Infinitely many weak solutions for a fourth-order equation on the whole space

Infinitely many weak solutions for a fourth-order equation on the whole space

Mohammad Reza Heidari Tavani

and Mehdi Khodabakhshi

1Department of Mathematics, Ramhormoz branch, Islamic Azad University, Ramhormoz, Iran

2Department of Mathematics and Computer Sciences, Amir Kabir University of Technology, Tehran, Iran

Received 30 June 2020, appeared 20 May 2021 Communicated by Gabriele Bonanno

Abstract. The existence of infinitely many weak solutions for a fourth-order equation on the whole space with a perturbed nonlinear term is investigated. Our approach is based on variational methods and critical point theory.

Keywords:weak solution, fourth-order equation, critical point theory, variational meth- ods.

2020 Mathematics Subject Classification: 34B40, 34B15, 47H14.

1 Introduction

In this paper we consider the following problem

u^iυ(x)−(q(x)u⁰(x))⁰+s(x)u(x) =λf(x,u(x)), a.e.x∈ _R, (1.1) where λ is a positive parameter and q,s ∈ L^∞(_R) with q₀ = ess inf_Rq > 0 and s₀ = ess inf_Rs>0. Here the function f :R²→_Ris an L¹-Carathéodory function.

As we know, differential equations have many applications in engineering and mechanical science. Many important engineering topics eventually lead to a differential equation. One of the most important and widely used types of such equations is the fourth-order differen- tial equation. These equations play an essential role in describing the large number of elastic deflections in beams. Due to the importance of these equations in applied sciences, many authors have studied different types of these equations and obtained important results. Re- search on the existence and multiplicity of solutions for different types of these equations can be seen in the work of many authors. For example, to study fourth-order two-point boundary value problems we refer the reader to references [3–5,8,10–12].

For instance in [3], the authors researched the following problem:

(u^iυ+Au⁰⁰+Bu= λf(t,u), t ∈[0, 1],

u(0) =u(1) =u⁰⁰(0) =u⁰⁰(1) =0 (1.2)

BCorresponding author. Email: m.reza.h56@gmail.com

where A and B are real constants and they achieved multiplicity results using variational methods and critical point theory. It should be noted that in the study of many important problems such as mathematical models of beam deflection, the differential equation is consid- ered at infinite interval. Also, because the operators used to solve equations such as (1.1) onR are not compact, so the study of such problems is very important. That is why some authors have turned their attention to the whole space. For example in [9], applying the critical point theory the author has studied the existence and multiplicity of solutions for the following problem:

u^iυ(x) +Au⁰⁰(x) +Bu(x) =λα(x).f(u(x)), a.e.x ∈ _R, (1.3) whereAis a real negative constant andBis a real positive constant,λis a positive parameter andα, f :R→ _Rare two functions such that α∈ L¹(_R), α(x)≥0, for a.e. x ∈_R,α6≡0 and also f is continuous and non-negative.

In this work, using a critical point theorem obtained in [2] which we recall in the next section (Theorem 2.7), we establish the existence of infinitely many weak solutions for the problem (1.1).

2 Preliminaries

Let us recall some basic concepts.

Definition 2.1. A function f :R²→_Ris said to be an L¹-Carathéodory function, if (C₁) the functionx7→ f(x,t)is measurable for everyt ∈_R,

(C₂) the functiont7→ f(x,t)is continuous for almost everyx ∈_R, (C3) for everyρ>0 there exists a functionlρ(x)∈ L¹(_R)such that

sup

|t|≤ρ

|f(x,t)| ≤lρ(x), for a.e.x∈_R.

DenoteW₀^2,2(_R)is the closure ofC₀^∞(_R)inW^2,2(_R)and according to the properties of the Sobolev spaces, we know thatW₀^2,2(_R) =W^2,2(_R), [1, Corollary 3.19].

We denote by| · |_t the usual norm onL^t(_R), for allt ∈ [1,+_∞]and it is well known that W^2,2(_R)is continuously embedded inL^∞(_R), [6, Corollary 9.13].

The Sobolev spaceW^2,2(_R)is equipped with the following norm kuk_W2,2(_R)=

_Z

R(|u⁰⁰(x)|²+|u⁰(x)|²+|u(x)|²)dx 1/2

, for allu∈W^2,2(_R). Also, we considerW^2,2(_R)with the norm

kuk= Z

R(|u⁰⁰(x)|²+q(x)|u⁰(x)|²+s(x)|u(x)|²)dx 1/2

, for allu∈W^2,2(_R). According to

(_min{_1,q₀,s₀})¹²kuk_W2,2(_R)≤ kuk ≤(_max{_1,|q|_∞,|s|_∞})¹²kuk_W2,2(_R), (2.1)

the norm k · kis equivalent to the k · k_W2,2(_R) norm. Since the embeddingW^2,2(_R)→ L^∞(_R) is continuous hence there exists a constantCq,s(depending on the functions q and s) such that

|u|_∞≤ C_q,skuk, ∀u∈W^2,2(_R).

In the following proposition, we provide an approximation for this constant.

Proposition 2.2. We have

|u|_∞ ≤ Cq,skuk (2.2)

where C_q,s = ¹

4|q|_∞|s|_∞

¹_{4 max{1,|q|∞,|s|∞}}

min{1,q0,s0}

¹₂ .

Proof. Letv∈W^1,1(_R), then from [7, p. 138, formula 4.64], one has

|v(x)| ≤ ¹ 2 Z

R|v⁰(t)|dt. (2.3)

Now if u ∈ W^2,2(_R) then v(x) = (|q|_∞|s|_∞)¹²|u(x)|² ∈ W^1,1(_R) and thus from (2.3) and Hölder’s inequality one has,

(|q|_∞|s|_∞)¹²|u(x)|²≤

Z

R(|q|_∞|s|_∞)¹²|u⁰(t)||u(t)|dt≤((|q|_∞)¹²|u⁰|₂)(|s|_∞¹²|u|₂) that is ,

|u(x)| ≤

1

|q|_∞|s|_∞ ¹₄

((|q|_∞)¹²|u⁰|₂)¹²(|s|_∞¹²|u|₂)¹². (2.4) Now according to x^ay^1−a ≤ a^a(1−a)^1−a(x+y), x,y ≥ 0, 0 < a < 1 [7, p. 130, formula 4.47], and classical inequalitya^1p +b^1p ≤2

(p−1)

p (a+b)^1p, from (2.1) and (2.4) one has

|u(x)| ≤

1

|q|_∞|s|_∞ ¹₄

1 2

¹₂ 1 2

¹_2Z

R|q|_∞|u⁰(t)|²dt ¹₂

+ _Z

R|s|_∞|u(t)|²dt ¹₂

≤

1

|q|_∞|s|_∞ ¹₄

1 2

¹₂ 1 2

¹₂ (2)¹²

_Z

R(|q|_∞|u⁰(t)|²+|s|_∞|u(t)|²)dt ¹₂

≤

1 4|q|_∞|s|_∞

¹_4Z

R(|u⁰⁰(t)|²+|q|_∞|u⁰(t)|²+|s|_∞|u(t)|²)dt ¹₂

≤

1 4|q|_∞|s|_∞

¹₄

max{1,|q|_∞,|s|_∞} min{1,q₀,s₀}

¹₂ Z

R(|u⁰⁰(t)|²+|u⁰(t)|²+|u(t)|²)dt ¹₂

which means that|u|_∞≤ Cq,skuk.

LetΦ, Ψ:W^2,2(_R)→_Rbe defined by Φ(u) = ¹

2kuk² = ¹ 2 Z

R(|u⁰⁰(x)|²+q(x)|u⁰(x)|²+s(x)|u(x)|²)dx (2.5) and

Ψ(u) =

Z

RF(x,u(x))dx (2.6)

for everyu∈W^2,2(_R)_where F(x,ξ) =Rξ

0 f(x,t)dtfor all (x,ξ)∈_R². It is well known thatΨ is a sequentially weakly upper semicontinuous whose differential at the pointu∈W^2,2(_R)is

Ψ⁰(u)(v) =

Z

Rf(x,u(x))v(x)dx.

It is clear that Φ is a strongly continuous and coercive functional. Also since the norm k · k on Hilbert space W^2,2(_R)is a weakly sequentially lower semi-continuous functional in W^2,2(_R) therefore Φ is a sequentially weakly lower semicontinuous functional on W^2,2(_R). Moreover,Φis continuously Gâteaux differentiable functional whose differential at the point u∈W^2,2(_R)is

Φ⁰(u)(v) =

Z

R(u⁰⁰(x)v⁰⁰(x) +q(x)u⁰(x)v⁰(x) +s(x)u(x)v(x))dx for everyv∈W^2,2(_R).

Definition 2.3. Let Φ and Ψ be defined as above. Put I_λ = _Φ−_λΨ, λ > 0. We say that u ∈ W^2,2(_R) is a critical point of Iλ when I_λ⁰(u) = 0_{W^2,2_(R)^∗_}, that is, I_λ⁰(u)(v) = 0 for all v∈W^2,2(_R).

Definition 2.4. A functionu :R → _R is a weak solution to the problem (1.1) if u ∈ W^2,2(_R)

and _Z

R u⁰⁰(x)v⁰⁰(x) +q(x)u⁰(x)v⁰(x) +s(x)u(x)v(x)−λf(x,u(x))v(x)dx=0, for allv∈W^2,2(_R).

Remark 2.5. We clearly observe that the weak solutions of the problem (1.1) are exactly the solutions of the equationI_λ⁰(u)(v) =_Φ⁰(u)(v)−_λΨ⁰(u)(v) =0.

Lemma 2.6. Suppose that f : R² → _R is a non-negative L¹-Carathéodory function. If u0 6≡ 0is a weak solution for problem(1.1)then u₀is non-negative.

Proof. From Remark2.5, one has Φ⁰(u0)(v)−_λΨ⁰(u0)(v) = 0 for allv ∈W^2,2(_R). Let v(x) =

¯

u₀ =max{−u₀(x), 0}and we assume that E={x ∈_R:u₀(x)<0}. Then we have Z

E∪E^c

(u₀⁰⁰(x)u¯⁰⁰₀(x) +q(x)u⁰₀(x)u¯₀⁰(x) +s(x)u₀(x)u¯₀(x))dx=

Z

Rλf(x,u₀(x))u¯₀(x)dx,

that is _Z

E

(−|u¯⁰⁰₀(x)|²−q(x)|u¯⁰₀(x)|²−s(x)|u¯₀(x)|²)dx≥0 which means thatku¯₀k=0 and henceu₀ ≥0 and the proof is complete .

Our main tool is the following critical point theorem.

Theorem 2.7([2, Theorem 2.1]). Let X be a reflexive real Banach space, letΦ,Ψ : X → _Rbe two Gâteaux differentiable functionals such that Φis sequentially weakly lower semicontinuous, strongly continuous, and coercive andΨ is sequentially weakly upper semicontinuous. For every r > inf_XΦ, 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_XΦand everyλ ∈ 0, _ϕ¹_(r)

, the restriction of the functional I_λ = _Φ−_λΨto Φ⁻¹(]−_∞,r[)admits a global minimum, which is a critical point (local minimum) of I_λin 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(u_n) = +_∞. (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, with limn→+_∞Φ(u_n) =_infXΦ, which weakly converges to a global minimum ofΦ.

3 Main results

Let

τ:= ⁵⁴⁰

86111(max{1,|q|_∞,|s|_∞})Cq,s2, (3.1) A:=lim inf

ρ→+_∞

R

Rsup_|t|≤ρF(x,t)dx

ρ² , (3.2)

and

B:=_{lim sup}

ρ→+∞

R⁵₈

3 8

F(x,ρ)dx

ρ² . (3.3)

Now we formulate our main result as follows.

Theorem 3.1. Let f :R²→_Rbe an L¹-Carathéodory function, and assume that (i) F(x,t)≥0for every(x,t)∈_R×]0,³₈[∪]⁵₈, 1[,

(ii) A<τB, whereτ,A and B are given by(3.1),(3.2)and(3.3)respectively.

Then for every

λ∈

#(max{1,|q|_∞,|s|_∞}) B

86111 1080, 1

2A C_q,s²

"

the problem(1.1)admits a sequence of many weak solutions which is unbounded in W^2,2(_R)_.

Proof. Fixλas in our conclusion. Our aim is to apply Theorem2.7, part(b)withX =W^2,2(_R), andΦ,Ψ are the functionals introduced in section 2. As shown in the previous section, the functionalsΦ and Ψ satisfy all regularity assumptions requested in Theorem 2.7. Now, we look on the existence of critical points of the functional I_λ in W^2,2(_R). To this end, we take {ρ_n}be a sequence of positive numbers such that limn→_∞ρ_n= +_∞and

nlim→_∞

R

Rsup_|t|≤ρnF(x,t)dx ρ²_n = A.

Setrn := ¹_{2 C}^ρn

q,s

2

, for everyn∈_N.

For eachu∈W^2,2(_R)and bearing (2.2) in mind, we see that Φ⁻¹(]−_∞,rn[) ={u∈W^2,2(_R);Φ(u)<rn}

=

u∈W^2,2(_R); 1

2kuk² < ¹ 2

ρn

C_q,s 2

=u∈ X; Cq,skuk<ρn ⊆u∈W^2,2(_R); |u|_∞ ≤ρn . Now, since 0∈_Φ⁻¹(]−_∞,r_n[)then we have the following inequalities:

ϕ(r_n) = inf

u∈_Φ⁻¹(]−_∞,rn[)

sup_v∈Φ−1(]−_∞,rn[)

R

RF(x,v(x))dx−R

RF(x,u(x))dx rn− ^ku₂^k2

≤ R

Rsup_|t|≤ρnF(x,t)dx

r_n =

R

Rsup_|t|≤ρnF(x,t)dx

1 2

ρn

Cq,s

2

=₂C_q,s² R

Rsup_|t|≤ρnF(x,t)dx ρn2 , for everyn∈N. Hence, it follows that

γ≤lim inf

n→_∞ Φ(r_n)≤2C_q,s²A< +_∞,

because condition (ii)_shows A < +∞. Now, we prove that the functional I_λ is unbounded from below. For our goal, let{ηn}be a sequence of positive numbers such that limn→_∞ηn = +_∞and

n→+lim_∞ R ⁵₈

3 8

F(x,ηn)dx

η_n² = B. (3.4)

Let{v_n}be a sequence inW^2,2(_R)which is defined by

v_n(x):=















−⁶⁴₉^ηn x²− ³₄x

, if x∈0,³₈ , ηn, if x∈³₈,⁵₈

,

−⁶⁴₉^ηn x²− ⁵₄x+¹₄, if x∈⁵₈, 1 ,

0, otherwise.

(3.5)

One can compute that

kv_nk_W2,2(_R)

2 = ⁸⁶¹¹¹ 540 η_n², and so from (2.1) we have

(min{1,q₀,s₀})⁸⁶¹¹¹

1080 η_n² ≤_Φ(v_n)≤(max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹

1080 η_n². (3.6) Also, by using condition(i), we infer

Z

RF(x,v_n(x))dx≥

Z ⁵

8 3 8

F(x,η_n)dx, for every n∈_N. Therefore, we have

I_λ(vn) =_Φ(vn)−λΨ(vn)≤(max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹

1080 ηn2−λ Z ⁵₈

3 8

F(x,ηn)dx, for every n∈_{N. If}B<+_{∞, let}

e∈

(_max{_1,|q|_∞,|s|_∞}) λB

86111 1080, 1

. By (3.4) there isN_e such that

Z ⁵

8 3 8

F(x,η_n)dx>eBη_n², (∀n> N_e)_. Consequently, one has

I_λ(vn)≤ (max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹

1080 ηn2−λeBηn2

= η_n²

(max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹

1080 −λeB

, for every n> Ne. Thus, it follows that

nlim→_∞Iλ(vn) =−_∞.

If B= +∞, then consider

M > (max{1,|q|_∞,|s|_∞}) λ

86111 1080. By (3.4) there isN(M)such that

Z ⁵

8 3 8

F(x,η_n)dx> Mη_n², (∀n> N(M)). So, we have

Iλ(vn)≤ (max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹

1080 η_n²−λMηn2

= ηn2

(max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹ 1080 −λM

,

for every n> N(M). Taking into account the choice of M, also in this case, one has

nlim→_∞I_λ(v_n) =−_∞.

Therefore according to Theorem2.7, the functional I_λ admits an unbounded sequence{un} ⊂ W^2,2(_R) of critical points. It means that, problem (1.1) admits a sequence of many weak solutions which is unbounded inW^2,2(_R)_.

Now we present the following example to illustrate Theorem3.1.

Example 3.2. LetF :R²→_Rbe the function defined as F(x,t):=







t⁵ 1−cos(ln|t|)

1+x² , if(x,t)∈_R×_R− {0} 0, if(x,t)∈_R× {0} and therefore we have

f(x,t):=







5t⁴ 1−cos(ln|t|)

+t⁴sin(ln|t|)

1+x² , if (x,t)∈_R×_R− {0}

0, if (x,t)∈_R× {0}.

We observe that

A:=lim inf

ρ→+_∞

R

Rsup_|t|≤ρF(x,t)dx

ρ² =0 (3.7)

and

B:=lim sup

ρ→+_∞

R⁵₈

3 8

F(x,ρ)dx

ρ² = +_∞. (3.8)

So, by Theorem3.1, for everyλ∈(0,+_∞)the problem







u^iυ(x)− (1+e^−x2)u⁰(x)⁰+ (π+tan⁻¹x)u(x)

=λ^5u

(x)⁴ 1−cos(ln|u(x)|)

+u(x)⁴sin(ln|u(x)|)

1+x² , a.e.x ∈ _R,

(3.9)

has a sequence of weak solutions which is unbounded inW^2,2(_R).

Note that, as in the previous example, under appropriate conditions, the existence of infinitely many weak solutions for problem (1.1) will be guaranteed for anyλ∈_R⁺. For this case, the following result is a consequence of Theorem3.1.

Corollary 3.3. Suppose that f : R² → _R is an L¹-Carathéodory function. Also, assume that the assumption (i)in Theorem 3.1 holds and A = _∞ and B = 0 where A and B are given by(3.2)and (3.3)respectively. Then, for everyλ>0,the problem(1.1)possesses a sequence of many weak solutions which is unbounded in W^2,2(_R)_.

A special case of Theorem3.1is given in the following corollary.

Corollary 3.4. Suppose that f : R² → _R is an L¹-Carathéodory function. Also, assume that the assumption(i)in Theorem3.1holds and

(i₁) (max{1,|q|_∞,|s|_∞})⁸⁶¹¹¹₁₀₈₀ < B, (i₂) A< ¹

2Cq,s2. Then, the problem

u^iυ(x)−(q(x)u⁰(x))⁰+s(x)u(x) = f(x,u(x)), a.e. x ∈ _R, (3.10) possesses a sequence of many weak solutions which is unbounded in W^2,2(_R)_.

Proof. The corollary is an immediate consequence of Theorem3.1whenλ=_1.

Remark 3.5. In Theorem 3.1, we can consider f(x,t) = β(x)g(t) where β,g : R → _R are two functions such that β ∈ L¹(_R), β ≥ 0 , for a.e. x ∈ _R, β 6≡ 0 and also g is continuous and non-negative. We set G(t) = Rt

0g(ξ)dξ for all t ∈ _{R. Since} G⁰(t) = g(t) ≥ 0 then G is non-decreasing function. Therefore (3.2) and (3.3) become the following simpler forms:

A:=lim inf

ρ→+_∞

R

Rsup_|t|≤ρF(x,t)dx

ρ² =lim inf

ρ→+_∞

G(ρ)|β|₁

ρ² (3.11)

and

B:=lim sup

ρ→+_∞

R⁵₈

3 8

F(x,ρ)dx

ρ² =lim sup

ρ→+_∞

G(ρ)R⁵₈

3 8

β(x)dx

ρ² . (3.12)

Now if we assume that A<τBwhere τ,AandBare given by (3.1), (3.11) and (3.12) respec- tively, then according to Theorem3.1 and Lemma2.6for every

λ∈

#(max{1,|q|_∞,|s|_∞}) B

86111 1080, 1

2A C_q,s²

"

the problem

u^iυ(x)−(q(x)u⁰(x))⁰+s(x)u(x) =λ β(x)g(u(x))_{, a.e.} x ∈ _{R, (3.13)} admits a sequence of many non-negative weak solutions which is unbounded inW^2,2(_R).

Using the conclusion (c) instead of (b) in Theorem 3.1, can be obtained a sequence of pairwise distinct weak solutions to the problem (1.1) which converges uniformly to zero. In this case, by replacing ρ → +_{∞ with} ρ → ₀⁺_, A and B will be converted to the following forms:

A⁰ :=lim inf

ρ→0⁺

R

Rsup_|t|≤ρF(x,t)dx

ρ² (3.14)

and

B⁰ :=lim sup

ρ→0⁺

R⁵₈

3 8

F(x,ρ)dx

ρ² . (3.15)

Therefore, we can present the other main result of this section as follows.

Corollary 3.6. Let f :R² →_Rbe an L¹-Carathéodory function, and assume that (i) F(x,t)≥0for every(x,t)∈_R×0,³₈

∪⁵₈, 1 ,

(ii) A⁰ < τB⁰, whereτ,A⁰ and B⁰ are given by(3.1),(3.14)and(3.15)respectively.

Then for every

λ∈

#(max{1,|q|_∞,|s|_∞}) B⁰

86111 1080 , 1

2A⁰C_q,s²

"

the problem (1.1) admits a sequence of many weak solutions which strongly converges to zero in W^2,2(_R).

We present the following example to illustrate Corollary3.6.

Example 3.7. Let α> ⁸⁶¹¹¹

√π 6

120R⁵₈

38

e^−x2dx

−1≈ 2673 be a real number andF :R² →_Rbe a function defined by

F(x,t):=

(e^−x2t²(1+αcos²(¹_t)), if(x,t)∈_R×]0,+_∞[ 0, if(x,t)∈_R×]−_{∞, 0}]. FromF(x,t) =Rt

0 f(x,ξ)dξ we have f(x,t):=

(e^−x2 2t+2αtcos²(¹_t) +αsin(²_t)_{, if}(x,t)∈_R×]_0,+_∞[

0, if(x,t)∈_R×]−_{∞, 0}].

It is clear that f :R² →_Ris anL¹-Carathéodory function.

Letq(x) =1+₁₊¹_x2 ands(x) =2+tanhx and therefore|q|_∞ =2 , q0=1 ,|s|_∞ =3 ,s0 =1 andτ= ¹²⁰

√6 86111. Puta_n= _2n+¹₁

2 π andb_n= _nπ¹ for every n∈_N, one has A⁰ := _{lim inf}

ρ→0⁺

sup_|t|≤ρt² 1+αcos^{2 1}_tR

Re^−x2dx ρ²

≤ lim

n→_∞

a_n²

1+αcos²

1 an

R

Re^−x2dx

a_n² =√

π (3.16)

and

B⁰ := lim sup

ρ→0⁺

R⁵₈

3 8

F(x,ρ)dx

ρ² ≥ lim

n→_∞

b_n²

1+αcos²

1 bn

R⁵₈

3 8

e^−x2dx b_n²

= (₁+a)

Z ⁵

8 3 8

e^−x2dx. (3.17)

Now, sinceα> ⁸⁶¹¹¹

√π 6

120R⁵₈

38

e^−x2dx

−1, we have

A⁰ ≤√

π< ¹²⁰

√6

86111(1+a)

Z ⁵

8 3 8

e^−x2dx≤ τB⁰

and so condition (ii) of the Theorem3.1 is satisfied. Now, according to the Theorem 3.1 for every

λ∈

# 86111

360(1+a)R⁵₈

38

e^−x2dx,1 3

r6 π

"

the problem

(u^iυ(x)− (1+₁₊¹_x2)u⁰(x)⁰+ (2+tanhx)u(x)

=λe^−x2 2u(x) +2αu(x)cos²( ¹

u(x)) +αsin( ²

u(x)), a.e. x ∈ _R, (3.18) admits a sequence of many weak solutions which is converges uniformly to zero.

4 Acknowledgments

The second author is sponsored by the National Science Foundation of Iran (INFS). Also the authors are very thankful for the many helpful suggestions and corrections given by the referees who reviewed this paper.

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