A Variational Approach for the Neumann Problem in Some FLRW Spacetimes

Cristian Bereanu; Pedro J. Torres

doi:10.1515/ans-2018-2030

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A Variational Approach for the Neumann Problem in Some FLRW Spacetimes

Published/Copyright: September 10, 2018

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From the journal Advanced Nonlinear Studies Volume 19 Issue 2

Abstract

In this paper, we study, using critical point theory for strongly indefinite functionals, the Neumann problem associated to some prescribed mean curvature problems in a FLRW spacetime with one spatial dimension. We assume that the warping function is even and positive and the prescribed mean curvature function is odd and sublinear. Then, we show that our problem has infinitely many solutions. The keypoint is that our problem has a Hamiltonian formulation. The main tool is an abstract result of Clark type for strongly indefinite functionals.

Keywords: Quasilinear Elliptic Equation; Critical Point Theory; Hamiltonian Systems; FLRW Spacetime; Prescribed Mean Curvature; Neumann Problem

MSC 2010: 35J25; 58E05; 70H05; 53C42; 53C50

1 Introduction

In this paper, we initiate the study by variational methods of the Neumann problem associated to the prescribed mean curvature equation in a certain family of Friedmann–Lemaitre–Robertson–Walker (FLRW) spacetimes. The FLRW metric models a spatially homogeneous and isotropic universe and has the expression

d ⁢ s 2 = - d ⁢ t 2 + f ⁢ ( t ) ⁢ d ⁢ x 2 ,

where f⁢(t) is a positive function of time called the scale factor or warping function in the related literature (see for example [15]).

The mathematical problem under consideration in this paper is described as follows. Let f∈C1⁢([-ε,ε]) be a positive function. For R>ρ≥0 and a given continuous function g:[ρ,R]×ℝ→ℝ, we look for solutions of the following ODE with Neumann conditions:

(1.1) - ( q ′ f ⁢ ( q ) 2 - q ′ ⁣ 2 ) ′ - f ⁢ ( q ) ⁢ f ′ ⁢ ( q ) f ⁢ ( q ) 2 - q ′ ⁣ 2 = f ⁢ ( q ) ⁢ g ⁢ ( r , q ) ,

| q ′ | < f ⁢ ( q ) in ⁢ [ ρ , R ] , q ′ ⁢ ( ρ ) = 0 = q ′ ⁢ ( R ) .

A solution of the above Neumann problem is a function q∈C1⁢([ρ,R]) such that

∥ q ∥ ∞ < ε , | q ′ | < f ⁢ ( q ) in ⁢ [ ρ , R ] , q ′ f ⁢ ( q ) 2 - q ′ ⁣ 2 ∈ C 1 ⁢ ( [ ρ , R ] ) ,

and the above problem holds true.

The study of Neumann problems in FLRW spacetimes started very recently with the paper [13] (see also [16]). The main result of these works is an existence result in a ball centered in the origin and a small radius, proved by using Leray–Schauder degree. In the present paper, following [12], we prove a multiplicity result assuming that the prescribed mean curvature function is odd and sublinear, and the warping function is even. To the best of our knowledge, it is the first time that variational methods are applied to this problem.

Consider G:[ρ,R]×ℝ→ℝ given by

G ⁢ ( r , q ) = ∫ 0 q f ⁢ ( t ) ⁢ g ⁢ ( r , t ) ⁢ d ⁢ t .

The main result of this paper is the following:

Theorem 1.

Assume that g⁢(r,⋅) is odd for all r∈[ρ,R], and f is even with

(1.2) f ⁢ ( 0 ) = max [ - ε , ε ] ⁡ f , f ⁢ ( 0 ) - f ⁢ ( q ) ≤ d ⁢ q 2 ,

for all q∈[-ε,ε] and some d>0. If

(1.3) lim q → 0 ⁡ G ⁢ ( r , q ) q 2 = + ∞ uniformly in ⁢ r ∈ [ ρ , R ] ,

then (1.1) has infinitely many solutions qk with ∥qk∥∞→0 as k→∞.

The following example illustrates the applicability of the result.

Example 1.

Consider the warping function (see [13, 15])

f ( t ) = β ( cos α t ) 2 / 3 ( α , β > 0 )

or the Minkowski case f⁢(t)=1 (see [4, 8, 9, 10]).

Assume that 1<s<2<t, λ>0 and μ∈ℝ. Then, the Neumann problem (1.1) with

g ⁢ ( u ) = λ ⁢ | u | s - 2 ⁢ u + μ ⁢ | u | t - 2 ⁢ u

has infinitely many solutions qk with ∥qk∥∞→0 as k→∞. See [1, 6] for the analogous result for the Laplacian operator.

For the proof, a key observation is that the above Neumann problem is equivalent to the Hamiltonian system

p ′ = - H q ⁢ ( r , p , q ) , q ′ = H p ⁢ ( r , p , q ) , p ⁢ ( ρ ) = 0 = p ⁢ ( R ) ,

where the Hamiltonian function H is given by

H ⁢ ( r , p , q ) = f ⁢ ( q ) ⁢ 1 + p 2 - f ⁢ ( 0 ) + G ⁢ ( r , q ) and p = q ′ f ⁢ ( q ) 2 - q ′ ⁣ 2 .

Once the Hamiltonian setting has been identified, our main tool is a modification of a Clark type result for strongly indefinite functionals given in [12, Theorem 1.2]. Using this abstract result, it is shown in [12] that a general Hamiltonian system has infinitely many periodic solutions if the Hamiltonian function is even in the spatial variable and sublinear around zero. We observe that our H defined above does not satisfy this condition due to the presence of the term f⁢(q)⁢1+p2-f⁢(0). An analogous situation occurs in [11] in the case of first and second order superquadratic Hamiltonian systems. To prove both cases simultaneously, the main idea in [11] is to use an auxiliar operator B of the form

B ⁢ ( p , q ) = ( μ s ⁢ p , μ t ⁢ q ) ,

together with a minimax theorem including B which is a version of Benci–Rabinowitz minimax theorem for strongly indefinite functionals (see [7, 14]). Notice that a different situation occurs for first and second order forced superlinear autonomous systems. It is proved separately in [3, 2] that this type of systems has infinitely many periodic solutions, and there is no proof unifying both cases. In this paper, we use, like in [11], an auxiliar operator Lμ and a critical point theorem including this operator which is a version of [12, Theorem 1.2]. Then, we will apply this abstract result to the action functional associated to the above Hamiltonian system.

Problem (1.1) corresponds to the prescribed mean curvature equation with Neumann conditions in a FLRW spacetime with spatial dimension n=1. The consideration of the Neumann problem in the ball for a general FLRW spacetime with arbitrary dimension n≥1 is an interesting open problem. The use of a theorem of Clark type imposes necessarily the even symmetry in the functional, which implies serious restrictions on the warping and the prescribed curvature functions. As a counterpart, for n=1, we find an infinite number of solutions, a result that seems quite new in this context. At this moment, we are not able to identify a variational structure for the case n≥2, which is another intriguing open question.

This paper is organized as follows. In Section 2, we prove the abstract version of [12, Theorem 1.2]. In Section 3, we show that our problem has a Hamiltonian formulation, and we introduce the functional setting. Section 4 is devoted to the proof of the main result.

2 An Abstract Result

Let E be a Hilbert space such that E=⊕m∈ℤEm, where dim⁡Em=1 for all m∈ℤ. We denote E±=⊕m≥1E±m. For any positive integer k, let Lk:E→E be a linear bijective operator such that (∥Lk∥) is a bounded sequence and

L k ⁢ ( ⊕ m = - n n E m ) = ⊕ m = - n n E m for all ⁢ n ≥ 1 .

On the other hand, let

X n = ⊕ m ≥ - n E m ( n ≥ 1 ) .

For I∈C1⁢(E,ℝ), we recall that I satisfies (PS) condition if any sequence (uk)⊂E for which (I⁢(uk)) is bounded and I′⁢(uk)→0 as k→∞ possesses a convergent subsequence. Also, I is said to satisfy the (PS)* condition with respect to the sequence of subspaces (Xn) if for any subsequence (nj) of (n), any sequence (unj) such that unj∈Xnj for all j, (I⁢(unj)) is bounded and ∥(I|Xnj)′⁢(unj)∥→0 as j→∞, contains a subsequence converging to a critical point of I.

In the next lemma, we denote Sρ={u∈E:∥u∥=ρ}.

Lemma 1.

Let I∈C1⁢(E,R) be even with I⁢(0)=0, I|E+ satisfies (PS) condition and I satisfies (PS)* condition with respect to the sequence of subspaces (Xn). Suppose that

the functional I | E + is bounded from below,
there exists ρ k , ε k > 0 with ρ k → 0 such that, for all k ≥ k 0 ,

sup L k ⁢ ( S ρ k ∩ ( ⊕ m ≤ k E m ) ) ⁡ I ≤ - ε k for some positive integer ⁢ k 0 .

Then, I possesses a sequence of critical points (uk) such that ∥uk∥→0 as k→∞.

Proof.

We will show, following [12], that at least one of the following propositions holds.

There exists a sequence of critical points (uk) such that I⁢(uk)<0 for all k and ∥uk∥→0 as k→∞.
There exists r>0 such that, for any 0<a≤r, there exists a critical point u∈Sa with I⁢(u)=0.

Assume by contradiction that both (i) and (ii) are false. It follows that there exists 0<r1<r0 such that if one of the following propositions holds true:

∥ u ∥ ≤ r 0 and I⁢(u)<0,
∥ u ∥ = r 1 and I⁢(u)=0,

then I′⁢(u)≠0. Using that I⁢(0)=0, we may assume that

(2.1) I ⁢ ( u ) > - 1 for all ⁢ ∥ u ∥ ≤ r 0 .

Let us denote IAc={u∈E:u∈A,I⁢(u)≤c}. Then, using the (PS)* condition, it follows that there exists a,b with 0<a<r1<b≤r0, ν1>0 and a positive integer n1 such that, for any n≥n1, one has that

(2.2) ∥ ( I | X n ) ′ ⁢ ( u ) ∥ ≥ ν 1 for all ⁢ u ∈ I X n 0 ⁢ with ⁢ a ≤ ∥ u ∥ ≤ b .

Now, for n≥n1, consider Kn={u∈Xn:(I|Xn)′⁢(u)=0}, and let Wn:Xn∖Kn→Xn be an odd pseudogradient vector field of I|Xn. For a fixed u∈Xn∖Kn, there exists a unique maximal solution

η n ⁢ ( ⋅ , u ) : [ 0 , T n ⁢ ( u ) [ → X n ∖ K n

of the following Cauchy problem in Xn∖Kn:

d d ⁢ t η ( t , u ) = - W n ( η ( t , u ) ) ( t ≥ 0 ) , η ( 0 , u ) = u .

We also define ηn⁢(t,u)=u for t≥0 and Tn⁢(u)=∞ if u∈Kn. Then, using (2.2), we deduce that if u∈IXn∖Kn0 and 0≤t1<t2<Tn⁢(u) are such that ∥ηn⁢(t1,u)∥=a, ∥ηn⁢(t2,u)∥=b and a<∥ηn⁢(t,u)∥<b for all t∈]t1,t2[, then

(2.3) I ⁢ ( η n ⁢ ( t 2 , u ) ) ≤ - μ 0 :- - ( b - a ) ⁢ ν 1 2 .

Next, using that I is even, I⁢(0)=0, I|E+ is bounded from below and satisfies (PS), it follows that

(2.4) k 0 :- γ ⁢ ( I E + - μ 0 ) < ∞ ,

where γ is the generalized genus (see [5]). Then, using that IE+-μ0 is closed in IXn-μ0 and (2.4), one has that

(2.5) γ ⁢ ( I X n - μ 0 ) ≤ γ ⁢ ( X n ∖ E + ) + γ ⁢ ( I E + - μ 0 ) = n + 1 + k 0 .

Let k1>k0 be such that ρk⁢∥Lk∥<a for all k≥k1. Fix k≥k1. Then, the choice of r0 and (PS)* implies that there exists ν2>0 and n2≥n1+k+1 such that, for any n≥n2, one has that

∥ ( I | X n ) ′ ⁢ ( u ) ∥ ≥ ν 2 for all ⁢ u ∈ I X n - ε k , ∥ u ∥ ≤ r 0 .

We denote

A k = L k ⁢ ( S ρ k ∩ ⊕ j = - n k E j ) ,

and note that Ak⊂⊕j=-nnEj⊂Xn.

One has that, for all u∈Ak, there exists sn⁢(u)∈]0,Tn⁢(u)[ such that ∥ηn⁢(sn⁢(u),u)∥>b. Suppose by contradiction that there exists u∈Ak such that ∥ηn⁢(t,u)∥≤b≤r0 for all t∈]0,Tn⁢(u)[. From (2.1), it follows that

- 1 < I ⁢ ( η n ⁢ ( t , u ) ) ≤ I ⁢ ( u ) ≤ - ε k for all ⁢ t ∈ ] 0 , T n ⁢ ( u ) [ .

Hence, for all t∈]0,Tn⁢(u)[, using the properties of Wn,

1 ≥ 1 - ε k ≥ I ⁢ ( u ) - I ⁢ ( η n ⁢ ( t , u ) ) ≥ ∫ 0 t ∥ ( I | X n ) ′ ⁢ ( η n ⁢ ( s , u ) ) ∥ 2 ⁢ d ⁢ s ≥ ν 2 2 ⁢ t ,

which implies that Tn⁢(u)≤ν2-2. It follows that

∥ η n ⁢ ( t 1 , u ) - η n ⁢ ( t 2 , u ) ∥ ≤ 2 ⁢ ( t 2 - t 1 ) 1 / 2 for all ⁢ 0 < t 1 < t 2 < T n ⁢ ( u ) ,

and there exists the limit u*=limt→Tn⁢(u)-⁡ηn⁢(t,u). One has that u*∈IXn∩Bb-εk, which implies that u*∈Xn∖Kn. This gives a contradiction with the maximality of the interval [0,Tn(u)[.

Now, consider u∈Ak. Using that

∥ u ∥ < a and ∥ η n ⁢ ( s n ⁢ ( u ) , u ) ∥ > b ,

it follows that there exists 0<t1<t2<sn⁢(u) such that ∥ηn⁢(t1,u)∥=a, ∥ηn⁢(t2,u)∥=b and a<∥ηn⁢(t,u)∥<b for all t∈]t1,t2[. Notice that u∈IBa-εk, which implies that u∈IXn∖Kn0. Hence, using (2.3), we deduce that

I ⁢ ( η n ⁢ ( s n ⁢ ( u ) , u ) ) < I ⁢ ( η n ⁢ ( t 2 , u ) ) ≤ - μ 0 .

Then, following [12], for all n≥n2, there exists an odd continuous function

h n : A k → I X n - μ 0 .

Finally, using (2.5), one has that

n + 1 + k = γ ⁢ ( S ρ k ∩ ⊕ j = - n k E j ) = γ ⁢ ( A k ) ≤ γ ⁢ ( h n ⁢ ( A k ) ) ≤ γ ⁢ ( I X n - μ 0 ) ≤ n + 1 + k 0 ,

which contradicts the fact that k>k0. The proof is completed. ∎

3 Hamiltonian Formulation and the Functional Setting

3.1 The Hamiltonian

The following result shows that the above Neumann nonlinear differential equation has a Hamiltonian structure. We have the following key lemma:

Lemma 2.

Consider the Hamiltonian function H:[ρ,R]×R×]-ε,ε[→R given by

H ⁢ ( r , p , q ) = f ⁢ ( q ) ⁢ 1 + p 2 - f ⁢ ( 0 ) + G ⁢ ( r , q )

and p,q∈C1⁢([ρ,R]) with ∥q∥∞<ε. The following statements are equivalent:

q is a solution of ( 1.1 ) and, for all r ∈ [ ρ , R ] ,

(3.1) p ⁢ ( r ) = q ′ ⁢ ( r ) f ⁢ ( q ⁢ ( r ) ) 2 - q ′ ⁣ 2 ⁢ ( r ) .
( p , q ) is a solution of the Hamiltonian system

(3.2) p ′ ( r ) = - H q ( r , p ( r ) , q ( r ) ) , q ′ ( r ) = H p ( r , p ( r ) , q ( r ) ) ( r ∈ [ ρ , R ] ) ,

p ⁢ ( ρ ) = 0 = p ⁢ ( R ) .

Proof.

We will prove that (ii) implies (i). The reversed implication is analogous and easier. One has that

p ′ ( r ) = - f ′ ( q ( r ) ) 1 + p 2 ⁢ ( r ) - f ( q ( r ) ) g ( r , q ( r ) ) and q ′ ( r ) = f ⁢ ( q ⁢ ( r ) ) ⁢ p ⁢ ( r ) 1 + p 2 ⁢ ( r ) ( r ∈ [ ρ , R ] ) .

Hence

q ′ ⁢ ( ρ ) = 0 = q ′ ⁢ ( R ) .

Then, using the second equation and that f is positive, it follows that

| q ′ ⁢ ( r ) | < f ⁢ ( q ⁢ ( r ) ) for all ⁢ r ∈ [ ρ , R ] ,

and (3.1) holds true. This, together with the second equation, implies that, for all r∈[ρ,R],

1 + p 2 ⁢ ( r ) = f ⁢ ( q ⁢ ( r ) ) f ⁢ ( q ⁢ ( r ) ) 2 - q ′ ⁣ 2 ⁢ ( r ) .

Then, the first equation gives the conclusion. ∎

3.2 The Functional

Consider ω=πR-ρ and, for any positive integer m,

e m ( r ) = sin ( m ω ( r - ρ ) ) ( r ∈ [ ρ , R ] ) .

Let H01/2 be the set of functions p∈L2⁢(ρ,R) having the Fourier expansion

p = ∑ m ≥ 1 p m ⁢ e m such that ∑ m ≥ 1 m ⁢ p m 2 < ∞ .

Taking φ∈H01/2 with φ=∑m≥1φm⁢em, we define the scalar product

( p | φ ) H 0 1 / 2 = ω ⁢ π 2 ∑ m ≥ 1 m p m φ m .

It is well known that H01/2 is a Hilbert space together with the above scalar product. Notice that H01/2 is compactly embedded into Ls⁢(0,π) for all 1≤s<∞. In particular, there is ls>0 such that

∥ p ∥ L s ≤ l s ⁢ ∥ p ∥ H 0 1 / 2 for all ⁢ p ∈ H 0 1 / 2 .

We need also a second fractional Sobolev space. For any nonnegative integer m, let

f m ( r ) = cos ( m ω ( r - ρ ) ) ( r ∈ [ ρ , R ] ) ,

and let H1/2 be the set of functions q∈L2⁢(ρ,R) having the Fourier expansion

q = ∑ m ≥ 0 q m ⁢ f m such that ∑ m ≥ 1 m ⁢ q m 2 < ∞ .

Taking ψ∈H1/2 with ψ=∑m≥0ψm⁢fm, we define the scalar product

( q | ψ ) H 1 / 2 = ( R - ρ ) q 0 ψ 0 + ω ⁢ π 2 ∑ m ≥ 1 m q m ψ m .

Like before, H1/2 is compactly embedded into Ls⁢(0,π) for all 1≤s<∞. In particular, there is ls>0 such that

∥ q ∥ L s ≤ l s ⁢ ∥ q ∥ H 1 / 2 for all ⁢ q ∈ H 1 / 2 .

Finally, consider the Hilbert space E=H01/2×H1/2 endowed with the usual scalar product

( z | w ) = ( p | φ ) H 0 1 / 2 + ( q | ψ ) H 1 / 2

for any z=(p,q) and w=(φ,ψ) in E.

In the sequel, we will write (⋅|⋅) and ∥⋅∥ for any scalar product and norm in the spaces defined above.

We introduce now a modified Hamiltonian function H^. Consider H:[ρ,R]×ℝ2→ℝ smooth, such that H=H on [ρ,R]×[-1,1]×[-ε2,ε2] and H=0 on [ρ,R]×(ℝ2∖[-2,2]×[-2,2]). The Hamiltonian H is defined as follows. Let f∈𝒞∞⁢(ℝ) be an even smooth positive function such that f=f on [-ε2,ε2], f=c>0 constant on ℝ∖]-ε,ε[, f≥c on ℝ, and let β∈𝒟⁢(ℝ) be an even smooth positive function such that β=1 on [-1,1], supp⁡β⊂]-2,2[ and β is nonincreasing on [1,2]. Consider H given by

H ⁢ ( r , p , q ) = β ⁢ ( p ) ⁢ [ f ⁢ ( q ) ⁢ 1 + p 2 - f ⁢ ( 0 ) ] + β ⁢ ( q ) ⁢ G ⁢ ( r , q )

for all (r,p,q)∈[ρ,R]×ℝ2. Now, we consider α∈𝒟⁢(ℝ2) an even smooth positive function such that supp⁡α⊂[-3,3]×[-3,3], α≤1 and α=1 on [-2,2]×[-2,2], then the Hamiltonian H^ is defined by

H ^ ⁢ ( r , z ) = α ⁢ ( z ) ⁢ H ⁢ ( r , z ) + ( 1 - α ⁢ ( z ) ) ⁢ c ^ ⁢ | z | 2

for all (r,z)∈[ρ,R]×ℝ2, where (2⁢l2⁢ω)-1>c^>0 is a small fixed constant. The reason for introducing H^ was that we could not verify (PS)* condition without the Hamiltonian growing at a prescribed rate at infinity.

Next, consider the functional

𝒥 : H 0 1 / 2 → ℝ : p ↦ ∫ ρ R 1 + p 2 ⁢ d ⁢ r .

Then, it is easy to prove that 𝒥∈C1⁢(H01/2,ℝ) and

𝒥 ′ ⁢ ( p ) ⁢ ( φ ) = ∫ ρ R p ⁢ φ 1 + p 2 ⁢ d ⁢ r for all ⁢ p , φ ∈ H 0 1 / 2 .

It follows that the following result holds true.

Lemma 3.

Assume that H^ satisfies the above conditions, and consider the functional

𝒥 ^ : E → ℝ : z ↦ ∫ ρ R H ^ ⁢ ( r , z ) ⁢ d ⁢ r .

One has that J^∈C1⁢(E,R) and

𝒥 ^ ′ ⁢ ( z ) ⁢ ( w ) = ∫ ρ R H ^ z ⁢ ( r , z ) ⋅ w ⁢ d ⁢ r

for all z,w∈E.

Consider now the Hamiltonian system associated to H^,

(3.3) p ′ ( r ) = - H ^ q ( r , p ( r ) , q ( r ) ) , q ′ ( r ) = H ^ p ( r , p ( r ) , q ( r ) ) ( r ∈ [ ρ , R ] ) ,

p ⁢ ( ρ ) = 0 = p ⁢ ( R ) .

We define the action functional associated to (3.3) as follows. Consider the continuous symmetric bilinear form

B : E × E → ℝ : ( z , w ) ↦ - π 2 ⁢ ∑ m ≥ 1 m ⁢ ( p m ⁢ ψ m + φ m ⁢ q m ) ,

where

z = ( ∑ m ≥ 1 p m ⁢ e m , ∑ m ≥ 0 q m ⁢ f m ) and w = ( ∑ m ≥ 1 φ m ⁢ e m , ∑ m ≥ 0 ψ m ⁢ f m ) .

For z,w∈H01×H1 with z=(p,q) and w=(φ,ψ), one has that

B ⁢ [ z , w ] = ∫ ρ R ( p ⁢ ψ ′ + φ ⁢ q ′ ) ⁢ d ⁢ r .

Consider now the quadratic form associated to B,

A : E → ℝ : z ↦ 1 2 ⁢ B ⁢ [ z , z ] .

One has that, for z∈H01×H1 with z=(p,q),

A ⁢ ( z ) = ∫ ρ R p ⁢ q ′ ⁢ d ⁢ r .

Next, consider

E^ = { ( p , q ) ∈ E : p = ∑ m ≥ 1 p m ⁢ e m , q = ∑ m ≥ 1 ( ∓ p m ) ⁢ f m } ,

E 0 = { ( 0 , q 0 ) : q 0 ∈ ℝ } ,

and notice that E=E-⊕E0⊕E+. It follows that, for

z = ( ∑ m ≥ 1 p m ⁢ e m , ∑ m ≥ 0 q m ⁢ f m ) ∈ E ,

one has z=z-+z0+z+, where z0=(0,q0) and

z ± = ( ∑ m ≥ 1 ( p m ∓ q m 2 ) ⁢ e m , ∑ m ≥ 1 ( q m ∓ p m 2 ) ⁢ f m ) .

This implies that

A ⁢ ( z ) = ( 2 ⁢ ω ) - 1 ⁢ ( ∥ z + ∥ 2 - ∥ z - ∥ 2 ) for all ⁢ z ∈ E .

Using Lemma 3 and a classical strategy (see [14]), it is not difficult to show the following:

Lemma 4.

Consider the action functional

I : E → ℝ : z ↦ A ⁢ ( z ) - ∫ ρ R H ^ ⁢ ( r , z ) ⁢ d ⁢ r .

Then, I∈C1⁢(E,R) with

I ′ ( z ) ( w ) = B [ z , w ] - ∫ ρ R H ^ z ( r , z ) ⋅ w d r ( z , w ∈ E ) .

Moreover, if z∈E with I′⁢(z)=0, then z=(p,q) is a solution of (3.3).

3.3 (PS) and (PS)*

Consider for any positive integer m,

E ± m = { ( p , q ) ∈ E : p = p m ⁢ e m , q = ( ∓ p m ) ⁢ f m } ,

and remark that

⊕ m = - k k E m = { ( p , q ) ∈ E : p = ∑ m = 1 k p m ⁢ e m , q = ∑ m = 0 k q m ⁢ f m } .

Lemma 5.

One has that I|E+ satisfies (PS) condition and I has (PS)* condition with respect to the sequence of subspaces

X n = ⊕ m ≥ - n E m ( n ≥ 1 ) .

Proof.

We will prove that I has (PS)* condition with respect to (Xn). Consider (nj) a subsequence of (n), and let zj∈Xnj be such that (I⁢(zj)) is bounded and ∥(I|Xnj)′⁢(zj)∥→0 as j→∞. Note that, for all j≥1,

B ⁢ [ z j , z j + ] = ω - 1 ⁢ ∥ z j + ∥ 2 , ∫ ρ R z j ⋅ z j + ⁢ d ⁢ r = ∫ ρ R | z j + | 2 ⁢ d ⁢ r .

It follows that

( I | X n j ) ′ ⁢ ( z j ) ⁢ [ z j + ] ≥ ω - 1 ⁢ ∥ z j + ∥ 2 - c 1 ⁢ ∫ ρ R | z j + | ⁢ d ⁢ r - 2 ⁢ c ^ ⁢ l 2 ⁢ ∥ z j + ∥ 2 ,

which implies that (∥zj+∥) is bounded. Similarly, one has that (∥zj-∥) is bounded. In particular, (∥zj++zj-∥L2) is bounded. On the other hand, using moreover that (I⁢(zj)) is bounded, we deduce that ∥zj∥L2 is bounded. It follows that (|zj0|) is bounded, and ∥zj∥ is bounded. Hence, passing to a subsequence, one has that zj⇀z weakly in E and zj→z in L2 and a.e. on [ρ,R]. We can assume also that zj0→z0. One has that

( I ′ ⁢ ( z j ) - I ′ ⁢ ( z ) ) ⁢ [ z j + - z + ] = ω - 1 ⁢ ∥ z j + - z + ∥ 2 + ∫ ρ R ( H ^ z ⁢ ( r , z ) - H ^ z ⁢ ( r , z j ) ) ⋅ [ z j + - z + ] ⁢ d ⁢ r ,

which implies that ∥zj+-z+∥→0 as j→∞. Now, let Pj be the orthogonal projection onto Xnj, and note that ∥z--Pj⁢z-∥→0 as j→∞. On the other hand, like above, one has that ∥zj--Pj⁢z-∥→0 as j→∞, and ∥zj--z-∥→0. Hence, we have that ∥z-zj∥→0, and z is a critical point of I. The fact that I|E+ has (PS) follows in the same way, and it is easier. ∎

4 Proof of the Main Result

One has that

1 + p 2 - 1 ≥ 1 4 ⁢ 2 p 2 ( ( r , p ) ∈ [ 0 , R ] × [ - 1 , 1 ] ) .

This, together with (1.2), implies that, for all (r,p,q)∈[ρ,R]×[-1,1]×[-ε2,ε2],

β ⁢ ( p ) ⁢ [ f ⁢ ( q ) ⁢ 1 + p 2 - f ⁢ ( 0 ) ] = f ⁢ ( q ) ⁢ [ 1 + p 2 - 1 ] - [ f ⁢ ( 0 ) - f ⁢ ( q ) ] ≥ c ⁢ 1 4 ⁢ 2 ⁢ p 2 - d ⁢ q 2 .

It follows that, for all (r,p,q)∈[ρ,R]×[-1,1]×[-1,1],

(4.1) H ^ ⁢ ( r , p , q ) ≥ c ⁢ 1 4 ⁢ 2 ⁢ p 2 - d ⁢ q 2 + G ⁢ ( r , q )

for some d>0. Then, from (1.3) and (4.1), it follows that there exist c1>0 such that, for any λ>0, there exists c2=c2⁢(λ) with

(4.2) H ^ ⁢ ( r , p , q ) ≥ c 1 ⁢ p 2 + λ ⁢ q 2 - c 2 ⁢ ( λ ) ⁢ q 4 for all ⁢ ( r , p , q ) ∈ [ ρ , R ] × ℝ 2 .

Consider μ>0 and the linear operator

L μ : E → E : ( p , q ) ↦ ( p , μ ⁢ q ) .

One has that ∥Lμ∥≤1+μ and

L μ ⁢ ( ⊕ m = - j j E m ) = ⊕ m = - j j E m for all ⁢ j ≥ 1 .

Let z=Lμ⁢(p,q), where (p,q)=(p-,q-)+(0,q0)+(p+,q+) and

( p - , q - ) = ( ∑ m ≥ 1 a m ⁢ e m , ∑ m ≥ 1 a m ⁢ f m ) , ( p + , q + ) = ( ∑ m = 1 k b m ⁢ e m , ∑ m = 1 k ( - b m ) ⁢ f m )

with k a positive fixed integer. It follows that z0=(0,μ⁢q0), and if we denote z±=(p±,q±), then

p ± = ∑ m = 1 k ( a m + b m ) ∓ μ ⁢ ( a m - b m ) 2 ⁢ e m + ∑ m ≥ k + 1 ( 1 ∓ μ ) ⁢ a m 2 ⁢ e m .

This implies that

A ⁢ ( z ) = μ ⁢ π 2 ⁢ ( ∑ m = 1 k m ⁢ b m 2 - ∑ m ≥ 1 m ⁢ a m 2 ) .

On the other hand,

∥ p ∥ L 2 2 = R - ρ 2 ⁢ ( ∑ m = 1 k ( a m + b m ) 2 + ∑ m ≥ k + 1 a m 2 ) ,

∥ q ∥ L 2 2 = R - ρ 2 ⁢ ( ∑ m = 1 k ( a m - b m ) 2 + ∑ m ≥ k + 1 a m 2 ) + ( R - ρ ) ⁢ q 0 2

and

∥ ( p , q ) ∥ 2 = ( R - ρ ) ⁢ q 0 2 + ω ⁢ π ⁢ ( ∑ m = 1 k m ⁢ b m 2 + ∑ m ≥ 1 m ⁢ a m 2 ) .

Then, from (4.2), it follows that

I ⁢ ( z ) ≤ A ⁢ ( z ) - c 1 ⁢ ∥ p ∥ L 2 2 - λ ⁢ μ 2 ⁢ ∥ q ∥ L 2 2 + c 2 ⁢ μ 4 ⁢ ∥ q ∥ L 4 4 ≤ μ ⁢ π 2 ( ∑ m = 1 k m b m 2 - ∑ m ≥ 1 m a m 2 ) - ( c 1 + λ μ 2 ) ( R - ρ ) ( ∑ m = 1 k b m 2 + ∑ m ≥ 1 a m 2 ) + ( μ 2 ⁢ λ - c 1 ) ⁢ ( R - ρ ) ⁢ ∑ m = 1 k a m ⁢ b m - λ ⁢ μ 2 ⁢ ( R - ρ ) ⁢ q 0 2 + c 2 ⁢ μ 4 ⁢ ∥ q ∥ L 4 4 .

Now, consider

μ k = 2 ⁢ c 1 ⁢ ( R - ρ ) k ⁢ π , λ k = c 1 / μ 2 .

Then, using that

k ⁢ ∑ m = 1 k b m 2 ≥ ∑ m = 1 k m ⁢ b m 2 ,

it follows that

μ k ⁢ π 2 ⁢ ∑ m = 1 k m ⁢ b m 2 - ( c 1 + λ k ⁢ μ 2 ) ⁢ ( R - ρ ) ⁢ ∑ m = 1 k b m 2 ≤ - μ k ⁢ π 2 ⁢ ∑ m = 1 k m ⁢ b m 2 .

Hence, there exists k0>0 such that, for all k≥k0, one has that

I ⁢ ( z ) ≤ - μ k ⁢ π 2 ⁢ ( ∑ m = 1 k m ⁢ b m 2 + ∑ m ≥ 1 m ⁢ a m 2 ) - c 1 ⁢ ( R - ρ ) ⁢ q 0 2 + c 2 ⁢ μ k 4 ⁢ l 4 ⁢ ∥ q ∥ H 1 / 2 4 ≤ - μ k 2 ⁢ ω ⁢ ∥ ( p , q ) ∥ 2 + c 2 ⁢ μ k 4 ⁢ l 4 ⁢ ∥ ( p , q ) ∥ 4 .

Consider ρk<1/k such that

- ( 2 ⁢ ω ) - 1 + c 2 ⁢ l 4 ⁢ μ k 3 ⁢ ρ k 2 < 0 .

Then, it follows that I satisfies (ii) from Lemma 1.

Next, one has that there exists c3>0 such that

H ^ ⁢ ( r , z ) ≤ c ^ ⁢ | z | 2 + c 3 for all ⁢ ( r , z ) ∈ [ ρ , R ] × ℝ 2 .

This implies that, for any z∈E+, one has that

I ⁢ ( z ) ≥ ( 2 ⁢ ω ) - 1 ⁢ ∥ z + ∥ 2 - c ^ ⁢ ∥ z ∥ L 2 2 - c 3 ⁢ ( R - ρ ) ≥ ( ( 2 ⁢ ω ) - 1 - c ^ ⁢ l 2 ) ⁢ ∥ z + ∥ 2 - c 3 ⁢ ( R - ρ ) ,

which, together with (2⁢l2⁢ω)-1>c^>0, implies that I satisfies (i) from Lemma 1. Then, using Lemma 5, we deduce that I satisfies the assumptions of Lemma 1, and I has a sequence (zk) of critical points such that ∥zk∥→0 as k→∞. Passing to a subsequence, one has that ∥zk∥L2→0 and zk→0 a.e. on [ρ,R] as k→∞. On the other hand, from Lemma 4, one has that zk=(pk,qk) is a solution of the Hamiltonian system (3.3). Then, integrating in (3.3) and using the convergence in L2 and a.e. on [ρ,R], it follows that ∥zk∥∞→0 as k→∞. This implies that zk=(pk,qk) is a solution of (3.2), and using Lemma 2, we deduce that qk is a solution of (1.1). Moreover, ∥qk∥∞→0 as k→∞.

The proof of Theorem 1 is complete.

Communicated by David Ruiz

Funding source: Ministerio de Economía, Industria y Competitividad, Gobierno de España

Award Identifier / Grant number: MTM2017-82348-C2-1-P

Funding statement: Pedro J. Torres was partially supported by Spanish MINECO and ERDF project MTM2017-82348-C2-1-P.

References

[1] A. Ambrosetti, H. Brezis and G. Cerami, Combined effects of concave and convex nonlinearities in some elliptic problems, J. Funct. Anal. 122 (1994), no. 2, 519–543. 10.1006/jfan.1994.1078Search in Google Scholar

[2] A. Bahri and H. Berestycki, Existence of forced oscillations for some nonlinear differential equations, Comm. Pure Appl. Math. 37 (1984), no. 4, 403–442. 10.21236/ADA124372Search in Google Scholar

[3] A. Bahri and H. Berestycki, Forced vibrations of superquadratic Hamiltonian systems, Acta Math. 152 (1984), no. 3–4, 143–197. 10.1007/BF02392196Search in Google Scholar

[4] R. Bartnik and L. Simon, Spacelike hypersurfaces with prescribed boundary values and mean curvature, Comm. Math. Phys. 87 (1982/83), no. 1, 131–152. 10.1007/BF01211061Search in Google Scholar

[5] T. Bartsch and Z. Liu, Location and critical groups of critical points in Banach spaces with an application to nonlinear eigenvalue problems, Adv. Differential Equations 9 (2004), no. 5–6, 645–676. 10.57262/ade/1355867939Search in Google Scholar

[6] T. Bartsch and M. Willem, On an elliptic equation with concave and convex nonlinearities, Proc. Amer. Math. Soc. 123 (1995), no. 11, 3555–3561. 10.1090/S0002-9939-1995-1301008-2Search in Google Scholar

[7] V. Benci and P. H. Rabinowitz, Critical point theorems for indefinite functionals, Invent. Math. 52 (1979), no. 3, 241–273. 10.1007/BF01389883Search in Google Scholar

[8] C. Bereanu, P. Jebelean and J. Mawhin, Variational methods for nonlinear perturbations of singular ϕ-Laplacians, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 22 (2011), no. 1, 89–111. 10.4171/RLM/589Search in Google Scholar

[9] C. Bereanu, P. Jebelean and J. Mawhin, Radial solutions of Neumann problems involving mean extrinsic curvature and periodic nonlinearities, Calc. Var. Partial Differential Equations 46 (2013), no. 1–2, 113–122. 10.1007/s00526-011-0476-xSearch in Google Scholar

[10] C. Bereanu, P. Jebelean and P. J. Torres, Positive radial solutions for Dirichlet problems with mean curvature operators in Minkowski space, J. Funct. Anal. 264 (2013), no. 1, 270–287. 10.1016/j.jfa.2012.10.010Search in Google Scholar

[11] P. L. Felmer, Periodic solutions of “superquadratic” Hamiltonian systems, J. Differential Equations 102 (1993), no. 1, 188–207. 10.1006/jdeq.1993.1027Search in Google Scholar

[12] Z. Liu and Z.-Q. Wang, On Clark’s theorem and its applications to partially sublinear problems, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), no. 5, 1015–1037. 10.1016/j.anihpc.2014.05.002Search in Google Scholar

[13] J. Mawhin and P. J. Torres, Prescribed mean curvature graphs with Neumann boundary conditions in some FLRW spacetimes, J. Differential Equations 261 (2016), no. 12, 7145–7156. 10.1016/j.jde.2016.09.013Search in Google Scholar

[14] P. H. Rabinowitz, Minimax Methods in Critical Point Theory with Applications to Differential Equations, CBMS Reg. Conf. Ser. Math. 65, American Mathematical Society, Providence, 1986. 10.1090/cbms/065Search in Google Scholar

[15] B. S. Ryden, Introduction to Cosmology, Pearson, London, 2014. Search in Google Scholar

[16] P. J. Torres, The prescribed mean curvature problem with Neumann boundary conditions in FLRW spacetimes, Rend. Istit. Mat. Univ. Trieste 49 (2017), 19–25. Search in Google Scholar

Received: 2018-04-22

Accepted: 2018-08-22

Published Online: 2018-09-10

Published in Print: 2019-05-01

This work is licensed under the Creative Commons Attribution 4.0 International License.

Articles in the same Issue

https://doi.org/10.1515/ans-2018-2030

Keywords for this article

Quasilinear Elliptic Equation; Critical Point Theory; Hamiltonian Systems; FLRW Spacetime; Prescribed Mean Curvature; Neumann Problem

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