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Advances in Nonlinear Analysis

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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
  • Email
  • Other articles by this author:
  • De Gruyter OnlineGoogle Scholar
/ Bui Kim My
Published Online: 2017-07-28 | DOI: https://doi.org/10.1515/anona-2016-0165


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


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:


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


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.,


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


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


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


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


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


(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


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


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


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


    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.,


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


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


with the norm


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:




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


is compact for every γ[1,pλ*).

We now prove the following important result.

Lemma 2.2.

The embedding


is continuous if 1γ2QQ-4.


For any uC0(Ω), we have






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


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


From (2.1)–(2.3) we obtain


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


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




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


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


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


On the other hand, by Proposition 2.1, we have




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


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


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


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


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


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-,



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




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


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


We say that

  • (i)

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


  • (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


    where rk is obtained in (Φ2).

  • (Φ4)


  • (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


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


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


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].


Integrating by parts, we have






Integrating by parts in I2,2 gives


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


It follows from (3.3)–(3.6) that


By similar computations, we also obtain


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


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


From (1.1) we have


From (3.2) and (3.10) we infer that


On the other hand, integrating by parts, we get




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


From (3.11) and (3.14) we get


This is equivalent to


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


Substituting this identity into (3.15), we obtain


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


We infer from (3.16) that


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


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


then problem (1.1) has infinitely many weak solutions.


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


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


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


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


we have




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


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


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


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)


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

Using (4.8)–(4.10), we get




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


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




we see that one of the following two inequalities occurs:


By (4.13) and (4.14),




On the other hand,


Therefore, for z+E+=α we obtain




we infer from (4.15) and (4.16) that


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


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


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.


uEsCandvEtCfor all (u,v)U.

We have


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


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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