Existence of solutions for a fourth-order boundary value problem on the half-line via critical point theory

Existence of solutions for a fourth-order boundary value problem on the half-line via critical point theory

Mabrouk Briki

, Toufik Moussaoui

and Donal O’Regan

1Laboratory of Fixed Point Theory and Applications, École Normale Supérieure, Kouba, Algiers, Algeria

2School of Mathematics, Statistics and Applied Mathematics, National University of Ireland, Galway, Ireland

Received 14 December 2015, appeared 2 May 2016 Communicated by Alberto Cabada

Abstract. In this paper, a fourth-order boundary value problem on the half-line is considered and existence of solutions is proved using a minimization principle and the mountain pass theorem.

Keywords: fourth-order BVPs, unbounded interval, critical point, minimization princi- ple, mountain-pass theorem.

2010 Mathematics Subject Classification: 35A15, 35B38.

1 Introduction

We consider the existence of solutions for the following fourth-order boundary value problem set on the half-line











u⁽⁴⁾(t)−u⁰⁰(t) +u(t) = f(t,u(t))_, t ∈[_0,+_∞)_, u(0) =u(+_∞) =0,

u⁰⁰(0) =u⁰⁰(+_∞) =0,

(1.1)

where f ∈ C([0,+_∞)×_R,R).

Many authors used critical point theory to establish the existence of solutions for fourth- order boundary value problems on bounded intervals (see for example [8,9,13]), but there are only a few papers that consider the above problem on the half-line using critical point theory.

We cite [5] where the authors consider the existence of solutions for a particular fourth-order BVP on the half-line using critical point theory.

We endow the following space

H₀²(0,+_∞) =ⁿu∈ L²(0,+_∞), u⁰ ∈L²(0,+_∞), u⁰⁰ ∈L²(0,+_∞), u(0) =0, u⁰(0) =0o

BCorresponding author. Email: moussaoui@ens-kouba.dz

with its natural norm kuk=

_{Z +∞}

0 u⁰⁰²(t)dt+

Z _+∞

0 u⁰²(t)dt+

Z _+∞

0 u²(t)dt ¹₂

.

Note that if u ∈ H₀²(0,+_∞), then u(+_∞) = 0, u⁰(+_∞) = 0, (see [3, Corollary 8.9]). Let p,q:[0,+_∞)−→(0,+_∞)be two continuously differentiable and bounded functions with

M₁ =max(kpk_L2,kp⁰k_L2)< +_∞, M₂ =max(kqk_L2,kq⁰k_L2)<+_∞. We also consider the following spaces

C_l,p[0,+_∞) =

u∈ C([0,+_∞),R): lim

t→+_∞p(t)u(t)exists

endowed with the norm

kuk_∞,p = sup

t∈[0,+_∞)

p(t)|u(t)|, and

C¹_l,p,q[0,+_∞) =

u∈C¹([0,+_∞),R): lim

t→+_∞p(t)u(t), lim

t→+_∞q(t)u⁰(t)exist

endowed with the natural norm kuk_∞,p,q= sup

t∈[0,+_∞)

p(t)|u(t)|+ sup

t∈[0,+_∞)

q(t)|u⁰(t)|. Let

C_l[0,+_∞) =

u∈C([0,+_∞),R): lim

t→+_∞u(t)exists

endowed with the normkuk_∞ =sup_t∈[0,+∞)|u(t)|.

To prove that H₀²(0,+_∞) embeds compactly in C_l,p,q¹ [0,+_∞), we need the following Corduneanu compactness criterion.

Lemma 1.1 ([4]). Let D ⊂ C_l([0,+_∞),R) be a bounded set. Then D is relatively compact if the following conditions hold:

(a) D is equicontinuous on any compact sub-interval ofR⁺,i.e.

∀J ⊂[0,+_∞) compact, ∀ε>0, ∃δ >0, ∀t₁,t₂ ∈ J :

|t₁−t₂|< δ=⇒ |u(t₁)−u(t₂)| ≤ε, ∀u∈ D;

(b) D is equiconvergent at+_∞i.e.,

∀ε>0, ∃T =T(ε)>0such that

∀t:t≥ T(ε) =⇒ |u(t)−u(+_∞)| ≤ε, ∀u∈ D.

Similar reasoning as in [6] yields the following compactness criterion in the space C¹_l,p,q([0,+_∞),R).

Lemma 1.2. Let D ⊂ C¹_l,p,q([0,+_∞),R) be a bounded set. Then D is relatively compact if the following conditions hold:

(a) D is equicontinuous on any compact sub-interval of[0,+_∞),i.e.

∀J ⊂[0,+_∞) compact, ∀ε >0,∃δ> 0, ∀t₁,t₂∈ J :

|t₁−t₂|<δ =⇒ |p(t₁)u(t₁)−p(t₂)u(t₂)| ≤ε, ∀u∈ D,

|t₁−t₂|<δ =⇒ |q(t₁)u⁰(t₁)−q(t₂)u⁰(t₂)| ≤ε, ∀u∈ D;

(b) D is equiconvergent at+_∞i.e.,

∀ε>0,∃T= T(ε)>0such that

∀t:t ≥T(ε) =⇒ |p(t)u(t)−(pu)(+_∞)| ≤ε, ∀u∈ D,

∀t:t ≥T(ε) =⇒ |q(t)u⁰(t)−(qu⁰)(+_∞)| ≤ε, ∀u∈ D.

Now we recall some essential facts from critical point theory (see [1,2,10]).

Definition 1.3. LetXbe a Banach space,Ω⊂ Xan open subset, andJ :Ω−→_Ra functional.

We say that J is Gâteaux differentiable atu∈_Ωif there exists A∈ X^∗such that limt→0

J(u+tv)−J(u)

t = Av,

for all v∈X. NowA, which is unique, is denoted by A= J_G⁰ (u).

The mapping which sends to everyu∈_Ωthe mapping J_G⁰ (u)is called the Gâteaux differ- ential of J and is denoted by J_G⁰ .

We say that J ∈C¹ if J is Gâteaux differential onΩand J_G⁰ is continuous at everyu∈_Ω.

Definition 1.4. Let X be a Banach space. A functional J : Ω −→ _R is called coercive if, for every sequence(u_k)_k∈N⊂ X,

ku_kk →+_∞=⇒ |J(u_k)| →+_∞.

Definition 1.5. Let X be a Banach space. A functional J : X −→ (−_∞,+_∞] is said to be sequentially weakly lower semi-continuous (swlscfor short) if

J(u)≤lim inf

n→+_∞ J(u_n) asu_n*uin X,n→_∞.

Lemma 1.6 (Minimization principle [2]). Let X be a reflexive Banach space and J a functional defined on X such that

(1) lim_kuk→+∞J(u) = +_∞(coercivity condition), (2) J is sequentially weakly lower semi-continuous.

Then J is lower bounded on X and achieves its lower bound at some point u0.

Definition 1.7. Let X be a real Banach space, J ∈ C¹(X,R). If any sequence (u_n) ⊂ X for which (J(u_n)) is bounded in R and J⁰(u_n) −→ 0 as n → +_{∞ in} X⁰ possesses a convergent subsequence, then we say thatJ satisfies the Palais–Smale condition (PS condition for brevity).

Lemma 1.8(Mountain Pass Theorem, [11, Theorem 2.2], [12, Theorem 3.1]). Let X be a Banach space, and let J ∈ C¹(X,R) satisfy J(0) = 0. Assume that J satisfies the (PS)condition and there exist positive numbersρandαsuch that

(1) J(u)≥αifkuk=ρ,

(2) there exists u₀∈ X such thatku₀k> ρand J(u₀)<α.

Then there exists a critical point. It is characterized by J⁰(u) =0, J(u) = inf

γ∈_Γmax

t∈[0,1]J(γ(t)), where

Γ= {γ∈C([_{0, 1}]_,X)_: γ(₀) =_0, γ(₁) =u₀}. 1.1 Variational setting

Take v ∈ H₀²(0,+_∞), and multiply the equation in Problem (1.1) by v and integrate over (0,+_∞), so we get

Z _+∞

0

u⁽⁴⁾(t)−u⁰⁰(t) +u(t)v(t)dt=

Z _+∞

0

f(t,u(t))v(t)dt.

Hence

Z +_∞

0 u⁰⁰(t)v⁰⁰(t) +u⁰(t)v⁰(t) +u(t)v(t)dt=

Z +_∞

0 f(t,u(t))v(t)dt.

This leads to the natural concept of a weak solution for Problem (1.1).

Definition 1.9. We say that a functionu∈ H₀²(_0,+_∞)is a weak solution of Problem (1.1) if Z _+∞

0 u⁰⁰(t)v⁰⁰(t) +u⁰(t)v⁰(t) +u(t)v(t)dt=

Z _+∞

0 f(t,u(t))v(t)dt, for allv∈ H₀²(0,+_∞).

In order to study Problem (1.1), we consider the functional J : H₀²(0,+_∞) −→_R defined by

J(u) = ¹

2kuk²−

Z _+∞

0

F(t,u(t))dt, where

F(t,u) =

Z _u

0 f(t,s)ds.

2 Some embedding results

We begin this section by proving some continuous and compact embeddings. Here p andq (andM₁,M₂) are as in Section 1.

Lemma 2.1. H₀²(0,+_∞)embeds continuously in C¹_l,p,q[0,+_∞).

Proof. Foru∈ H₀²(0,+_∞), we have

|p(t)u(t)|=|p(+_∞)u(+_∞)−p(t)u(t)|

=

Z _+∞

t

(pu)⁰(s)ds

≤

Z +_∞

t p⁰(s)u(s)ds

+

Z +_∞

t p(s)u⁰(s)ds

≤

_{Z +∞}

0

p⁰²(s)ds

¹_{2 Z +∞}

0

u²(s)ds ¹₂

+

_{Z +∞}

0

p²(s)ds

¹_{2 Z +∞}

0

u⁰²(s)ds ¹₂

≤ max(kp⁰k_L2,kpk_L2)kuk

≤ M₁kuk, and

|q(t)u⁰(t)|= |q(+_∞)u⁰(+_∞)−q(t)u⁰(t)|

=

Z _+∞

t

(qu⁰)⁰(s)ds

≤

Z _+∞

t

q⁰(s)u⁰(s)ds

+

Z _+∞

t

q(s)u⁰⁰(s)ds

≤

_{Z +∞}

0 q⁰²(s)ds

¹_{2Z +∞}

0 u⁰²(s)ds ¹₂

+

_{Z +∞}

0 q²(s)ds

¹_{2 Z +∞}

0 u⁰⁰²(s)ds ¹₂

≤ max(kq⁰k_L2,kqk_L2)kuk

≤ M₂kuk.

Hencekuk_∞,p,q≤ Mkuk, with M =max(M₁,M2).

The following compactness embedding is an important result.

Lemma 2.2. The embedding H₀²(0,+_∞),→C¹_l,p,q[0,+_∞)is compact.

Proof. LetD⊂ H²₀(0,+_∞)be a bounded set. Then it is bounded inC_l,p,q¹ [0,+_∞)by Lemma2.1.

Let R>0 be such that for allu∈ D,kuk ≤R. We will apply Lemma 1.2.

(a) D is equicontinuous on every compact interval of [0,+_∞). Let u ∈ D and t₁,t₂ ∈ J ⊂ [0,+_∞)where J is a compact sub-interval. Using the Cauchy–Schwarz inequality, we have

|p(t₁)u(t₁)−p(t₂)u(t₂)|=

Z _t1

t2

(pu)⁰(s)ds

=

Z _t1

t₂ p⁰(s)u(s) +u⁰(s)p(s)ds

≤ _{Z t}

1

t2

p⁰²(s)ds

¹_{2Z t}

1

t2

u²(s)ds ¹₂

+ _{Z t}

1

t2

p²(s)ds

¹_{2Z t}

1

t2

u⁰²(s)ds ¹₂

≤ max

"

Z _t1

t2

p⁰²(_s)_ds ¹₂

, Z _t1

t2

p²(_s)_ds ¹₂#

kuk

≤Rmax

"

_{Z t}

1

t2

p⁰²(s)ds ¹₂

, _{Z t}

1

t2

p²(s)ds ¹₂#

−→0,

as|t₁−t₂| →0, and

|q(t₁)u⁰(t₁)−q(t₂)u⁰(t₂)|=

Z _t1

t2

(qu⁰)⁰(s)ds

=

Z _t1

t2

q⁰(s)u⁰(s) +q(s)u⁰⁰(s)ds

≤ Z _t1

t2

q⁰²(s)ds

¹₂ Z _t1

t2

u⁰²(s)ds ¹₂

+ _{Z t}

1

t2

q²(s)ds

¹_{2 Z t}

1

t2

u⁰⁰²(s)ds ¹₂

≤ max

"

Z _t1

t2

q⁰²(s)ds ¹₂

, Z _t1

t2

q²(s)ds ¹₂#

kuk

≤ Rmax

"

_{Z t}

1

t₂ q⁰²(s)ds ¹₂

, _{Z t}

1

t₂ q²(s)ds ¹₂#

−→0, as|t₁−t₂| →0.

(b) Dis equiconvergent at +_{∞. For}t ∈ [_0,+_∞)_andu ∈ D, using the fact that (pu)(+_∞) = 0,(qu⁰)(+_∞) =0 (note thatu(_∞) =0,u⁰(_∞) =0 andp,qare bounded) and using the Cauchy–

Schwarz inequality, we have

|(pu)(t)−(pu)(+_∞)|=

Z +_∞ t

(pu)⁰(s)ds

=

Z _+∞

t p⁰(s)u(s) +u⁰(s)p(s)ds

≤max

"

_{Z +∞}

t p⁰²(s)ds ¹₂

,

_{Z +∞}

t p²(s)ds ¹₂#

kuk

≤Rmax

"_{Z +∞}

t p⁰²(s)ds ¹₂

, _{Z +∞}

t p²(s)ds ¹₂#

−→_0, ast→+_∞, and

|(qu⁰)(t)−(qu⁰)(+_∞)|=

Z _+∞

t

(qu⁰)⁰(s)ds

=

Z _+∞

t q⁰(s)u⁰(s) +q(s)u⁰⁰(s)ds

≤max

"

_{Z +∞}

t q⁰²(s)ds ¹₂

,

_{Z +∞}

t q²(s)ds ¹₂#

kuk

≤ Rmax

"

Z _+∞

t q⁰²(s)ds ¹₂

, Z _+∞

t q²(s)ds ¹₂#

−→0, ast→+_∞.

Corollary 2.3. C¹_l,p,q[0,+_∞)embeds continuously in C_l,p[0,+_∞).

Corollary 2.4. The embedding H₀²(0,+_∞),→C_l,p[0,+_∞)is continuous and compact.

3 Existence results

Here p(and M₁) are as in Section 1.

Theorem 3.1. Assume that F satisfy the following conditions.

(F1) There exist two constants1<α<β<2and two functions a,b with _p^aα∈L¹([0,+_∞),[0,+_∞)),

b

p^β ∈ L¹([0,+_∞),[0,+_∞))such that

|F(t,x)| ≤a(t)|x|^α, ∀(t,x)∈[0,+_∞)×_R,|x| ≤1 and

|F(t,x)| ≤b(t)|x|^β, ∀(t,x)∈[0,+_∞)×_R, |x|>1.

(F2) There exist an open bounded set I ⊂[0,+_∞)and two constantsη>0and0<γ<2such that F(t,x)≥ η|x|^γ, ∀(t,x)∈ I×_R, |x| ≤1.

Then Problem(1.1)has at least one nontrivial weak solution.

Proof.

Claim 1. We first show that J is well defined.

Let

Ω1={t ≥0, |u(t)| ≤1}, Ω2 ={t≥0, |u(t)|>1}. Givenu∈ H₀²(0,+_∞), it follows from(F1)and Corollary2.4that

Z _+∞

0

|F(t,u(t))|dt=

Z

Ω1

|F(t,u(t))|dt+

Z

Ω2

|F(t,u(t))|dt

≤

Z

Ω1

a(t)|u(t)|^αdt+

Z

Ω2

b(t)|u(t)|^βdt

≤

Z

Ω1

a(t)

p^α(t)|p(t)u(t)|^αdt+

Z

Ω2

b(t)

p^β(t)|p(t)u(t)|^βdt

≤

a p^α

L¹

kuk^α_∞,p+

b p^β

L¹

kuk_∞,p^β

≤ M^α₁

a p^α

L¹

kuk^α+M^β₁

b p^β

L¹

kuk^β. Thus

|J(u)| ≤ ¹

2kuk²+M^α₁

a p^α

L¹

kuk^α+M^β₁

b p^β

L¹

kuk^β <+_∞.

Claim 2. J is coercive.

From(F1)and Corollary2.4, we have J(u) = ¹

2kuk²−

Z

Ω1

F(t,u(t))dt−

Z

Ω2

F(t,u(t))dt

≥ ¹

2kuk²−M₁^α

a p^α

L¹

kuk^α−M₁^β

b p^β

L¹

kuk^β.

(3.1)

Now since 0<α<β<2, then (3.1) implies that

kuk→+lim_∞J(u) = +_∞.

Consequently,J is coercive.

Claim 3. J is sequentially weakly lower semi-continuous.

Let (u_n) be a sequence in H²₀(0,+_∞) such that u_n * u as n −→ +_∞ in H₀²(0,+_∞). Then there exists a constant A > 0 such that kunk ≤ A, for all n ≥ 0 and kuk ≤ A. Now (see Corollary 2.4) (p(t)u_n(t)) converges to (p(t)u(t)) as n −→ +_∞ for t ∈ [0,+_∞). Since F is continuous, we haveF(t,u_n(t))−→F(t,u(t))asn−→+_∞, and using(F1)we have

|F(t,u_n(t))| ≤a(t)|u_n(t)|^α+b(t)|u_n(t)|^β

≤ ^a(t)

p^α(t)|p(t)un(t)|^α+ ^b(t)

p^β(t)|p(t)un(t)|^β

≤ ^a(t)

p^α(t)ku_nk_∞,p^α + ^b(t)

p^β(t)ku_nk_∞,p^β

≤ ^a(t) p^α(t)^M

α

1ku_nk^α+ ^b(t) p^β(t)^M

β 1ku_nk^β

≤ ^a(t) p^α(t)^M

α

1A^α+ ^b(t) p^β(t)^M

β 1A^β, so from the Lebesgue Dominated Convergence Theorem we have

n→+lim_∞ Z _+∞

0 F(t,un(t))dt=

Z _+∞

0 F(t,u(t))dt.

The norm in the reflexive Banach space is sequentially weakly lower semi-continuous, so lim inf

n→+_∞ kunk ≥ kuk. Thus one has

lim inf

n→+_∞ J(u_n) =_{lim inf}

n→+_∞

1

2ku_nk²−

Z _+∞

0

F(t,u_n(t))dt

≥ ¹

2kuk²−

Z +_∞

0 F(t,u(t))dt= J(u). Then, J is sequentially weakly lower semi-continuous.

From Lemma1.6, J has a minimum pointu0which is a critical point of J.

Claim 4. We show that u₀6=0.

Letu₁ ∈H₀²(0,+_∞)\ {0}and|u₁(t)| ≤1, for all t∈ I. Then from(F2), we have J(su₁) = ^s

2

2ku₁k²−

Z _+∞

0 F(t,su₁(t))dt

≤ ^s

2

2ku₁k²−

Z

I

η|su₁(t)|^γdt

≤ ^s

2

2ku₁k²−s^γη Z

I

|u₁(t)|^γdt, 0<s<1.

Since 0 < γ < 2, it follows that J(su₁) < 0 for s > 0 small enough. Hence J(u0) < 0, and thereforeu₀ is a nontrivial critical point of J.

Finally, it is easy to see that under (F1), the functional J is Gâteaux differentiable and the Gâteaux derivative at a pointu ∈Xis

(J⁰(u),v) =

Z _+∞

0 u⁰⁰(t)v⁰⁰(t) +u⁰(t)v⁰(t) +u(t)v(t)dt−

Z _+∞

0 f(t,u(t))v(t)dt, (3.2)

for all v∈ H₀²(0,+_∞). Thereforeuis a weak solution of Problem (1.1).

Theorem 3.2. Assume that f satisfies the following assumptions.

(F3) There exist nonnegative functionsϕ,g such that g∈C(_R,[0,+_∞))with

|f(t,x)| ≤ ϕ(t)g(x), for all t∈[0,+_∞) and all x ∈_R,

and for any constant R>0there exists a nonnegative functionψ_Rwith ϕψ_R∈ L¹(0,+_∞)and sup

g

y p(t)

:y∈[−R,R]

≤ψ_R(t) for a.e. t≥0.

(F4)

1 a(t)^{F t,}

1 p(t)^x

=o(|x|²) as x−→0 uniformly in t∈[0,+_∞)for some function a∈ L¹(0,+_∞)∩C[0,+_∞).

(F5) There exists a positive function c₁ and a nonnegative function c₂ with c₁,c₂ ∈ L¹(0,∞), and µ>₂such that

(a)F(t,x)≥ c₁(t)|x|^µ−c₂(t)_, for t ≥_0, ∀x ∈_R\ {₀}_, (b)µF(t,x)≤ x f(t,x), for t≥0, ∀x∈_R.

Then Problem(1.1)has at least one nontrivial weak solution.

Proof. We have J(0) =0.

Claim 1. J satisfies the(PS)condition.

Assume that (un)_n∈N ⊂ H²₀(0,+_∞) is a sequence such that (J(un))_n∈N is bounded and J⁰(u_n)−→0 asn−→+∞. Then there exists a constantd>0 such that

|J(u_n)| ≤d, kJ⁰(u_n)k_E⁰ ≤dµ, ∀n∈_N.

From(F5)(b)we have

2d+2dku_nk ≥2J(u_n)− ² µ

(J⁰(u_n)_,u_n)

≥

1− ² µ

ku_nk²+2 _{Z +∞}

0

1

µu_n(t)f(t,u_n(t))−F(t,u_n(t))

dt

≥

1− ² µ

ku_nk².

Sinceµ>2, then(u_n)_n∈Nis bounded inH²₀(0,+_∞).

Now, we show that(un)converges strongly to someuinH²₀(0,+_∞). Since(un)is bounded inH₀²(0,+_∞), there exists a subsequence of(u_n)still denoted by(u_n)such that(u_n)converges weakly to some uin H₀²(0,+_∞). There exists a constant c> 0 such thatku_nk ≤c. Now (see Corollary2.4)(p(t)un(t))converges to p(t)u(t)on [0,+_∞). We have f(t,un(t))−→ f(t,u(t)) and

|f(t,u_n(t))|=

f(t, 1

p(t)^p(t)u_n(t))

≤ ϕ(t)g 1

p(t)^p(t)u_n(t)

≤ ϕ(t)ψ_cM1(t),

and using the Lebesgue Dominated Convergence Theorem, we have

n→+lim_∞ Z _+∞

0

(f(t,u_n(t))− f(t,u(t))) (u_n(t)−u(t))dt=_{0. (3.3)} Since lim_n→+_∞J⁰(u_n) =0 and(u_n)converges weakly to someu, we have

n→+lim_∞hJ⁰(u_n)−J⁰(u),u_n−ui=0. (3.4) It follows from (3.2) that

(J⁰(un)−J⁰(u),un−u) =kun−uk²−

Z +_∞ 0

(f(t,un(t))− f(t,u(t)))(un(t)−u(t))dt.

Hence limn→+_∞kun−uk=0. Thus(un)converges strongly tou in H₀²(0,+_∞), so J satis- fies the(PS)condition.

Claim 2. J satisfies assumption(1)of Lemma1.8.

Let 0< ε< _| ¹

a|_L1M₁². From(F4), there exists 0<δ <1 such that

1 a(t)^{F t,}

1 p(t)^x

≤ ^ε

2|x|², for t∈ [0,+_∞)and|x| ≤δ.

Using Corollary2.4, we have Z _+∞

0

|F(t,u(t))dt|=

Z _+∞

0

F

t, 1

p(t)^p(t)u(t)

dt

≤

Z _+∞

0

ε

2|a(t)|p²(t)|u(t)|²dt

≤ ^ε

2M₁²|a|_L1kuk²_, wheneverkuk_∞,p ≤δ.

Let 0< ρ≤ _M^δ

1 andα= ¹₂(₁−ε|a|_L1M²₁)ρ². Then for kuk= ρ(notekuk_∞,p ≤δ), we have J(u) = ¹

2kuk²−

Z +_∞

0 F(t,u(t))dt

≥ ¹

2(₁−ε|a|_L1M₁²)kuk²=_α, so assumption (1) in Lemma1.8 is satisfied.

Claim 3. J satisfies assumption(2)of Lemma1.8.

By(F5)(a)we have for somev₀ ∈ H₀²(_0,+_∞)_, v₀6=0, J(ξv₀) = ¹

2ξ²kv₀k²−

Z _+∞

0 F(t,ξv₀(t))dt

≤ ¹

2ξ²kv₀k²− |ξ|^µ

Z _+∞

0

c₁(t)|v₀(t)|^µdt+

Z _+∞

0

c₂(t)dt.

Now sinceµ >2, then foru₀ = ξv₀, J(u₀)≤ 0, asξ →+∞, so assumption (2) in Lemma1.8 is satisfied. From Lemma1.8, J possesses a critical point which is a nontrivial weak solution of Problem (1.1).

As an example of the above theorem, take f(t,x) = ⁵₂exp(−t)|x|¹²x. To see this take c₁(t) =exp(−t), c₂(t) =0,

µ= ⁵

2, a(t) = ¹

(1+t)^{2, p}(t) = ¹ 1+t, ϕ(t) = ⁵

2e^−t, g(x) =|x|³² and ψ_R(t) = (1+t)³²R³².

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