Infinitely many solutions for a class of quasilinear two-point boundary value systems

Infinitely many solutions for a class of quasilinear two-point boundary value systems

Giuseppina D’Aguì

, Shapour Heidarkhani

and Angela Sciammetta

1DICIEAMA, University of Messina, C/da di Dio, 98166 Messina, Italy

2Department of Mathematics, Faculty of Sciences, Razi University, 67149 Kermanshah, Iran

3DIECII, University of Messina, C/da di Dio, 98166 Messina, Italy

Received 7 January 2014, appeared 18 February 2015 Communicated by Gabriele Villari

Abstract. The existence of infinitely many solutions for a class of Dirichlet quasilinear elliptic systems is established. The approach is based on variational methods.

Keywords: infinitely many solutions, doubly eigenvalue Dirichlet quasilinear system, critical point theory, variational methods.

2010 Mathematics Subject Classification: 34B10, 34B15.

1 Introduction

The aim of this paper is to investigate the existence of infinitely many weak solutions for the following doubly eigenvalue quasilinear two-point boundary value system

(−(p_i−1)|u⁰_i(x)|^pi−2u⁰⁰_i (x) = (λFu_i(x,u₁, . . . ,un)+µGu_i(x,u₁, . . . ,un))h_i(x,u⁰_i) in(a,b)

u_i(a) =u_i(b) =0, 1≤i≤n, (D_λ,µ)

where p_i > 1 for 1 ≤ i ≤ n, λ > 0, µ ≥ 0 are real numbers, a,b ∈ _R with a < b, F: [a,b]×_Rⁿ → _R is a function such that F ∈ C¹([a,b]×_Rⁿ) and F(x, 0, . . . , 0) = 0 for all x∈[a,b]_,G: [a,b]×_Rⁿ →_Ris a function such thatG∈C¹([a,b]×_Rⁿ)_andG(x, 0, . . . , 0) =₀ for all x ∈ [a,b] and h_i : [a,b]×_R →]0,+_∞[ is a bounded and continuous function with m_i := inf_{(x,t)∈[a,b]×R}h_i(x,t) > 0. Here, F_ui andG_ui denote respectively the partial derivatives of FandGwith respect to u_i for 1≤ i≤n.

On the existence of multiple solutions for two-point boundary value problems of the type (D_λ,µ), several results are known whenn = 1, see for example [2,3,18,23] and the references cited therein. Existence results for nonlinear elliptic systems with Dirichlet boundary condi- tions have also received a great deal of interest in recent years; see, for instance, the papers [11,13,19,20,22].

BCorresponding author. Email: s.heidarkhani@razi.ac.ir

For a discussion about the existence of infinitely many solutions for boundary value problems, using Ricceri’s variational principle [26] and its variants ([5, Theorem 2.1] and [24, Theorem 1.1]) we refer the reader to the papers [1,4,6–10,12,14–17,21,27]. We also refer the reader, for instance, to the papers [25,28] where the existence of infinitely many solutions for boundary value problems has been studies by using different approach.

In the present paper, employing a smooth version of [5, Theorem 2.1], under some hy- potheses on the behavior of the nonlinear terms at infinity, under conditions on the potential ofh_i for 1≤i≤n, we determine the exact collections of the parametersλandµin which the system (D_λ,µ) admits infinitely many weak solutions (Theorem3.1). We also list some conse- quences of Theorem3.1and one example. Here, due to the facts, no symmetric assumptions are requested on the nonlinearities, the infinitely many solutions are local minima of the en- ergy functionals associated to the problem, and the nonlinearities depend on the termh_i(x,u_i⁰) being h_i a continuous bounded function and u_i⁰ is the weak derivative of the component u_i of the weak solution u = (u₁,u2, . . . ,un)of the system (D_λ,µ), the application of variational methods to investigate the system (D_λ,µ) is not standard.

A special case of our main result is the following theorem.

Theorem 1.1. Let f1,f2:R²→_Rbe two positive C⁰(_R²)-functions such that the differential 1-form w := f₁(σ,ν)dσ+ f₂(σ,ν)dνis integrable and let F be a primitive of w such that F(0, 0) =0. Fix two integers p,q>2, with p ≤q, and assume that

lim inf

ξ→+_∞

F(ξ,ξ)

ξ^p =0 and lim sup

ξ→+_∞

F(ξ,ξ)

ξ^q = +_∞.

Then, for every nonnegative arbitrary C¹(_R²)function G:R²→_Rsatisfying the condition G_∞^∗ :=lim sup

ξ→+_∞

G(ξ,ξ)

ξ^p < +_∞, and for everyµ∈[0,µ_G[where

µ_G := ¹

p 2^p +₂^qp

^p_q G^∗_∞

,

the system







−(p−1)|u⁰₁(x)|^p−2u₁⁰⁰(x) = f₁(u₁,u₂) +µG_u1(u₁,u₂) in(0, 1),

−(q−1)|u⁰₂(x)|^q−2u⁰⁰₂(x) = f₂(u₁,u₂) +µG_u2(u₁,u₂) in(0, 1), u₁(0) =u₁(1) =u₂(0) =u₂(1) =0,

admits a sequence of pairwise distinct positive weak solutions.

2 Preliminaries

Our main tool to investigate the existence of infinitely many weak solutions for the system (D_λ,µ) is a smooth version of Theorem 2.1 of [5] that we recall here.

Theorem 2.1. Let X be a reflexive real Banach space, letΦ,Ψ: X−→_Rbe two Gâteaux differentiable functionals such thatΦis sequentially weakly lower semicontinuous, strongly continuous, and coercive andΨis sequentially weakly upper semicontinuous. For every r>_infXΦ, let us put

ϕ(r):= inf

u∈_Φ⁻¹(]−_∞,r[)

sup_v∈Φ−1(]−_∞,r])Ψ(v)−_Ψ(u) r−_Φ(u) ^, and

γ:=lim inf

r→+_∞ ϕ(r), δ:= lim inf

r→(infXΦ)⁺ϕ(r). Then, one has

(a) for every r > inf_XΦ and every λ ∈]0, ¹

ϕ(r)[, the restriction of the functional I_λ = _Φ−λΨ to Φ⁻¹(]−_∞,r[)admits a global minimum, which is a critical point (local minimum) of I_λ in X.

(b) Ifγ<+_∞then, for eachλ∈]0,_γ¹[, the following alternative holds:

either

(b₁) I_λpossesses a global minimum, or

(b₂) there is a sequence{u_n}of critical points (local minima) of I_λ such that

n→+lim_∞Φ(un) = +_∞.

(c) Ifδ <+_∞then, for eachλ∈]0,¹_δ[, the following alternative holds:

either

(c₁) there is a global minimum ofΦwhich is a local minimum of I_λ, or

(c2) there is a sequence of pairwise distinct critical points (local minima) of I_λ which weakly converges to a global minimum ofΦ.

LetXbe the Cartesian product ofnSobolev spacesW₀^1,p1([a,b]),W₀^1,p2([a,b]), . . . ,W₀^1,pn([a,b]), i.e.,X =_∏ⁿ_i=1W₀^1,pi([a,b]), equipped with the norm

k(u₁,u2, . . . ,un)k=

∑

ku⁰_ik_pi, for every(u₁,u2, . . . ,un)∈X, where

ku⁰_ik_pi = _{Z b}

a

|u⁰_i(x)|^pidx 1/pi

, i=1, . . . ,n.

Since p_i >1 fori=1, . . . ,n,Xis compactly embedded in (C([a,b]))ⁿ. In the sequel, let p=min{p_i; 1≤i≤n}, p=max{p_i; 1≤i≤n},

m_i := inf

(x,t)∈[a,b]×_Rh_i(x,t)>0 for 1≤i≤n, M_i := sup

(x,t)∈[a,b]×_R

h_i(x,t) for 1≤i≤n,

M := max{M_i; 1 ≤ i≤ n}andm := min{m_i; 1≤ i≤ n}. Then, M ≥ M_i ≥ m_i >m >0 for eachi=_{1, . . . ,}n.

In order to apply Theorem2.1 we set H_i(x,t) =

Z _t

0

^Z τ 0

(p_i−₁)|δ|^pi−2 h_i(x,δ) ^dδ

dτ,

for 1 ≤ i ≤ n and for all (x,t) ∈ [a,b]×R, and consider the functionals Φ, Ψ: X → _{R for} eachu= (u₁, . . . ,un)∈ X, as follows

Φ(u) =

∑

Z _b

a H_i(x,u⁰_i(x))dx, and

Ψ(u) =

Z _b

a F(x,u₁(x), . . . ,u_n(x))dx+ ^µ λ

Z _b

a G(x,u₁(x), . . . ,u_n(x))dx.

It is well known thatΨ is a Gâteaux differentiable functional and sequentially weakly lower semicontinuous whose Gâteaux derivative at the point u ∈ X is the functional Ψ⁰(u) ∈ X^∗, given by

Ψ⁰(u)(v) =

Z _b

a

∑

Fu_i(x,u₁(x), . . . ,un(x))v_i(x)dx+ ^µ λ

Z _b

a

∑

Gu_i(x,u₁(x), . . . ,un(x))v_i(x)dx, for every v = (v₁, . . . ,v_n) ∈ X, and Ψ⁰: X → X^∗ is a compact operator. Moreover, Φ is a Gâteaux differentiable functional whose Gâteaux derivative at the pointu∈ Xis the functional Φ⁰(u)∈X^∗, given by

Φ⁰(u₁, . . . ,un)(v₁, . . . ,vn) =

∑

Z _b

a

Z _u⁰

i(x) 0

(p_i−1)|τ|^pi−2 h_i(x,τ) ^dτ

v⁰_i(x)dx,

for everyv= (v₁, . . . ,v_n)∈X. Furthermore,Φis sequentially weakly lower semicontinuous.

By a classical solution of the system (D_λ,µ), we mean a functionu= (u₁, . . . ,u_n)such that, for i = _{1, . . . ,}n, u_i ∈ C¹[a,b]_, u⁰_i ∈ AC[a,b]_{, and} u satisfies (D_λ,µ). We say that a function u= (u₁, . . . ,un)∈ Xis a weak solution of the system (D_λ,µ) if

∑

Z _b

a

_{Z u}⁰

i(x) 0

(p_i−1)|τ|^pi−2 h_i(x,τ) ^dτ

v⁰_i(x)dx−λ Z _b

a

∑

Fu_i(x,u₁(x), . . . ,un(x))v_i(x)dx

−µ Z _b

a

∑

G_ui(x,u₁(x), . . . ,u_n(x))v_i(x)dx=0, for everyv= (v₁, . . . ,vn)∈X.

3 Main results

In this section, we present our main results. To be precise, we establish an existence result of infinitely many solutions to problem (D_λ,µ).

For allξ >0 we denote byK(ξ)_{the set}

(t₁, . . . ,t_n)∈_Rⁿ:

∑

|t_i| ≤ξ

. (3.1)

Let

p^∗ =

(p if b−a≥1, p if 0<b−a<1.

Put

A:= lim inf

ξ→+_∞ Z _b

a max

(t1,...,tn)∈K(ξ)F(x,t₁, . . . ,t_n)dx

ξ^p ,

B:= lim sup

(t1,...,tn)→_∞ Z _b−^b−a

4

a+^b−₄^a

F(x,t₁, . . . ,t_n)dx

∑

D_i(t_i)

,

where D_i(t_i):=

Z _a+^b−₄^a

a H_i x,t_i(p_i−1)(x−a)^pi−2 (^b−₄^a)^pi−1

! dx+

Z _b

b−^b−₄^a H_i x,−^ti(p_i−1)(b−x)^pi−2 (^b−₄^a)^pi−1

! dx, for each t_i ∈ _{R, for all}i=_{1, . . . ,}n,

λ₁:= ¹

B and λ₂:=





∑

p_i(b−a)^p∗−1M 2^p

_pi¹





p

A . (3.2)

Theorem 3.1. Assume that (A1) F(x,t₁, . . . ,t_n)≥0for each

x ∈

a,a+ ^b−a 4

∪

b− ^b−a 4 ,b

, t_i ∈_R, ∀i=_{1, . . . ,n,} (A2)

lim inf

ξ→+_∞ Z _b

a max

(t1,...,tn)∈K(ξ)F(x,t₁, . . . ,t_n)dx ξ^p

<

∑

p_i(b−a)^p∗−1M 2^p

¹

pi!p

lim sup

(t1,...,tn)→_∞

Z _b−^b−a

4

a+^b−₄^a F(x,t₁, . . . ,tn)dx

∑

D_i(t_i)

.

Then, for eachλ∈]λ₁,λ₂[and for every nonnegative arbitrary function G: [a,b]×_Rⁿ→_Rwhich is measurable in[a,b]and of class C¹(_Rⁿ)satisfying the condition

G_∞ :=lim sup

ξ→+_∞ Z _b

a max

(t1,...,tn)∈K(ξ)G(x,t₁, . . . ,t_n)dx

ξ^p <+_∞, (3.3)

and for everyµ∈[_0,µ_G,λ[where

µ_G,λ := 1−λ

∑

p_i(b−a)^p∗−1M 2^p

_pi¹ !p

A

∑

p_i(b−a)^p∗−1M 2^p

_pi¹!p

G_∞ ,

the system(D_λ,µ)has an unbounded sequence of weak solutions in X.

Proof. Our aim is to apply Theorem2.1. To this end, fixλ,µandGsatisfying our assumptions.

LetXbe the Sobolev space ∏ⁿi=1W₀^1,pi([a,b]). For anyu∈X, set Φ(u) =

∑

Z _b

a H_i(x,u⁰_i(x))dx, (3.4) and

Ψ(u) =

Z _b

a F(x,u₁(x), . . . ,u_n(x))dx+ ^µ λ

Z _b

a G(x,u₁(x), . . . ,u_n(x))dx. (3.5) It is well known that they satisfy all regularity assumptions requested in Theorem 2.1 and that the critical points inX of the functionalI_λ = _Φ−_λΨ are precisely the weak solutions of problem (D_λ,µ).

Let{ξ_k}be a real sequence of positive numbers such that lim

k→+_∞ξ_k = +_{∞, and} A= lim

k→+_∞ Z _b

a max

(t1,...,tn)∈K(ξk)F(x,t₁, . . . ,t_n)dx ξ_k^p

. Put

S=

∑

p_i(b−a)^p∗−1M 2^p

¹

pi!p

, and

r_k = ^ξ

p k

S,

for allk ∈ _{N. Since 0}< h_i(x,t)≤ Mfor each(x,t)∈ [a,b]×_Rfori=_{1, . . . ,}n, from (3.4) we see that

1 M

∑

ku⁰_ik^p_pⁱ_i

p_i ≤_Φ(u₁, ...,u_n)≤ ¹ m

∑

ku⁰_ik^p_pⁱ_i

p_i for all u= (u₁, ...,u_n)∈X. (3.6) Taking into account that

xmax∈[a,b]|u_i(x)| ≤ (b−a)^pi

−1 pi

2 ku⁰_ik_pi, for eachu_i ∈W₀^1,pi([a,b])(see [28]), we have

xmax∈[a,b]

∑

|u_i(x)|^pi

p_i ≤ (b−a)^p∗−1 2^p

∑

ku⁰_ik^p_pⁱ_i

p_i , (3.7)

for eachu= (u₁,u2, . . . ,un)∈X. This, for eachr>0, along with (3.6), ensures that Φ⁻¹(]−_∞,r])⊆

(

u ∈X; max

∑

|u_i(x)|^pi

p_i ≤ (b−a)^p∗−1Mr

2^p for each x∈[a,b] )

.

Therefore, one has

ϕ(r_k)≤

sup

v∈_Φ⁻¹(]−_∞,rk])

Ψ(v) r_k

≤S Z _b

a max

(t1,...,tn)∈K(ξ_k)F(x,t₁, . . . ,t_n)dx+ ^µ λ

Z _b

a max

(t1,...,tn)∈K(ξ_k)G(x,t₁, . . . ,t_n)dx ξ_k^p

≤S Z _b

a max

(t1,...,tn)∈K(ξk)F(x,t₁, . . . ,t_n)dx ξ_k^p

+Sµ λ

Z _b

a max

(t1,...,tn)∈K(ξk)G(x,t₁, . . . ,t_n)dx ξ_k^p

,

(3.8)

for all k∈_N. Therefore, from assumption (A2) and the condition (3.3) one has γ≤lim inf

k→+_∞ ϕ(r_k)≤SA+Sµ

λG_∞ <+_{∞. (3.9)}

Now, let {(η_i,k)} ⊆_Rⁿbe positive real sequences such thatη_i,k > 0 for alli=1, . . . ,nand for allk∈_{N, and}

k→+lim_{∞ n}

i

∑

η²_i,k ¹₂

= +_∞.

Put

B:= lim

k→+_∞

Z _b−^b−₄^a

a+^b−₄^a F(x,η_1,k, . . . ,η_n,k)dx

∑

D_i(η_i,k)

. (3.10)

Let{w_k = (w_1,k(x), ...,w_n,k(x))}be a sequence inX defined by

w_i,k(x) =

















 4

b−a pi−1

η_i,k(x−a)^pi−1 if a≤ x<a+ ^b−a 4 ,

η_i,k if a+^b−a

4 ≤x≤ b−^b−a 4 , 4

b−a pi−1

η_i,k(b−x)^pi−1 if b−^b−a

4 <x≤ b,

(3.11)

for each i=1, . . . ,n. Clearlyw_k(x)∈_∏ⁿ_i=1W₀^1,pi([a,b])for each k∈_N.

Hence, we have

Φ(w_k) =

∑

Z _b

a H_i(x,w⁰_i,k)dx

=

∑

"

Z _a+^b−₄^a

a H_i

x,η_i,k(p_i−1)(x−a)^pi−2 (^b−₄^a)^pi−1

dx+

Z _b−^b−₄^a

a+^b−₄^a H_i(x, 0)dx +

Z _b

b−^b−₄^a H_i

x,−^ηi,k(p_i−1)(b−x)^pi−2 (^b−₄^a)^pi−1

dx

#

=

∑

D_i(η_i,k)_.

(3.12)

On the other hand, since Gis nonnegative and bearing assumption (A1) in mind, from (3.5) one has

Ψ(w_k) =

Z _b

a F(x,η_1,k, . . . ,η_n,k)dx+ ^µ λ

Z _b

a G(x,η_1,k, . . . ,η_n,k)dx

≥

Z _b

a F(x,η_1,k, . . . ,η_n,k)dx

≥

Z _b−^b−a

4

a+^b−₄^a F(x,η_1,k, . . . ,η_n,k)dx,

(3.13)

and so

I_λ(w_k) =_Φ(w_k)−_λΨ(w_k)≤

∑

D_i(η_i,k)−λ

Z _b−^b−a

4

a+^b−₄^a F(x,η_1,k, . . . ,η_n,k)dx.

Now, consider the following cases.

IfB<+_{∞, let}e∈0,B− ¹_λ. From (3.10), there existsνesuch that Z _b−^b−a

4

a+^b−₄^a F(x,η_1,k, . . . ,η_n,k)dx>(B−e)

∑

D_i(η_i,k), for all k >νe, and so

I_λ(w_k)<

∑

D_i(η_i,k)−λ(B−e)

∑

D_i(η_i,k) =

∑

D_i(η_i,k) [1−λ(B−e)]. Since 1−λ(B−e)<0, and taking into account (3.6) and (3.12) one has

k→+lim_∞I_λ(w_k) =−_∞.

IfB= +_{∞, fix} M> _λ¹. From (3.10), there existsνM such that Z _b−^b−₄^a

a+^b−₄^a F(x,η_1,k, . . . ,η_n,k)dx> M

∑

D_i(η_i,k), for all k >ν_M, and moreover,

I_λ(w_k)<

∑

D_i(η_i,k)[1−λM]. Since 1−λM<0, and arguing as before, we have

k→+lim_∞I_λ(w_k) =−_∞.

Taking into account that

1 B, S

A

⊆

0, 1 γ

,

and that I_λ does not possess a global minimum, from part (b) of Theorem 2.1, there exists an unbounded sequence{u_k} of critical points which are the weak solutions of (D_λ,µ). So, our conclusion is achieved.

Proof of Theorem1.1. Since f₁ and f₂ are positive, then Fis nonnegative inR²+. Moreover, one has that the functionst₁→ F(t₁,t2),t2∈_{R, and}t2 →F(t₁,t2),t₁∈_Rare increasing inRand, hence, max_{(t1,t2)∈K(ξ)}F(t₁,t₂)≤ F(ξ,ξ)for everyξ ∈ _R⁺. Therefore,

lim inf

ξ→+_∞ Z ₁

0 max

(t₁,t₂)∈K(ξ)F(t₁,t₂)dx ξ^p

≤lim inf

ξ→+_∞ Z ₁

0 F(ξ,ξ)dx ξ^p

=lim inf

ξ→+_∞

F(ξ,ξ) ξ^p

=0.

On the other hand, one has

h₁(u⁰₁) =1 and h₁(u⁰₁) =1.

By simple calculations, we see that

H₁(t₁) = |t1|^p

p and H2(t2) = |t2|^q q . Moreover,

D₁(t₁) = ⁴

p−1|t₁|^p p

(p−1)^p−2+ (1−p)^p−2, and

D₂(t₂) = ⁴

q−1|t₂|^q q

(q−1)^q−2+ (1−q)^q−2. Since p≤q, one has

D₁(t₁) +D₂(t₂)≤ ⁴

q−1|t₁|^p

p [(q−1)^q−2+ (1+q)^q−2] +⁴

q−1|t2|^q q

(q−1)^q−2+ (1+q)^q−2

≤ ⁴

q−1[(q−1)^q−2+ (1+q)^q−2]

p (|t₁|^p+|t₂|^q)

≤ ⁴

q−1[(q−1)^q−2+ (1+q)^q−2]

p (|t₁|^q+|t₂|^q). Then

+_∞= ^p

4^q−1[(q−1)^q−2+ (1+q)^q−2] 1

2lim sup

ξ→+_∞

F(ξ,ξ) ξ^q

≤ lim sup

ξ→+_∞

F(ξ,ξ)

D₁(ξ) +D₂(ξ) ≤ lim sup

(t1,t2)→_∞

F(t₁,t₂) D₁(t₁) +D₂(t₂)^. Now, arguing as before we obtain

G^∗_∞ =lim sup

ξ→+_∞ Z ₁

0 max

(t1,t2)∈K(ξ)G(_t1,t2)_dx

ξ^p ≤lim sup

ξ→+_∞

G(ξ,ξ)

ξ^p ≤+_∞.

Therefore, since one has also that

µ_G= ¹

p 2^p

+ ^q

2^{p p}_q

G_∞^∗ ,

λ₁=0,

and

λ₂ = +_∞.

Theorem3.1, taking into account the positivity of f andg, ensures the conclusion.

We now exhibit an example in which the hypotheses of Theorem3.1 are satisfied.

Example 3.2. Put p₁ = p₂ = 2, [a,b] = [0, 1]and consider the increasing sequence of positive real numbers given by

a₁ =2, a_k+1= ka²_k+2, for everyk∈_{N. Let} F: R² →_Rbe a function such that

F(t₁,t₂) =







(a_k+₁)⁴e⁻

1

1−[(t1−_ak+1)²+(t2−_ak+1)²]+1

(t₁,t₂)∈ ^[

k≥1

B((a_k+₁,a_k+₁), 1),

0 otherwise,

whereB((a_k+1,a_k+1), 1)denotes the open unit ball of center(a_k+1,a_k+1)and radius 1.

Now, put

h₁(y) =h₂(y) = ¹ 2+cosy, for eachy∈R. By simple calculations, we see that

H₁(y) = H₂(y) =y²−cosy+1, for eachy∈_{R, and}

D₁(t₁) = ^16t

21−cos(4t₁) +1

2 and D₂(t₂) = ^16t

22−cos(4t2) +1

2 .

By definition, F is nonnegative and F(0, 0) = 0. Further it is a simple matter to verify that F ∈ C¹(_R²). We will denote by f₁ and f₂ respectively the partial derivative of F respect to t₁ andt2. Now, for everyk ∈ N, the restrictionF(t₁,t2)|_{B((ak+1,ak+1),1)} attains its maximum in (a_k+1,a_k+1)and one hasF(a_k+1,a_k+1) = (a_k+1)⁴. Clearly

lim sup

(t1,t2)→_∞ Z ³

4 1 4

F(t₁,t2)dx D₁(t₁) +D₂(t₂) = ¹

2 lim sup

(t1,t2)→_∞

F(t₁,t₂)

D₁(t₁) +D₂(t₂) = +_∞, since

k→+lim_∞

F(a_k+1,a_k+1)

D₁(a_k+1) +D₂(a_k+1) = lim

k→+_∞

a⁴_k+1

16a²_k+1−cos(4a_k+1) +1 = +_∞.

On the other hand, by settingξ_k =a_k+1−1 for everyk ∈_{N, one has}

|t1|+|maxt2|≤ξ_k

F(t₁,t2) =a⁴_k, ∀k∈_N.

Then

k→+lim_∞

|t₁|+|maxt₂|≤ξ_k

F(t₁,t₂) (a_k+1−₁)² =0,

since

lim inf

ξ→+_∞ Z ₁

0 max

|t1|+|t2|≤ξ

F(t₁,t₂)dx

ξ² =0.

Hence, condition (A2) is provided.

Now, letG: R² →_Rbe a function defined by

G(t₁,t2) =1−cos(t₁t2).

By definitionG∈C¹(_R²)andG(0, 0) =0. For any sequence{ρ_k}_k∈Nsuch that lim

k→+_∞ρ_k = +_∞, since |t₁|+|t₂| ≤ρ_k, one has

|t1|+|maxt2|≤ρ_k

G(t₁,t2)≤2.

Then,

0≤ G_∞=_{lim sup}

ξ→+_∞ Z ₁

0 max

|t1|+|t2|≤ξ

G(t₁,t₂)dx ξ²

≤_0.

All hypotheses of Theorem3.1are satisfied. Then for all(λ,µ)∈]0,+_∞[×[0,+_∞[, the system















−u⁰⁰₁(x) =λf₁(u₁,u₂) +µG_u1(u₁,u₂) ¹

2+cos(u⁰₁(x))^,

−u⁰⁰₂(x) =λf₂(u₁,u₂) +µG_u2(u₁,u₂) ¹

2+cos(u⁰₂(x))^, u₁(0) =u₁(1) =u₂(0) =u₂(1) =0,

admits a sequence of weak solutions which is unbounded inW₀^1,2([0, 1])×W₀^1,2([0, 1]).

Remark 3.3. Under the conditions A= 0 andB= +_{∞, Theorem}3.1concludes that for every λ>0 and for each

µ∈















0, 1

∑

p_i(b−a)^p∗−1M 2^p

¹

pi!p

G_∞













 ,

the system (D_λ,µ) admits infinitely many weak solutions inX. Moreover, ifG_∞ =0, the result holds for every λ>0 andµ≥0.

Remark 3.4. Put

cλ₁ =λ₁, and

cλ₂ = ¹

k→+lim_∞ Z _b

a sup

(t1,...,tn)∈K(b_k)

F(x,t₁, . . . ,t_n)dx−

Z _b−^b−a

4

a+^b−₄^a F(x,a_1,k, . . . ,a_n,k)dx b_k^p

ⁿ

i

∑

p_i(b−a)^p∗−1M 2^p

¹

pip −

∑

D_i(a_i,k)

.

We explicitly observe that assumption (A2) in Theorem3.1could be replaced by the following more general condition

(A3) there existn+1 sequences{a_i,k}fori=1, . . . ,nand{b_k}with

∑

D_i(a_i,k)< ^b

p k

∑

p_i(b−a)^p∗−1M 2^p

_pi¹ !p,

for everyk∈_{Nand limk→+∞}b_k = +_{∞such that}

k→+lim_∞ Z _b

a max

(t1,...,tn)∈K(b_k)F(x,t₁, . . . ,t_n)dx−

Z _b−^b−a

4

a+^b−₄^a F(x,a_1,k, . . . ,a_n,k)dx b^p_k

∑

p_i(b−a)^p∗−1M 2^p

_pi¹ !p −

∑

D_i(a_i,k)

<

∑

p_i(b−a)^p∗−1M 2^p

_pi¹ !p

lim sup

(t1,...,tn)→_∞

Z _b−^b−a

4

a+^b−₄^a F(x,t₁, . . . ,t_n)dx

∑

D_i(t_i)

,

whereK(b_k) ={(t₁, . . . ,t_n)|_∑iⁿ₌₁|t_i| ≤b_k}(see (3.1)).

Obviously, from (A3) we obtain (A2), by choosing a_i,k = 0 for all k ∈ N. Moreover, if we assume (A3) instead of (A2) and set

r_k = ^b

p k

∑

p_i(b−a)^p∗−1M 2^p

_pi¹!p,

for allk∈N, by the same arguing as inside in Theorem3.1, we obtain

ϕ(r_k) = inf

u∈_Φ⁻¹(]−_∞,rk[)

sup

v∈_Φ⁻¹(]−_∞,rk])

Ψ(v)

−_Ψ(u) r_k−_Φ(u)

≤

sup

v∈_Φ⁻¹(]−_∞,rk])

Ψ(v)− Z _b

a F(x,u₁(x), . . . ,un(x))dx+ ^µ λ

Z _b

a G(x,u₁(x), . . . ,un(x))dx

r_k−

∑

Z _b

a H_i(x,u⁰_i(x))dx

≤

Z _b

a max

(t1,...,tn)∈K(bk)F(x,t₁, . . . ,t_n)dx−

Z _b−^b−a

4

a+^b−₄^a F(x,a_1,k, . . . ,a_n,k)dx b_k^p

∑

p_i(b−a)^p∗−1M 2^p

_pi¹ !p −

∑

D_i(a_i,k)

.

We have the same conclusion as in Theorem3.1with Λreplaced byΛ⁰ := cλ₁,cλ₂ .

Here, we point out two simple consequences of Theorem3.1.

Corollary 3.5. Assume that (A1) holds and

(B1) lim inf

ξ→+_∞ Z _b

a max

(t1,...,tn)∈K(ξ)F(x,t₁, . . . ,t_n)dx ξ^p

<

∑

p_i(b−a)^p∗−1M 2^p

_pi¹!p

,

(B2) lim sup

(t1,...,tn)→_∞

Z _b−^b−a

4

a+^b−₄^a F(x,t₁, . . . ,tn)dx

∑

D_i(t_i)

>1,

where∑ⁿi=1D_i(t_i)is given as in assumption (A2).

Then, for every nonnegative function G: [a,b]×_Rⁿ →_Rwhich is measurable in[a,b]and of class C¹(_Rⁿ)satisfying(3.3)and for everyµ∈[0,µ_G[where

µ_G := 1−

∑

p_i(b−a)^p∗−1M 2^p

_pi¹!p

A

∑

p_i(b−a)^p∗−1M 2^p

_pi¹!p

G_∞ ,

the system

(−(p_i−1)|u⁰_i(x)|^pi−2u⁰⁰_i (x) = (F_ui(x,u₁, . . . ,u_n) +µG_ui(x,u₁, . . . ,u_n))h_i(x,u⁰_i), x∈ (a,b), u_i(a) =u_i(b) =0,

for1≤i≤n, has an unbounded sequence of weak solutions in X.

Theorem 3.6. Assume that the assumptions (A1) and (A2) in Theorem3.1hold.

Then, for eachλ∈]λ₁,λ2[whereλ₁andλ2 are given in(3.1), the system (−(p_i−1)|u⁰_i(x)|^pi−2u_i⁰⁰(x) =λF_ui(x,u₁, . . . ,u_n), x∈(a,b),

u_i(a) =u_i(b) =0,

for1≤i≤n, has an unbounded sequence of weak solutions in X.

Remark 3.7. We observe that in Theorem 3.1 we can replace ξ → +_∞ with ξ → 0⁺, and arguing in the same way as in the proof of Theorem 3.1, but using conclusion (c) of Theorem 2.1, the system(D_λ,µ)has a sequence of weak solutions, which strongly converges to 0 inX.

References

[1] G. A. Afrouzi, A. Hadjian, Infinitely many solutions for a class of Dirichlet quasilinear elliptic systems,J. Math. Anal. Appl.393(2012), 265–272.MR2921667

[2] G. A. Afrouzi, S. Heidarkhani, Three solutions for a quasilinear boundary value prob- lem,Nonlinear Anal.69(2008), 3330–3336.MR2450542