Existence of homoclinic orbits for unbounded time-dependent p-Laplacian systems

Existence of homoclinic orbits for unbounded time-dependent p-Laplacian systems

Adel Daouas

High School of Sciences and Technology, Hammam Sousse, 4011, Tunisia Received 12 March 2016, appeared 14 September 2016

Communicated by Dimitri Mugnai

Abstract. In this paper, we consider the following ordinaryp-Laplacian system

d

dt |u˙(t)|^p−2u˙(t)− ∇K(t,u(t)) +∇W(t,u(t)) = f(t), (HS) wheret∈ Rand p >1. Using the Mountain Pass Theorem, we establish the existence of a nontrivial homoclinic solution for (HS) under new assumptions on the growth of the potential which allow W(t,x) to be either super p-linear or asymptotically p- linear at infinity. Also, contrary to previous works,W(t,x)will be neither periodic nor bounded with respect to the variablet. Recent results in the literature are generalized even if p=2.

Keywords: homoclinic solutions, Hamiltonian systems, Mountain Pass Theorem, p-Laplacian systems.

2010 Mathematics Subject Classification: 34C37, 37J45, 70H05.

1 Introduction

Consider the ordinary p-Laplacian system d

dt |u˙(t)|^p−2u˙(t)− ∇K(t,u(t)) +∇W(t,u(t)) = f(t) (HS) where t ∈ _R, p > 1, K, W : R×_R^N → _R are C¹-maps and f : R −→ _R^N is a continuous and bounded function. We will say that a solutionu of (HS) is a nontrivial homoclinic (to 0) ifu 6≡0 andu(t)−→0 ast−→ ±_∞.

When p=2, (HS) reduces to the following second order Hamiltonian system

¨

u(t)− ∇K(t,u(t)) +∇W(t,u(t)) = f(t). (1.1) Homoclinic orbits were introduced by Poincaré more than a century ago, and since then, they became a fundamental tool in the study of chaos. Their existence has been extensively

BEmail: daouas−adel@yahoo.fr

investigated in the last two decades in many papers via critical point theory. In particular, the following second-order systems were considered in many works (see [1,3–8,11,13,16,17,19, 23,26])

¨

u(t)−L(t)u(t) +∇W(t,u(t)) =0 (1.2) where L(t) is a symmetric matrix valued function. Later, the authors of [7] introduced the more general system (1.1) where the quadratic function(L(t)x,x)is replaced byK(t,x).

Most of the previous works treat the superquadratic case under the global Ambrosetti–

Rabinowitz condition, i.e.,there existsµ>2such that

0<µW(t,x)≤(∇W(t,x)_,x)_, for all(t,x)∈_R×_R^N\{₀}_{. (}AR) Moreover, they suppose that L(t)andW(t,x)are either periodic intor independent of t.

In the case where W(t,x) and L(t) or K(t,x) are neither autonomous nor periodic,

∇W(t,x)is usually bounded with respect to the first variable. Indeed, a variant of the follow- ing condition is used:

There is a function W ∈C(_R^N_,R)such that

|W(t,x)|+|∇W(t,x)| ≤ |W(x)|, for all(t,x)∈_R×_R^N. (1.3) In a recent paper the authors of [9] studied the problem (HS) under new superquadratic conditions which allowW to be neither periodic nor bounded in t. Particularly, they suppose

K(t,x) =a(t)|x|^p, with a(t)−→+_∞as|t| −→+_∞ and

W =W₁−W₂ ∈C¹(_R^N,R),

where the functionsW₁,W2 satisfy some (AR)-type conditions to be either increasing or de- creasing (see [9], Lemma 2.5).

Motivated by the above mentioned works, in the present paper we study the existence of homoclinic solutions for (HS) under more general conditions which cover the case of un- bounded potentials with respect to the variablet. Here, to overcome the difficulty due to the unboundedness of the domain, a homoclinic solution will be obtained as a limit of a sequence of solutions of some nil-boundary-value problem. The existence of such sequence of solutions is guaranteed through a standard version of the Mountain Pass Theorem. Furthermore, the forcing term f satisfies an easier condition compared to that given in [12,20] mainly. Our results complete and improve recent works in the literature even in the casep=_2.

Precisely, we suppose:

(H₁) there exist γ∈ (1,p]anda>0 such that

a|x|^γ ≤K(t,x)≤(x,∇K(t,x))≤ pK(t,x), for all(t,x)∈_R×_R^N, (H₂) W(t, 0) =0 and∇W(t,x) =o(|x|^p−1), as|x| −→0 uniformly int∈_R,

(_H3) there existsT0>0 such that lim inf

|x|−→_∞

W(t,x)

|x|^p > ^π

p

pT₀^p +m₁, uniformly int ∈[−T0,T0], wherem₁=_sup{K(t,x)|t ∈[−T₀,T₀]_,|x|=₁}_,

(H₄) there exist constantsµ> p, 0≤b<µ−pandβ∈ L¹(_R,R+) such that µW(t,x)−(∇W(t,x),x)≤bK(t,x) +β(t), for all (t,x)∈_R×_R^N. Remark 1.1. Note that, by(H₂), there exists 0<ρ₀ <1 such that

|W(t,x)| ≤ ^a

p|x|^p, ∀ |x| ≤ρ0, t∈_R. (1.4) Now, we state our main results.

Theorem 1.2. Assume that W and K satisfy(H₁)–(H₄)and (H₅) 0<

Z

R|f(t)|^qdt<

min{1,a(p−1)}

p

q

(ρ₀/2)^p,where 1 q+ ¹

p =1.

Then(HS)possesses a nontrivial homoclinic solution u∈W^1,p(_R,R^N). Example 1.3. Consider the functions

W(t,x) = |x|³[e^t2(|x|2−1)−1]

t²+1 , K(t,x) = (2+sint)|x|²+cos²t |x|^5/2.

A straightforward computation shows thatW andKsatisfy the assumptions of Theorem1.2, with γ= 2,p =5/2,µ=3 butW does not satisfy neither (AR) nor (1.3). Moreover, contrary to [6,24], we have inf_t∈R,|x|=1W(t,x) =sup_t∈R,|x|=1W(t,x) =0. Note also thatW changes sign near the origin. Hence, Theorem1.2extends and completes the results in [3,6,12,16,19,24,25].

Corollary 1.4. Assume that W and K satisfy(H₁)–(H₃),(H₅)and (H₄⁰) there exist constantsµ> p, 0≤c<a(µ−p)such that

µW(t,x)≤ (∇W(t,x),x) +c|x|^γ, ∀(t,x)∈_R×_R^N. Then(HS)possesses a nontrivial homoclinic solution u∈W^1,p(_R,R^N).

Remark 1.5. It is easy to see that (H₄⁰) _implies (H₄). However, the condition (H₄⁰)_{is weaker} than(H₄)in [12]. So, Corollary 1.4significantly improves Theorem 1.1 in [12].

Corollary 1.6. Assume that W and K satisfy(H₁),(H₂),(H₄),(H₅)and

(H₃⁰) _{lim inf}

|x|−→_∞

W(t,x)

|x|^p >m₁, uniformly in t∈_R.

Then(HS)possesses a nontrivial homoclinic solution u∈W^1,p(_R,R^N).

2 Preliminary results

Consider for each T>0 the following problem





 d

dt |u˙(t)|^p−2u˙(t)− ∇K(t,u(t)) +∇W(t,u(t)) = f_T(t), fort ∈[−T,T] u(−T) =u(T) =0,

(2.1)

where f_T is the function defined onRby f_T(t):=

(f(t) fort ∈[−T,T], 0 fort∈ _R\[−T,T]. Let

E_T :=W^1,p([−T,T],R^N) =ⁿu:[−T,T]−→_R^N is absolutely continuous function, u(−T) =u(T) =0 and ˙u∈ L^p([−T,T],R^N)^o

equipped with the norm

kuk_ET = _{Z T}

−T

|u˙(t)|^p+|u(t)|^pdt ¹_p

.

Furthermore, forα≥1, let L^α_T = L^α([−T,T],R^N)andL^∞_T = L^∞([−T,T],R^N)with their usual norms. Letη_T : ET −→[0,+_∞)given by

ηT(u) = Z _T

−T

|u˙(t)|^p+pK(t,u(t))dt 1/p

, andI_T :E_T −→R, be defined by

I_T(u) = ¹

pη_T^p(u)−

Z _T

−TW(t,u(t))dt+

Z _T

−T

(f_T(t),u(t))dt. (2.2) ThenI_T ∈C¹(E_T,R)and it’s easy to show that for all u,v∈ E_T, we have

I_T⁰(u)v=

Z _T

−T

h(|u˙|^p−2u˙(t)_{, ˙}v(t)) + (∇K(t,u(t))_,v(t))−(∇W(t,u(t))_,v(t))ⁱdt +

Z _T

−T

(f_T(t)_,v(t))dt.

By(H₁), we obtain, for allu∈ E_T I_T⁰(u)u≤η_T^p(u)−

Z _T

−T

(∇W(t,u(t)),u(t))dt+

Z _T

−T

(f_T(t),u(t))dt. (2.3) It is well known that critical points of I_T are classical solutions of the problem (2.1), (see [2,14]). We will obtain a critical point of IT by using a standard version of the Mountain Pass Theorem. It provides the minimax characterization for the critical value which is important for what follows. For completeness, we give this theorem.

Theorem 2.1([18]). Let E be a real Banach space and I: E−→_Rbe a C¹-smooth functional satisfies the Palais–Smale condition. If I satisfies the following conditions:

(I₁) I(0) =0,

(I2) there exist constantsρ,α>0such that I|_∂B

ρ(₀) ≥α,

(I₃) there exists e ∈ E\B^¯_ρ(0) such that I(e) ≤ 0, where B_ρ(0) is an open ball in E of radius ρ centered at0,

then I possesses a critical value c ≥αgiven by c= inf

g∈_Γmax

s∈[0,1]I(g(s)) where

Γ={g∈C([0, 1],E);g(0) =0,g(1) =e}.

Next we need an extension to the p-case of the following proposition first proved by Rabinowitz in [16].

Lemma 2.2([12,22]). Let u:R−→_R^N be a continuous map such thatu˙ ∈ L_loc^p (_R,R^N). Then, for all t∈_{R, we have}

|u(t)| ≤2^p

−1 p

"

Z _t+¹₂

t−¹₂ |u(s)|^p+|u˙(s)|^pds

#¹_p

. (2.4)

Corollary 2.3. For all u∈ E_T the following inequality holds:

kuk_L∞

T ≤2^p

−1 p

1+

1 2T

1/p

kuk_ET. (2.5)

Remark 2.4. Note that for T≥ ¹₂ we have 2^p

−1

p 1+ [_2T¹ ]^1/p ≤2. So, from (2.5), we get kuk_L∞

T ≤2kuk_ET, for allu∈ E_T. (2.6) Subsequently, we may assume this condition fulfilled.

Lemma 2.5. Assume that(H₁)holds, then for all t∈ _R, we have K(t,x)≤ K

t, x

|x|

|x|^p, if|x| ≥1. (2.7) The proof of Lemma2.5is a routine so we omit it.

3 Proof of Theorem 1.2

Lemma 3.1. Under the assumptions of Theorem1.2, the problem(2.1)possesses a nontrivial solution u_T ∈ E_T.

Proof. It suffices to prove that the functional I_T satisfies all the assumptions of the Mountain Pass Theorem.

Step 1. The functional IT satisfies the (PS)-condition, i.e., for every constant c and sequence {un} ⊂ E such that I_T(un) −→ c and I⁰_T(un) −→ 0 as n −→ _∞, {un} has a convergent subsequence. Indeed, let {u_n} ⊂ E_T is a (PS)-sequence of I_T. By (2.2) and (2.3) there exists MT >0 such that

M_T(₁+ku_nk_ET)≥µI_T(u_n)−I⁰_T(u_n)u_n

≥ µ

p −1

η_T^p(u_n) +

Z _T

−T

h(∇W(t,u_n(t)),u_n(t))−µW(t,u_n(t))ⁱdt + (µ−1)

Z _T

−T

(f_T(t),un(t))dt.

(3.1)

Using(H₄)and Hölder’s inequality, from (3.1), we get µ

p −1

η_T^p(un)≤M_T(1+kunk_ET) +b Z _T

−TK(t,un(t))dt+

Z _T

−Tβ(t)dt + (µ−1)kfk_Lq(_R,R^N)ku_nk_ET

≤MT(1+kunk_ET) + ^b

pη_T^p(un) +

Z _T

−Tβ(t)dt+ (µ−1)kfk_Lq(_R,R^N)kunk_ET, which yields

µ−b p −1

η_T^p(u_n)≤ M_T(1+ku_nk_ET) +

Z _∞

−_∞β(t)dt+ (µ−1)kfk_Lq(_R,R^N)ku_nk_ET. (3.2) Without loss of generality, we can assume thatku_nk_ET 6=0. Then from(H₁)and (2.6) we get

η_T^p(u_n) =

Z _T

−T

h|u˙_n(t)|^p+pK(t,u_n(t))ⁱdt

≥

Z _T

−T

|u˙(t)|^pdt+pa Z _T

−T

|u_n(t)|^γdt

≥

Z _T

−T

|u˙(t)|^pdt+pa(2ku_nk_ET)^γ−p

Z _T

−T

|u_n(t)|^pdt

≥minn

1,pa(2ku_nk_ET)^γ−poku_nk_E^p

T

≥minn ku_nk_E^p

T,pa2^γ−pku_nk^γ_E

T

o .

(3.3)

Combining (3.2) and (3.3) we obtain µ−b−p

p

min{ku_nk^p_E

T,pa2^γ−paku_nk^γ_E

T}

≤M_T(1+ku_nk_ET) +

Z _∞

−_∞β(t)dt+ (µ−1)kfk_Lq(_R,R^N)ku_nk_ET.

From(H₄), we know thatµ−b−p>0, then the sequence{un}is bounded inET. In a similar way to Lemma 2 in [19], we can prove that {u_n}has a convergent subsequence inE_T. Hence I_T satisfies the (PS)-condition.

Step 2.The functional I_T satisfies the condition (I₂) of the Mountain Pass Theorem.

Letρ= ^ρ0

2 andq∈ E_T, such thatkuk_ET =ρ, then 0<kuk_L∞

T ≤ρ₀. By (1.4) we have Z _T

−TW(t,u(t))dt≤ ^a p

Z _T

−T

|u(t)|^pdt. (3.4)

On the other hand, sinceγ≤ p, by(H₁), we have Z _T

−TK(t,u(t))dt≥ a Z _T

−T

|u(t)|^γdt≥ a Z _T

−T

|u(t)|^pdt. (3.5)

Then, by (2.2), (3.4), (3.5) and Hölder’s inequality it follows that I_T(u)≥ ¹

pη_T^p(u)− ^a p

Z _T

−T

|u(t)|^pdt− kfk_Lq(_R,R^N)kuk_ET

≥ ¹ p

Z _T

−T

h|u˙(t)|^p+pK(t,u(t))ⁱdt− ^a p

Z _T

−T

|u(t)|^pdt− kfk_Lq(_R,R^N)kuk_ET

≥ ¹ p

Z _T

−T

|u˙(t)|^pdt+ ^a(p−1) p

Z _T

−T

|u(t)|^pdt− kfk_Lq(_R,R^N)kuk_ET

≥_min 1

p,a(p−1) p

kuk_E^p

T − kfk_Lq(_R,R^N)kuk_ET

≥ ^min{1,a(p−1)}

p ρ^p− kfk_Lq(_R,R^N)ρ=:α.

(3.6)

From(H₅), it follows thatα>0 and (3.6) shows thatkuk_ET = ρimplies I_T(u)≥ α.

Step 3. The functionalI satisfies the condition (I₃) of the Mountain Pass Theorem.

Letm₂=_supK(t,x)|t ∈[−T₀,T₀]_,|x| ≤1 . From (2.7), it is easy to see that

K(t,x)≤m₁|x|^p+m2, for allt∈ [−T0,T0], x ∈_R^N. (3.7) Furthermore, by(H3), there existsε0>0 andr>0 such that

W(t,x)

|x|^p ≥ ^π

p+ε₀

pT₀^p +m₁, for allt∈ [−T₀,T₀], |x|>r.

Letδ =max (^πp+ε0

pT₀^p +m₁)|x|^p−W(t,x)|t∈[−T0,T0],|x| ≤r , hence we have W(t,x)≥ ^πp+ε₀

pT₀^p +m₁

!

|x|^p−δ, for all t∈[−T₀,T₀], x∈ _R^N. (3.8) For T≥ T₀, define

e(t) = (

ξ|sin(ωt)|e₁ if t∈ [−T₀,T₀]

0 if t∈ [−T,T]\[−T₀,T₀]. (3.9) whereω = _T^π

0 ande₁= (1, 0, . . . , 0)∈_R^N. Then by (2.2) and (3.7)–(3.9), we obtain IT(_e) =

Z _T

−T

1

p|e˙(_t)|^p+_K(_t,e(_t))−W(_t,e(_t))

dt+

Z _T

−T

(_fT(_t)_,e(_t))_dt

=

Z _T0

−T₀

1

p|e˙(t)|^p+K(t,e(t))−W(t,e(t))

dt+

Z _T0

−T₀

(f_T0(t),e(t))dt

≤ |ξ|^pω^p p

Z _T0

−T0

|cos(ωt)|^pdt+m₁|ξ|^p

Z _T0

−T0

|sin(ωt)|^pdt

− ^π

p+ε₀ pT₀^p +m₁

!

|ξ|^p

Z _T0

−T0

|sin(ωt)|^pdt +|ξ|kfk_L^q

_{Z T}

0

−T0

|sin(ωt)|^pdt 1/p

+2T₀(δ+m₂)

≤ − ^ε0 pT₀^p|ξ|^p

Z _T0

−T0

|sin(ωt)|^pdt+|ξ|kfk_L^q _{Z T0}

−T0

|sin(ωt)|^pdt 1/p

+2T₀(δ+m₂)−→ −_∞, asξ −→_∞.

Thus, we can chooseξ large enough such thatkek_ET >ρand I_T(e)<0.

For our setting, clearly IT(0) = 0, then, by application of the Mountain Pass Theorem, there exists a critical pointu_T ∈ E_T of I_T such thatI_T(u_T)≥αfor all T≥ T₀.

Lemma 3.2. u_T is bounded uniformly for T≥T₀. Proof. Define the set of paths

ΓT ={g∈ C([0, 1],E_T)|g(0) =0, g(1) =e}. By Lemma3.1, we know that there is a solutionuT of (2.1) at which

ginf∈_ΓT max

s∈[0,1]I_T(g(s))≡ N_T is achieved. Let now Te > T, then ΓT ⊂ _Γ

Te, since any function in E_T can be regarded as belonging toE_Teif one extends it by zero in[−T,e Te]\[−T,T]. Therefore, for all solutionu_T of (2.1), we get

I_T(u_T) = N_T ≤ N_T0 uniformly inT ≥T₀. (3.10) Using the fact that I⁰_T(u_T) = 0 and (3.10), the rest of the proof is identical to Step 1 in Lemma3.1. Hence there exists a constantM₀ >0, independent ofT such that

ku_Tk_ET ≤ M₀, for allT ≥T₀. This ends the proof of Lemma3.2.

Now, take an increasing sequence Tn −→_∞ with T₁ > T0 and consider the problem (2.1) on the interval [−T_n,T_n]. By the conclusion of Lemma 3.1 and Lemma 3.2, there exists a nontrivial solutionu_n :=u_Tn of (2.1) satisfying

ku_nk_ETn ≤ M₀, for alln∈_N. (3.11) Lemma 3.3. Let (un)_n∈N be the sequence given above. Then there exists a subsequence (un_j)_j∈N convergent to a certain function u₀in C_loc¹ (_R,R^N).

Proof. First of all from (2.6) and (3.11), we have kunk_L^∞

Tn ≤2M0≡ M₁ (3.12)

and

Z _Tn

−Tn

|u˙n(t)|^p 1/p

≤ kunk_ETn ≤ M0 (3.13)

for alln∈N. By Hölder’s inequality and (3.11), fort₁,t₂∈ [−T₁,T₁]_witht₁< t₂,

|un(t2)−un(t₁)|=

Z _t2

t1

˙ un(t)dt

≤(t2−t₁)^1/q Z _T1

−T1

|u˙n(t)|^p 1/p

≤ M0(t2−t₁)^1/q. (3.14) From (3.12) and (3.14) the sequence {u_n}_n∈N is equicontinuous and uniformly bounded on [−T₁,T₁]. By the Arzelà–Ascoli Theorem, there exists a uniformly convergent subsequence {u¹_nj}_j∈Nof {u_n}_n∈Non [−T₁,T₁]and we can choosen₁>1.

Consider {u¹_nj}_j∈N on [−T2,T2]. By the Arzelà–Ascoli Theorem, there exists again a uni- formly convergent subsequence {u²_nj}_j∈N of {u¹_nj}_j∈N on [−T₂,T₂] with n₁ in u²_n1 satisfies

n₁>2. Repeat this procedure for all i ≥ 1 and take the diagonal subsequence of {uⁱ_nj}_j∈N, i ≥ 1, which consists of u¹_n1,u²_n2,u³_n3, . . . It follows that this diagonal subsequence converges uniformly on any bounded interval to a certain functionu₀.

Next, we denote u_n instead of u_n^j_j. Let I be a bounded interval, there exists n₀ such that I ⊂ [−Tn₀,Tn₀]. Using (2.1), for allt∈ I, we have

|^d

dt(|u˙n(t)|^p−2u˙n(t)| ≤ |∇K(t,un(t))|+|∇W(t,un(t))|+|fn(t)|

≤ |∇K(t,un(t))|+|∇W(t,un(t))|+|f(t)|,

for n≥ n₀ where here and subsequently fn = f_Tn. Since f is bounded , by (3.12) there exists M₂>0 (dependent on I) such that

sup

t∈I

d

dt(|u˙n(t)|^p−2u˙n(t))

≤ M2, ∀n≥n0. (3.15)

From the Mean Value Theorem it follows that for every n ∈ _Nand t ∈ _R there exists τ_n ∈ [t−1,t]such that

˙

u_n(τ_n) =

Z _t

t−1u˙_n(s)ds=u_n(t)−u_n(t−1). Combining the above with (3.12) and (3.15) we obtain

|u˙_n(t)|^p−2u˙_n(t)=

Z _t

τn

d

dt(|u˙_n(s)|^p−2u˙_n(s)ds+|u˙_n(τ_n)|^p−2u˙_n(τ_n)

≤

Z _t

t−1

d

dt(|u˙n(s)|^p−2u˙n(s)

ds+|u˙n(τn)|^p−1≤ M₂+ (2M₁)^p−1≡ M₃^p−1, and hence

sup

t∈I

|u˙n(t)| ≤ M3, ∀ n≥n0. (3.16) Now we prove that the sequence{u˙_n}_n∈Nis equicontinuous on I. If not, there existe>0, two sequences {t¹_i}_i∈N⊂ I, {t²_i}_i∈N⊂ I and a sequence{n_i}_i∈N of integers such that

0<t²_i −t¹_i < ¹

i, |un_i(t²_i)−un_i(t¹_i)| ≥e, and n_i ≥n₀, i∈_N. (3.17) Since the sequences {u_ni(t¹_i)} and {u_ni(t¹_i)} are bounded, passing, if necessary, to subse- quences, one can assume that

un_i(t¹_i)−→α₁, and un_i(t²_i)−→α₂, asi−→_∞. (3.18) Combining (3.17) and (3.18), we get

|α₂−α₁| ≥e. (3.19)

On the other hand, from (3.15) and (3.17), we have

|u_ni(t²_i)|^p−2u_ni(t²_i)− |u_ni(t¹_i)|^p−2u_ni(t¹_i) =

Z _t²

i

t¹_i

d

dt(|u˙_n(s)|^p−2u˙_n(s)ds

≤

Z _t²_i

t¹_i

d

dt(|u˙n(s)|^p−2u˙n(s)

ds

≤ M₂(t²_i −t¹_i)≤ ^M2

i , i∈_N.

(3.20)

Passing to the limit in (3.20) and using (3.18), we obtain

|α₂|^p−2α₂− |α₁|^p−2α₁ =0

and consequently α₁ = α₂, which contradicts (3.19). Thus, {u˙_n}_n∈N is equicontinuous. By (3.16),{u˙_n}_n∈N is also uniformly bounded on I, the Arzelà–Ascoli Theorem proves the exis- tence of a subsequence convergent to a certain functionv. Since the interval I is arbitrary we conclude that according to a subsequence

u_nj −→u, as j−→_∞ inC_loc¹ (_R,R^N). Lemma3.3 is proved.

Lemma 3.4. Let u0:R−→_R^Nbe the function given by Lemma3.3. Then u0is the desired homoclinic solution of (HS).

Proof. The first step is to show thatu₀is a solution of (HS). Let(u_nj)_j∈Nbe the sequence given by Lemma3.3, then

d

dt(|u˙_nj(t)|^p−2u˙_nj(t))− ∇K(t,u_nj(t)) +∇W(t,u_nj(t)) = f_nj(t)

for every j∈ _N, and t ∈ [−T_nj,T_nj]. Take a,b ∈ _Rwith a < b. There exists j₀ ∈ _N such that for allj> j0 one has[a,b]⊂[−Tn_j,Tn_j]and

d

dt(|u˙_nj(t)|^p−2u˙_nj(t)) =∇K(t,u_nj(t))− ∇W(t,u_nj(t)) + f(t), ∀t ∈[a,b]. (3.21) Integrating (3.21) from atot ∈[a,b], we obtain

|u˙n_j(_t)|^p−2u˙n_j(_t)− |u˙n_j(_a)|^p−2u˙n_j(_a)

=

Z _t

a

h∇K(s,u_nj(s))− ∇W(s,u_nj(s)) + f(s)ⁱds, ∀ t∈ [a,b]. (3.22) Sinceu_nj −→u₀ uniformly on[a,b], and ˙u_nj −→u˙₀ uniformly on[a,b]as j−→∞, then from (3.22), we get

|u˙₀(t)|^p−2u˙₀(t)− |u˙₀(a)|^p−2u˙₀(a)

=

Z _t

a

h∇K(s,u₀(s))− ∇W(s,u₀(s)) + f(s)ⁱds, ∀ t∈ [a,b]. (3.23) Sinceaandbare arbitrary, we receive from (3.23) thatu0 satisfies(HS).

Now we prove thatu₀(t) −→0, as|t| −→ _∞. First of all note that, from (3.11), forl ∈ _N there existsj0 ∈_Nsuch that for allj>j0, we have

Z _T

nl

−T_nl

(|u_nj(t)|^p+|u˙_nj(t)|^p)dt≤ ku_njk_E^p

Tnj ≤ M₀^p. Lettingj−→_{∞, we get}

Z _T

nl

−T_nl

(|u₀(t)|^p+|u˙₀(t)|^p)dt≤ M₀^p,

and now, lettingl−→∞, we obtain Z +_∞

−_∞ (|u₀(t)|^p+|u˙₀(t)|^p)dt≤ M₀^p,

and so _Z

|t|≥r

(|u₀(t)|^p+|u˙₀(t)|^p)dt−→0, asr −→_∞. (3.24) From (2.4) and (3.24), we receive our claim.

Finally, it is obvious thatu₀ is nontrivial since, from(H₅), we have f 6≡0 and the proof of Theorem1.2is complete.

Remark 3.5. Under the assumptions of Theorem1.2 and (H₆)there is R>0 such that

∇K(t,x)−→0 as|x| −→0 uniformly int ∈(−_∞,−R]∪[R,+_∞),

the homoclinic solution u0 obtained above satisfies ˙u0(t) −→ 0, as |t| −→ ∞. The proof is analogous to [21].

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