) of the Cauchy problem (1.1)

and (1.2) blows up in finite time. So we can suppose that E(t) ≥ 0 for all t > 0, which leads to a constant control of the rateof the energy decrease. That is, from the energy identity, we get∫ t

0‖φt(s)‖m+1Lm+1 ds = E(0)− E(t) ≤ d. (4.5)

Put F(t) = ‖φ(t, ·)‖2L2. By simple calculations, we have by (3.2) that

F ′′(t) = 2‖φt‖2L2 − 2K(φ)− 2Re∫φφt |φt |

m−1 dx. (4.6)

In the following, we discuss three cases:

(1) m = p;(2) m = 1;(3) 1 < m < p.

For case (1),m = p. In this case, using the Hölder inequality and Young inequality we get∣∣∣∣∫ φφt |φt |m−1 dx

∣∣∣∣ ≤ ‖φ‖Lm+1‖φt‖mLm+1≤ ε‖φ‖m+1Lm+1 + c(ε)‖φt‖

m+1Lm+1

= ε‖φ‖m+1Lm+1 + c(ε)‖φt‖p+1Lp+1 . (4.7)

For case (2),m = 1. In this case, from K(φ) < 0, we have∫V (x)|φ|2 dx < ‖φ‖p+1

Lp+1.

Using the Young inequality and assumption (A), we obtain∣∣∣∣∫ φφt dx∣∣∣∣ ≤ εV‖φ‖2L2 + c(ε)‖φt‖2L2≤ ε‖φ‖

p+1Lp+1+ c(ε)‖φt‖2L2 . (4.8)

For case (3), 1 < m < p. In this case, applying the Hölder inequality and the so-called interpolation inequality, we obtain byφ ∈ Lp(RN) that∣∣∣∣∫ φφt |φt |

m−1 dx∣∣∣∣ ≤ ‖φ‖Lm+1‖φt‖mLm+1≤ ‖φ‖θL2‖φ‖

1−θLp+1‖φt‖

mLm+1 , (4.9)

Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707 705

where 1m+1 =

θ2 +

1−θp+1 , that is, θ = (

1m+1 −

1p+1 )/(

12 −

1p+1 ). From K(φ) < 0, we have∫

V (x)|φ|2 dx < ‖φ‖p+1Lp+1

,

which together with assumption (A) yields that

‖φ‖θL2‖φ‖1−θLp+1‖φt‖

mLm+1 ≤

1

Vθ2‖φ‖

θ2 (p+1)

Lp+1‖φ‖1−θ

Lp+1‖φt‖

mLm+1

=1

Vθ2‖φ‖

1−θ+ θ2 (p+1)

Lp+1 ‖φt‖mLm+1

≤1

Vθ2‖φt‖

mLm+1‖φ‖

1− p+1m+1−θ+θ(p+1)2

Lp+1 ‖φ‖p+1m+1Lp+1 . (4.10)

Since 1− p+1m+1 − θ +

θ(p+1)2 = 0, we can estimate the right-hand side of (4.10). Using the Young inequality, we obtain

‖φ‖θL2‖φ‖1−θLp+1‖φt‖

mLm+1 ≤ ε‖φ‖

p+1Lp+1+ c(ε)‖φt‖m+1L(m+1)

. (4.11)

From (4.6)–(4.11), we have

F ′′(t)+ c(ε)‖φt‖m+1Lm+1≥ 2‖φt‖2L2 − 2‖∇φ‖

2L2 − 2

∫V (x)|φ|2 dx+ (2− 2ε)‖φ‖p+1

Lp+1+ δ(E(t)− E(0))

=

(2+

δ

2

)‖φt‖

2L2 +

(δ

2− 2

)‖∇φ‖2L2

+

(δ

2− 2

)∫V (x)|φ|2 dx+

(2−

δ

p+ 1− 2ε

)‖φ‖

p+1Lp+1− δE(0), (4.12)

where δ satisfies4(p+ 1)J(D)

(p+ 1)J(D)− (p− 1)E(0)< δ < 2(p+ 1),

which is possible since E(0) < J(D) and guarantees that δ > 4. Thus from Lemma 4.2, we obtain(δ

2− 2

)(‖∇φ‖2L2 +

∫V (x)|φ|2 dx

)− δE(0) ≥

(δ

2− 2

)2J(D)(p+ 1)p− 1

− δE(0) ≥ 0. (4.13)

Once the constant δ is fixed, we can choose the constant ε so small that

C1 = 2−δ

p+ 1− 2ε > 0.

Finally, using inequalities (4.11) and (4.12), and Lemma 4.2 we get

F ′′(t)+ c(ε)‖φt‖m+1Lm+1≥ C1‖φ‖

p+1Lp+1

≥ C1

(‖∇φ‖2L2 +

∫V (x)|φ|2 dx

)≥ C1

2(p+ 1)J(D)p− 1

. (4.14)

Integrating (4.14) over [0, t] and taking into account (4.5), we arrive at

F ′(t) ≥ C1(p+ 1)J(D)p− 1

t − c(ε)d+ F ′(0),

which concludes that there exists a t1 such that F ′(t)|t=t1 > 0, then F(t) is increasing for t > t1 (with the interval ofexistence). Since K(φ) < 0, there exists a t2 such that ‖φ(t, x)‖

p+1Lp+1is increasing for t > t2. When t is large enough, the

quantity ‖φ(t, x)t‖m+1Lm+1is small enough. Otherwise, assume that there is t∗ such that ‖φ(t, ·)t‖m+1Lm+1

> ‖φ(t∗, ·)t‖m+1Lm+1for all

t > t∗. By integrating the inequality, we get a contradiction from (4.5) and E(t) ≥ 0.Thus in these cases, the quantity(

δ

2− 2

)‖∇φ‖2L2 +

(δ

2− 2

)∫V (x)|φ|2 dx+

(2−

δ

p+ 1− 2ε

)‖φ‖

p+1Lp+1− δE(0)− c(ε)‖φt‖m+1Lm+1

706 Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707

will eventually become positive. Therefore for t large enough, from (4.12) and (4.13) we get

F ′′(t) ≥(2+

δ

2

)‖φt‖

2L2 . (4.15)

Using the Hölder inequality, one has

F(t)F ′′(t) ≥4+ δ8[F ′(t)]2. (4.16)

Since

[F−δ−48 (t)]′′ = −

δ − 48F−

δ+128 (t)

[F(t)F ′′(t)−

4+ δ8F ′(t)2

],

from (4.16) we see that [F−δ−48 (t)]′′ ≤ 0. Therefore F−

δ−48 (t) is concave for sufficiently large t , and there exists a finite time

T ∗ such that

limt→T∗

F−δ−48 (t) = 0. (4.17)

From assumption (A), we get∫V (x)|φ|2 dx ≥ V

∫|φ|2 dx. (4.18)

Thus one has T <∞ and

limt→T−‖φ‖H = ∞. (4.19)

Now we completed the proof of Theorem 2.2. �

5. Global existence

In this section, we begin to prove Theorem 2.3.Proof of Theorem 2.3. From (2.5), (2.6) and (2.9), it follows that E(0) < J(D). In the following we show that φ0(x) satisfies

K(φ0) =∫|∇φ0|

2 dx+∫V (x)|φ0|2 dx−

∫|φ0|

p+1 dx > 0, (5.1)

which will be proved by contradiction. If (5.1) is not true, then we have K(φ0) ≤ 0. Thus there exists 0 < µ ≤ 1 such thatφ0 6= 0 and

K(µφ0) = µ2∫|∇φ0|

2 dx+ µ2∫V (x)|φ0|2 dx− µp+1

∫|φ0|

p+1 dx = 0

which imply µφ0 ∈ M .On the other hand, for 0 < µ ≤ 1, (2.9) and (3.1) yield

J(µφ0) < µ2∫|∇φ0|

2 dx+ µ2∫V (x)|φ0|2 dx

≤p− 12(p+ 1)

∫|D|p+1 dx

= J(D)

which is contradictory to Lemma 3.1.Therefore, by K(φ0) > 0 and Lemma 4.1, we get K(φ) > 0 and E(t) ≤ E(0) < J(D). Thus

E(t)−1p+ 1

K(φ) ≤ E(0),

namely

12

∫|φt |

2 dx+p− 12(p+ 1)

(∫|∇φ(t)|2 dx+

∫V (x)|φ(t)|2 dx

)≤ E(0). (5.2)

Therefore we have established the boundedness of φ(t, x) in H for t ∈ [0, T ) and thus the solution φ(t, x) of (1.1) and (1.2)exists globally on t ∈ [0,∞).From (5.1), E(0) < J(D) and Lemma 3.1, we get (2.10).Now the proof of Theorem 2.3 is completed. �

Y. Jiang et al. / Nonlinear Analysis 71 (2009) 697–707 707

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