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Advances in Nonlinear Analysis

Editor-in-Chief: Radulescu, Vicentiu / Squassina, Marco


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Existence and non-existence of solutions to a Hamiltonian strongly degenerate elliptic system

Cung The Anh
  • Corresponding author
  • Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy, Cau Giay, Hanoi, Vietnam
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/ Bui Kim My
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/anona-2016-0165

Abstract

We study the non-existence and existence of infinitely many solutions to the semilinear degenerate elliptic system

{-Δλu=|v|p-1vin Ω,-Δλv=|u|q-1uin Ω,u=v=0on Ω,

in a bounded domain ΩN with smooth boundary Ω. Here p,q>1, and Δλ is the strongly degenerate operator of the form

Δλu=j=1Nxj(λj2(x)uxj),

where λ(x)=(λ1(x),,λN(x)) satisfies certain conditions.

Keywords: Existence; non-existence; Pohozaev-type identity; variational method; hyperbola critical; strongly degenerate; Hamiltonian system

MSC 2010: 35J70; 35D30

1 Introduction

In this paper, we study the existence, multiplicity and non-existence of solutions to the following semilinear degenerate elliptic system of Hamiltonian type:

{-Δλu=|v|p-1v,xΩ,-Δλv=|u|q-1u,xΩ,u=v=0,xΩ,(1.1)

where p,q>1, and Ω is a bounded domain in N with smooth boundary Ω. Here Δλ is the strongly degenerate elliptic operator of the form

Δλ=j=1Nxj(λj2(x)uxj),

where x=(x1,,xN)N, and the λi:N, i=1,,N, satisfy some certain conditions. This operator was first introduced by Franchi and Lanconelli [8], and recently reconsidered and named Δλ-Laplacians by Kogoj and Lanconelli in [10] under an additional assumption that the operator is homogeneous of degree two with respect to a group dilation (δt)t0 in N. We denote by Q the homogeneous dimension of N with respect to the group of dilations {δt}t>0, i.e.,

Q:=ϵ1++ϵN,

see Section 2.1 for more details. The homogeneous dimension Q plays a crucial role, both in the geometry and the functional associated to the operator Δλ.

In the case of a single equation

{-Δλu=f(u)in Ω,u=0on Ω,

the existence, non-existence and regularity of weak solutions to this problem were proved in [10]. In particular, the authors established a Pohozaev-type identity, and then used it to prove the non-existence result. Later, the existence of weak solutions to this problem was proved in [12] for the case of non-homogeneous Dirichlet boundary conditions, and in [1] for the case where the nonlinearity does not satisfy the Ambrosetti–Rabinowitz condition. See also the previous works [18, 17] for related results.

For system (1.1), as is seen in the case of the Laplace operator [4, 5, 7, 3, 14, 15, 16] and very recent results on Liouville-type theorems [2], we know that the critical hyperbola is

1p+1+1q+1=Q-2Q.

For exponents (p,q) lying on or above this curve, that is,

1p+1+1q+1Q-2Q,

we prove the non-existence of positive classical solutions to (1.1) in δt-starshaped bounded domains by establishing a new Pohozaev-type identity. This new identity turns out to be a generalization of the Pohozaev-type identity in the scalar case in [10]. For (p,q) below the critical hyperbola, we prove the existence of infinitely many weak solutions. To do this, we will use the variational method and the Fountain Theorem of Bartsch and de Figueiredo. It is natural to view a weak solution to (1.1) as a critical point of the corresponding functional

Φ(u,v)=Ωλuλvdx-1p+1Ω|v|p+1𝑑x-1q+1Ω|u|q+1𝑑x.

A natural energy space for problem (1.1) is the Hilbert space

W̊(Ω)λ1,2×W̊(Ω)λ1,2.

However, this choice of energy space will impose a strict restriction on p,q, namely p,qQ+2Q-2, due to the Sobolev-type embedding

W̊(Ω)λ1,2L2QQ-2(Ω)

(see Proposition 2.1 below). To overcome this difficulty, following the approach introduced in [7, 9], we will use the fractional Sobolev-type spaces defined by using Fourier expansions on the eigenfunctions of -Δλ (see Section 2 for details). One notes that now one of the nonlinearities may have a larger growth than |s|(Q+2)/(Q-2) provided the other nonlinearity has a suitably lower growth. Another possible way to overcome this difficulty is to use the Orlicz-space approach [6]. It is worth noticing that the results obtained in this paper are the generalizations of the corresponding results for the Laplace operator in [7, 3, 15, 13, 9, 19].

In this paper, to simplify the exposition, we only state the theorems and give the proofs for the “model problem” (1.1), although these results can be extended to a slightly more general system of the form

{-Δλu=g(v),xΩ,-Δλv=f(u),xΩ,u=v=0,xΩ,

under some suitable assumptions of f and g.

This paper is organized as follows: In Section 2, we recall some known results and prove some important embeddings which are necessary for studying our problem. In Section 3, we prove the non-existence of positive classical solutions to (1.1) by establishing a new Pohozaev-type identity. The existence of infinitely many nontrivial weak solutions to the problem is shown in Section 4 by using the variational method on fractional Sobolev-type spaces.

2 Preliminary results

2.1 The Δλ-Laplace operator and related function spaces

As in [10], we consider the strongly degenerate operator of the form

Δλu=j=1Nxj(λj2(x)uxj),

where x=(x1,,xN)N, the λj:N are continuous and λj>0, j=1,,N, in C1(N), where

={(x1,,xN)N:i=1Nxi=0}.

We assume the following conditions:

  • (i)

    λ1(x)1, λi(x)=λi(x1,,xi-1), i=2,,N.

  • (ii)

    λi(x)=λi(x*) holds for every xN and i=1,,N, where

    x*=(|x1|,,|xN|)if x=(x1,,xN).

  • (iii)

    There exists a constant ρ0 such that

    0xkxkλi(x)ρλi(x)for all k{1,,i-1},i=2,,N,

    and for every x+N:={(x1,,xN)N:xi0 for all i=1,,N}.

  • (iv)

    For t>0 there exists a group of dilations

    δt:NN,δt(x)=δt(x1,,xN)=(tϵ1x1,,tϵNxN),

    where 1ϵ1ϵ2ϵN, such that λi is δt-homogeneous of degree ϵi-1, i.e.,

    λi(δt(x))=tϵi-1λi(x)for all xN,t>0,i=1,,N.

    This implies that the operator Δλ is δt-homogeneous of degree two, i.e.,

    Δλ(u(δt(x)))=t2(Δλu)(δt(x))for all uC(N).

We denote by Q the homogeneous dimension of N with respect to the group of dilations {δt}t>0, i.e.,

Q:=ϵ1++ϵN.

This operator Δλ is called the Δλ-Laplacian; for more details on properties and examples of this operator we refer the reader to the papers [8, 10, 11].

We now recall some function spaces related to the Δλ-Laplace operator. Denote by W̊(Ω)λ1,p, p1, the closure of C0(Ω) in the norm

uW̊λ1,p=(Ω|λu|p𝑑x)1p.

We define Wλ2,p(Ω) as the space of all functions u such that

uLp(Ω),λi(x)uxiLp(Ω),λi(x)xi(λj(x)uxj)Lp(Ω),i,j=1,2,,N,

with the norm

uWλ2,p=(Ω[|u|p+|λu|p+i,j=1N[λi(x)xi(λj(x)uxj)]p]𝑑x)1p.

It is easy to see that Wλ2,p(Ω) and W̊(Ω)λ1,p are Banach spaces. In particular, Wλ2,2(Ω) and W̊(Ω)λ1,2 are Hilbert spaces with the following inner products:

(u,v)Wλ2,2=(u,v)L2+i=1N(λiuxi,λivxi)L2+i,j=1N(λixi(λjuxj),λixi(λjvxj))L2

and

(u,v)W̊λ1,2=i=1N(λiuxi,λivxi)L2.

The following result was established in [10].

Proposition 2.1.

The embedding

W̊(Ω)λ1,pLpλ*(Ω),where pλ*:=pQQ-p,

is continuous. Moreover, the embedding

W̊(Ω)λ1,pLγ(Ω)

is compact for every γ[1,pλ*).

We now prove the following important result.

Lemma 2.2.

The embedding

Wλ2,2(Ω)W̊(Ω)λ1,2Lγ(Ω)

is continuous if 1γ2QQ-4.

Proof.

For any uC0(Ω), we have

uWλ2,2=(uL22+j=1NλjuxjL22+i,j=1Nλixi(λjuxj)L22)12

and

λixi(λjuxj)L2(Ω),i,j=1,2,,N.

Thus,

λjuxjW̊λ1,22=λ(λjuxj)L22=i=1NΩ|λixi(λjuxj)|2𝑑x<+.(2.1)

Hence, λjuxjW̊(Ω)λ1,2, j=1,,N, and by Proposition 2.2 we have W̊(Ω)λ1,2L2λ*(Ω), where 2λ*:=2QQ-2, so

λjuxjL2λ*CλjuxjW̊λ1,2.(2.2)

Therefore, we get λjuxjL2λ*(Ω), j=1,,N, and by the definition of the space W̊(Ω)λ1,2λ* we infer that uW̊(Ω)λ1,2λ*. Applying Proposition 2.2 once again, we obtain

uL(2λ*Q)/(Q-2λ*)CuW̊λ1,2λ*.(2.3)

From (2.1)–(2.3) we obtain

uL2QQ-4CuW̊λ1,2λ*=λuL2λ*CλuW̊λ1,2C(i,j=1NΩ|λixi(λjuxj)|2𝑑x)12C(uL22+j=1NλjuxjL22+i,j=1Nλixi(λjuxj)L22)12=CuWλ2,2.

2.2 Functional setting of the problem

We now define some functional spaces which are used to study problem (1.1).

We consider the operator

A:Wλ2,2(Ω)W̊(Ω)λ1,2L2(Ω),

where A=-Δλ with the homogeneous Dirichlet boundary condition. Then A is linear, positive, self-adjoint and has a compact inverse. Consequently, there exists an orthonormal basis of L2(Ω) consisting of eigenfunctions φjW̊(Ω)λ1,2, j=1,2,, of A with eigenvalues

0<μ1μ2andμj+ as j+.

We denote Es=D(As), with s>0, the space with the inner product

(u,v)Es=ΩAsuAsv𝑑x,u,vEs,

where

D(As)={φ=j=1ajφj:aj,j=1μj2saj2<+},Asφ=j=1ajμjsφj.

We notice that, as a consequence of Lemma 2.2 and interpolation theorems, we have the following important embeddings which play an important role for our investigation.

Lemma 2.3.

The embeddings

EsLq+1(Ω)𝑎𝑛𝑑EtLp+1(Ω)

are continuous if 1q+112-2sQ and 1p+112-2tQ, respectively. Moreover, these embeddings are compact if the corresponding inequalities are strict.

Proof.

By Lemma 2.2 and since 2QQ-4>2, we obtain

uL2CuWλ2,2.

On the other hand, by Proposition 2.1, we have

uL2CuW̊λ1,2.

Therefore,

uL2Cmax{uW̊λ1,2,uWλ2,2}=CuD(A).

Hence, by interpolation, the injection

Es=[D(A),W̊(Ω)λ1,2]2s[L2(Ω),L2λ*(Ω)]2s=Lq+1(Ω),with 1q+1=1-2s2+2s2λ*,

is continuous. Moreover, this embedding is compact if

1q+1>1-2s2+2s2λ*=12-2sQ.

This completes the proof of the first statement. The second one is proved similarly. ∎

For s,t0 such that s+t=1, we consider E=Es×Et, a Hilbert space with the inner product

(z,η)E=(u,φ)Es+(v,ψ)Etfor z=(u,v),η=(φ,ψ)E,

and the bilinear form

B((u,v),(φ,ψ))=Ω(AsuAtψ+AsφAtv)𝑑x.

It is easy to see that B is symmetric and continuous and there exists a self-adjoint bounded linear operator L:EE with

B(z,η)=(Lz,z)for all z,ηE.

Associated to B and L we define the quadratic form P:E by

P(z)=12(Lz,z)E=ΩAsuAtv𝑑xfor all z=(u,v)E.

Using arguments as in [7], one can prove that the operator L only has two eigenvalues ±1, and the eigenspaces E+ and E- are

E+={(u,As-tu):uEs}andE-={(u,-As-tu):uEs}.

Let {ej}j=1 be an orthonormal basis of Es. Thus {As-tej}j=1 is also an orthonormal basis of Et. We denote

Ens:=span{ej:j=1,,n}andEnt:=span{As-tej:j=1,,n}.

We can check that E is presented as a direct sum E=E+E-, En=En+En-=Ens×Ent, and

n=1En+¯=E+,n=1En-¯=E-.

Therefore, for every z=z++z-E=E+E-, we have z+=(u,As-tu) and z-=(u,-As-tu). This implies that

B(z+,z-)=0for all z+E+,z-E-,

and

P(z+)-P(z-)=12zE2.

Now we define the functional Φ:E=Es×Et associated to the problem (1.1) by

Φ(z)=P(z)-ΩH(u,v)𝑑x,

where

H(u,v)=|v|p+1p+1+|u|q+1q+1.

One can check that Φ is well-defined on E and ΦC1(E,) with

Φ(u,v)(ϕ,ψ)=Ω(AsuAtψ+AtvAsϕ)𝑑x+Ω(uqϕ+vpψ)𝑑x.

One also sees that the critical points of Φ are the weak solutions of problem (1.1) in the following sense.

Definition 2.4.

We say that z=(u,v)E=Es×Et is a weak solution of (1.1) if

ΩAsuAtψ𝑑x+Ωvpψ𝑑x=0for all ψEt,ΩAtvAsϕ𝑑x+Ωuqϕ𝑑x=0for all ϕEs.

2.3 Some results of abstract critical points theory

We will use the Fountain Theorem established by Bartsch and de Figueiredo in [3] to prove the existence of infinitely many weak solutions to problem (1.1).

Definition 2.5.

Let X be a Hilbert space and a functional ΦC1(X,). Given a sequence =(Xn) of finite dimensional subspaces of X such that XnXn+1, n=1,2,, and

n=1Xn¯=X.

We say that

  • (i)

    a sequence (zk)X with zkXnk, nk, is a (PS)c-sequence if

    Φ(zk)cand(1+zk)(Φ|Xnk)(zk)0;

  • (ii)

    Φ satisfies (PS)c at level c if every sequence (PS)c-sequence has a subsequence converging to a critical point of Φ.

Theorem 2.6 ([3]).

Assume Φ:ER is C1(E,R) and satisfies the following conditions:

  • (Φ1)

    Φ satisfies (PS)c with =(En), n=1,2, and c>0.

  • (Φ2)

    There exists a sequence rk>0, k=1,2, , such that for some k2,

    bk:=inf{Φ(z):zE+,zEk-1,z=rk}+as k.

  • (Φ3)

    There exists a sequence of isomorphisms Tk:EE, k=1,2, , with Tk(En)=En for all k and n , and there exists a sequence Rk>0, k=1,2, , such that, for z=z++z-Ek+E- and Rk=max{z+,z-} , one has

    Tkz>rk𝑎𝑛𝑑Φ(Tkz)<0,

    where rk is obtained in (Φ2).

  • (Φ4)

    dk:=sup{Φ(Tk(z++z-)):z+Ek+,z-E-,z+,z-Rk}<+.

  • (Φ5)

    Φ is even, i.e., Φ(z)=Φ(-z).

Then Φ has an unbounded sequence of critical values.

As is noticed in [3], condition (Φ4) holds if the functional Φ maps bounded sets in E to bounded sets in .

3 Non-existence of positive classical solutions

In this section, we prove the non-existence result for our problem when the domain Ω is δt-starshaped in the following sense.

Definition 3.1 ([10]).

A domain Ω is called δt-starshaped with respect to the origin if 0Ω and T,ν0 at every point of Ω, where ν is the outward normal vector and , denotes the inner product in N.

As mentioned above, we consider the vector field

T:=i=1Nϵixixi,(3.1)

and this field is the generator of the group of dilation (δt)t0 (here, a function u is δt-homogeneous of degree m if and only if Tu=mu).

As in [10], we will denote by Λ2(Ω¯) the linear space of the functions uC(Ω¯) such that Xju and Xj2u, for j=1,,N, exist in the weak sense of distributions in Ω and can be continuously extended to Ω¯. Here Xj:=λjxj.

Lemma 3.2.

For any u,vΛ2(Ω¯), we have

Ω[T(u)Δλv+T(v)Δλu]𝑑x=Ω[T(u)λv,νλ+T(v)λu,νλ]𝑑S-Ωλu,λvT,ν𝑑S+(Q-2)Ωλu,λv𝑑x,(3.2)

where T is the vector field in (3.1), ν is the outward normal to Ω and

νλ=(λ1ν1,,λNνN),λ=(λ1x1,,λNxN).

Remark 3.3.

If u=v, we recover from (3.2) the Pohozaev-type identity in the scalar case, which was obtained in [10, Theorem 2.1].

Proof.

Integrating by parts, we have

ΩT(u)Δλv𝑑x=i,j=1NΩϵixiuxiλj2vxjνj𝑑S-Ωλj2vxjxj(ϵixiuxi)𝑑x=I1+I2,(3.3)

where

I1=i,j=1NΩϵixiuxiλj2vxjνj𝑑S=i=1NΩϵixiuxiλv,νλ𝑑S=ΩT(u)λv,νλ𝑑S(3.4)

and

I2=-i,j=1NΩλj2vxj(ϵixiuxi)𝑑x=-i,j=1NΩ[δijϵiλj2vxjuxi+ϵixiλj2vxj2uxixj]𝑑x=-j=1NΩϵjλj2vxjuxj𝑑x-i,j=1NΩϵixiλj2vxj2uxixj𝑑x=I2,1+I2,2.

Integrating by parts in I2,2 gives

I2,2=-i,j=1NΩϵixiλj2vxj2uxixj𝑑x=-Ωλu,λvT,ν𝑑S+QΩλu,λv𝑑x+i,j=1NΩϵixiλj2uxj2vxjxi𝑑x+j=1NΩuxjvxjTλj2𝑑x.(3.5)

Since λj is δt-homogeneous of degree ϵj-1, we have Tλj2=2λjTλj=2(ϵj-1)λj2. Therefore, from (2.1) we get

I2,2=-Ωλu,λvT,ν𝑑S+QΩλu,λv𝑑x+i,j=1NΩϵixiλj2uxj2vxjxi𝑑x+j=1NΩ2(ϵj-1)uxjvxjλj2𝑑x=-Ωλu,λvT,ν𝑑S+(Q-2)Ωλu,λv𝑑x+i,j=1NΩϵixiλj2uxj2vxjxi𝑑x+2j=1NΩϵjuxjvxjλj2𝑑x.(3.6)

It follows from (3.3)–(3.6) that

ΩT(u)Δλv𝑑x=ΩT(u)λv,νλ𝑑S-Ωλu,λvT,ν𝑑S+(Q-2)Ωλu,λv𝑑x-I2,1+i,j=1NΩϵixiλj2uxj2vxjxi𝑑x.(3.7)

By similar computations, we also obtain

ΩT(v)Δλu𝑑x=ΩT(v)λu,νλ𝑑S-j=1NΩϵjλj2vxjuxj𝑑x-i,j=1NΩϵixiλj2uxj2vxixj𝑑x=ΩT(v)λu,νλ𝑑S+I2,1-i,j=1NΩϵixiλj2uxj2vxixj𝑑x.(3.8)

Combining (3.7) and (3.8), we obtain (3.2). ∎

The following theorem is the main result of this section.

Theorem 3.4.

Assume N3, and let p,q>1 satisfy

1p+1+1q+1Q-2Q.(3.9)

If Ω is bounded and δt-starshaped with respect to the origin, then problem (1.1) has no nontrivial nonnegative solution uΛ2(Ω¯).

Proof.

From (1.1) we have

Ω[T(v)Δλu+T(u)Δλv]𝑑x=-Ω[T(v)vp+T(u)uq]𝑑x=-i=1NΩ(|v|p+1p+1+|u|q+1q+1)ϵixi𝑑S+i=1NΩϵi(|v|p+1p+1+|u|q+1q+1)𝑑x=QΩ(|v|p+1p+1+|u|q+1q+1)𝑑x.(3.10)

From (3.2) and (3.10) we infer that

QΩ(|v|p+1p+1+|u|q+1q+1)𝑑x=Ω[λu,νλT(v)+λv,νλT(u)]𝑑S-Ωλu,λvT,ν𝑑S+(Q-2)Ωλu,λv𝑑x.(3.11)

On the other hand, integrating by parts, we get

Ω|v|q+1𝑑x=-ΩvΔλu𝑑x=Ωλu,λv𝑑x,(3.12)

and

Ω|u|q+1𝑑x=-ΩuΔλv𝑑x=Ωλu,λv𝑑x.(3.13)

Therefore, we obtain from (3.12) and (3.13) the following identity:

Ω[(1-θ)|u|q+1+θ|v|q+1]𝑑x=Ωλu,λv𝑑x.(3.14)

From (3.11) and (3.14) we get

QΩ(|v|p+1p+1+|u|q+1q+1)𝑑x-(Q-2)Ω[(1-θ)|u|q+1+θ|v|q+1]𝑑x=Ω[λu,νλT(v)+λv,νλT(u)]𝑑S-Ωλu,λvT,ν𝑑S.

This is equivalent to

Ω[(Qp+1-θ(Q-2))|v|p+1+(Qq+1-(1-θ)(Q-2))|u|q+1]𝑑x=Ω[λu,νλT(v)+λv,νλT(u)]𝑑S-Ωλu,λvT,ν𝑑S.(3.15)

Now, since u,vC1(Ω¯Π), the condition u=v=0 on Ω implies that uxj=uννj, vxj=vννj at any point of ΩΠ for any j=1,,N. Therefore, on ΩΠ we have

[λu,νλT(v)+λv,νλT(u)]-λu,λvT,ν=i,j=1N[λj2uxjνjϵixivxi+λj2vxjνjϵixiuxi-λj2uxjvxjϵixiνi]=i,j=1Nuνvνλj2νj2ϵixiνi=uνvν|νλ|2T,ν.

Substituting this identity into (3.15), we obtain

Ω[(Qp+1-θ(Q-2))|v|p+1+(Qq+1-(1-θ)(Q-2))|u|q+1]𝑑x=Ωuνvν|νλ|2T,ν𝑑S.(3.16)

Choosing θ=Q(Q-2)(p+1) yields Qp+1-θ(Q-2)=0, and from (3.9) we get

Qq+1-(1-θ)(Q-2)0.

We infer from (3.16) that

Ωuνvν|νλ|2T,ν𝑑S0.(3.17)

Since Ω is δt-starshaped, we have T,ν0 on Ω; along with uν,vν0, we arrive at

Ωuνvν|νλ|2T,ν𝑑S0.(3.18)

Then from (3.17) and (3.18) we get

Ωuνvν|νλ|2T,ν𝑑S=0at any point of Ω.

We have

Qq+1-(1-θ)(Q-2)<0if 1p+1+1q+1<Q-2Q,

and from (3.16) we get u0 and hence v0.

We have

Qq+1-(1-θ)(Q-2)=0if 1p+1+1q+1=Q-2Q.

Since T,ν0 on Ω, we have uν=0 or vν=0 at some point of Ω. Moreover, -Δλu,-Δλv,u,v0 in Ω and u=v=0 on Ω. This implies that u0 or v0; hence uv0. ∎

4 Existence of infinitely many weak solutions

Theorem 4.1.

If p,q>1, Ω is a smooth and bounded domain in RN and

1p+1+1q+1>Q-2Q,(4.1)

then problem (1.1) has infinitely many weak solutions.

Proof.

We notice that, by the fact that H(-u,-v)=H(u,v) for all (u,v)×, with

H(u,v)=|v|p+1p+1+|u|q+1q+1,

the functional I(u,v) is even, and therefore (Φ5) is satisfied. We will check conditions (Φ1)–(Φ4) in Theorem 2.6, and the proof is divided into four steps.

Step 1: Checking condition (Φ1). By using [3, Remark 2.1], it suffices to check that a (PS)c-sequence in E is bounded. This is the content of the following lemma.

Step 2: Checking condition (Φ2). We claim that if ΦC1(E,) and (4.1) is satisfied, then there exists a sequence rk>0, k=1,2,, such that for some k2,

bk:=inf{Φ(z):zE+,zEk-1,z=rk}+as k.(4.6)

Indeed, for any z=(u,v)E we have

ΩH(u,v)𝑑x=Ω(1q+1|u|q+1+1p+1|v|p+1)𝑑xC(uLq+1q+1+vLp+1p+1+1)C(uLq+1r+vLp+1r+1)(r=max{p+1,q+1}>2)=C(zLq+1×Lp+1r+1).

We prove that

μk:=sup{zLq+1×Lp+1:zE,zEk-1,zE=1}0as k.(4.7)

Indeed, since 0<μk+1μk, i.e., {μk} is decreasing and bounded from below, there exists μ0 such that

μkμ0as k.

On the other hand, for every k0 there exists zjE such that zjEkj-1, zjE=1 and

zjLq+1×Lp+1>μk2.

Since E is a Hilbert space, there exist zE and a subsequence relabeled {zj} such that

zjzin E,

that is,

(zj,η)(z,η)for all ηE.

Moreover, for

η=(ϕ,ψ)E=Es×Et

we have

ϕ=j=1cjejandψ=j=1djAs-tej.

Thus,

(z,η)=limj(zj,η)=limj[(uj,ϕ)Es+(vj,ψ)Et]=limj[k=1ck(uj,ek)+k=1dk(vj,As-tek)]=0.

This implies that

zj0in E,

and since the embedding ELq+1×Lp+1 is compact, we get

zj0in E.

Therefore, we obtain μ=0, i.e., μk0 as k. Thus, (4.7) is proved.

Next, for zE+, zEk-1, we have

Φ(z)=12zE2-Ω(|u|q+1q+1+|v|p+1p+1)𝑑x12zE2-C(zLq+1×Lp+1r+1)=12zE2-CzzELq+1×Lp+1rzEr-C12zE2-CμkrzEr-C=(12-CμkrzEr-2)zE2-C.

Choosing zE=rk with rk=(Crμkr)-1/(r-2)>0, k, we have

Φ(z)(12-1r)rk2-C+.

Therefore, we obtain (4.6) and this implies that condition (Φ2) holds.

Step 3: Checking condition (Φ3). We prove that there exists a sequence αk>0, k, such that (Φ3) is satisfied with Tk:=Tαk and Rk:=αk.

Indeed, for each α>0, we consider Tα:EE being the isomorphism defined by Tα=(αpu,αqv). It is easy to see that TαEn=En for all n=1,2,.

First, with z=z++z-Ek+E-, we denote z-=z1-+z2-, where z1-Ek-, z2-Ek-, and further z¯=z++z1-, z=(u,v) and z±=(u±,v±). Then we have u2-u¯ in L2, and hence, using the Hölder inequality and since Ω is bounded and q+1,p+1>2, we have

u¯L22=|(u,u¯)|L2uL2u¯L2uLq+1u¯Lq+1.(4.8)

Now, for each k, since Eks and Ekt are finite-dimensional subspaces, all norms corresponding on Eks and Ekt are equivalent, so there exist positive constants σk, σk, τk, τk such that

uL2σkuEs,uEsσkuLq+1  for all uEks,(4.9)

and

vL2τkvEt,vEtτkvLp+1  for all vEkt.(4.10)

Using (4.8)–(4.10), we get

uLq+1u¯L22u¯Lq+1-1σk2u¯Es2u¯Lq+1-1σk2σku¯Es2u¯Es-1=σ¯ku¯Es(4.11)

and

vLp+1τ¯kv¯Et,(4.12)

where σ¯k=σk2σk and τ¯k=τk2τk.

For u¯=u++u1- and v¯=As-t(u+-u1-), one has u¯Es=u++u1-Es and

v¯Et=AtAs-t(u+-u1-)L2=As(u+-u1-)L2=u+-u1-Es.

By the definition of the Hamiltonian function H(u,v) and by (4.11)–(4.12), for any α>0 we have

ΩH(Tαz)𝑑x=ΩH(αpu,αqv)𝑑xΩ(1q+1αp(q+1)|u|q+1+1p+1αq(p+1)|v|p+1)𝑑xc(αp(q+1)uLq+1q+1+αq(p+1)vLp+1p+1)c(αp(q+1)σ¯kq+1u¯Esq+1+αq(p+1)τ¯kp+1v¯Etp+1).(4.13)

Since

u+Es=12z+E+=α2,

we see that one of the following two inequalities occurs:

u++u1-Esα2oru+-u1-Esα2.(4.14)

By (4.13) and (4.14),

ΩH(αpu,αqv)𝑑xcδkα(p+1)(q+1),

where

δk=min{(σk2σk2)q+1,(τk2τk2)p+1}.

On the other hand,

P(Tαz)=ΩAs(αpu)At(αqv)𝑑x=αq+pP(z)=αq+p12zE2=12αq+p(z+E+2-z-E-2).(4.15)

Therefore, for z+E+=α we obtain

P(Tαz)12αq+p+2.(4.16)

Since

Φ(Tαz)=P(Tαz)-ΩH(Tαz)𝑑x,

we infer from (4.15) and (4.16) that

Φ(Tαz)12αq+p+2-δkα(q+1)(p+1).

Since p,q>1 and (q+1)(p+1)>q+p+2, there exists an α0(k)>0 such that for all αk>α0(k) one has Φ(Tαk)<0. Moreover, we also obtain

TαzE=ΩAs(αpu)At(αqv)𝑑x=αp+qzE2αmin{p,q}zE2.

This implies that, for αk=max{z+E+2,z-E-2},

TαkzEαkmin{p,q}(z+E+2-z-E-2)αkmin{p,q}+2for all αk>α0(k).

Thus we can choose αk to obtain

Φ(Tαkz)<0andTαkzErk

for rk>0 is given. Therefore, condition (Φ3) in Theorem 2.6 is satisfied.

Step 4: Checking condition (Φ4). Let U be any bounded subset of E, i.e.,

(u,v)Es×Et2=uEs2+vEt2Cfor all (u,v)U.

Then

uEsCandvEtCfor all (u,v)U.

We have

|Φ(u,v)|Ω|AsuAtv|𝑑x+Ω(|v|p+1p+1+|u|q+1q+1)𝑑xAsuL2AtvL2+1p+1vLp+1p+1+1q+1uLq+1q+1.

Since AsuL2=uEs, AtvL2=vEt and the embeddings EsLq+1 and EtLp+1 are compact, we get

|Φ(u,v)|uEsvEt+c(uEsq+1+vEtp+1).

This proves condition (Φ4).

The proof of Theorem 4.1 is therefore complete. ∎

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About the article

Received: 2016-07-22

Accepted: 2017-06-09

Published Online: 2017-07-28


Funding Source: National Foundation for Science and Technology Development

Award identifier / Grant number: 101.02-2015.10

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2015.10.


Citation Information: Advances in Nonlinear Analysis, Volume 8, Issue 1, Pages 661–678, ISSN (Online) 2191-950X, ISSN (Print) 2191-9496, DOI: https://doi.org/10.1515/anona-2016-0165.

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© 2019 Walter de Gruyter GmbH, Berlin/Boston. This work is licensed under the Creative Commons Attribution 4.0 Public License. BY 4.0

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