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This paper is devoted to studying constraint minimizers for a class of elliptic equations with two nonlocal terms. Using the methods of constrained variation and energy estimation, we analyze the existence, non-existence, and limit behavior of minimizers for the related minimization problem. Our work extends and enriches the study of bi-nonlocal problems.
Citation: Xincai Zhu, Yajie Zhu. Existence and limit behavior of constraint minimizers for elliptic equations with two nonlocal terms[J]. Electronic Research Archive, 2024, 32(8): 4991-5009. doi: 10.3934/era.2024230
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This paper is devoted to studying constraint minimizers for a class of elliptic equations with two nonlocal terms. Using the methods of constrained variation and energy estimation, we analyze the existence, non-existence, and limit behavior of minimizers for the related minimization problem. Our work extends and enriches the study of bi-nonlocal problems.
The Schrödinger-Virasoro algebra is an infinite-dimensional Lie algebra that was introduced (see, e.g., [10]) in the context of non-equilibrium statistical physics. In [21], the author give a representation of the Schrödinger-Virasoro algebra by using vertex algebras, and introduced an extension of the Schrödinger-Virasoro algebra. To be precise, for
{Li,Hj,Ii|i∈Z,j∈ε+Z} |
and Lie brackets
[Lm,Ln]=(m−n)Lm+n,[Lm,Hn]=(12m−n)Hm+n,[Lm,In]=−nIm+n,[Hm,Hn]=(m−n)Im+n,[Hm,In]=[Im,In]=0. |
The Lie algebra
Post-Lie algebras were introduced around 2007 by B. Vallette [22], who found the structure in a purely operadic manner as the Koszul dual of a commutative trialgebra. Post-Lie algebras have arose the interest of a great many authors, see [4,5,12,13]. One of the most important problems in the study of post-Lie algebras is to find the post-Lie algebra structures on the (given) Lie algebras. In [13,18,20], the authors determined all post-Lie algebra structures on
In this paper, we shall study the graded post-Lie algebra structures on the Schrödinger-Virasoro algebra. We only study the twisted Schrödinger-Virasoro algebra
Throughout this paper, we denote by
The paper is organized as follows. In Section 2, we give general results on post-Lie algebras and some lemmas which will be used to our proof. In Section 3, we completely characterize the graded post-Lie algebra structures on Schrödinger-Virasoro algebra
We will give the essential definitions and results as follows.
Definition 2.1. A post-Lie algebra
[x,y]▹z=x▹(y▹z)−y▹(x▹z)−⟨x,y⟩▹z, | (1) |
x▹[y,z]=[x▹y,z]+[y,x▹z] | (2) |
for all
Suppose that
τ(x▹1y)=τ(x)▹2τ(y),∀x,y∈L. |
Remark 1. The left multiplications of the post-Lie algebra
Lemma 2.2. [15] Denote by
Der(S)=Inn(S)⊕CD1⊕CD2⊕CD3 |
where
D1(Ln)=0,D1(Hn)=Hn,D1(In)=2In,D2(Ln)=nIn,D2(Hn)=0,D2(In)=0,D3(Ln)=In,D3(Hn)=0,D3(In)=0. |
Since the Schrödinger-Virasoro algebra
Lm▹Ln=ϕ(m,n)Lm+n, | (3) |
Lm▹Hn=φ(m,n)Hm+n, | (4) |
Lm▹In=χ(m,n)Im+n, | (5) |
Hm▹Ln=ψ(m,n)Hm+n, | (6) |
Hm▹Hn=ξ(m,n)Im+n, | (7) |
Im▹Ln=θ(m,n)Im+n, | (8) |
Hm▹In=Im▹Hn=Im▹In=0, | (9) |
for all
We start with the crucial lemma.
Lemma 3.1. There exists a graded post-Lie algebra structure on
ϕ(m,n)=(m−n)f(m), | (10) |
φ(m,n)=(m2−n)f(m)+δm,0μ, | (11) |
χ(m,n)=−nf(m)+2δm,0μ, | (12) |
ψ(m,n)=−(n2−m)h(m), | (13) |
ξ(m,n)=(m−n)h(m), | (14) |
θ(m,n)=mg(m)+δm,0na, | (15) |
(m−n)(f(m+n)(1+f(m)+f(n))−f(n)f(m))=0, | (16) |
(m−n)δm+n,0μ(1+f(m)+f(n))=0, | (17) |
(m2−n)(h(m+n)(1+f(m)+h(n))−f(m)h(n))=0, | (18) |
nδm+n,0a(1+f(m)+g(n))=0, | (19) |
n(m+n)(g(m+n)(1+f(m)+g(n))−f(m)g(n)) =δn,0m2a(f(m)−g(m)), | (20) |
(m−n)δm+n,0a(1+h(m)+h(n))=0, | (21) |
(m−n)(g(m+n)(1+h(m)+h(n))−h(m)h(n))=0. | (22) |
Proof. Suppose that there exists a graded post-Lie algebra structure satisfying (3)-(9) on
x▹y=(adψ(x)+α(x)D1+β(x)D2+γ(x)D3)(y)=[ψ(x),y]+α(x)D1(y)+β(x)D2(y)+γ(x)D3(y) |
where
Lm▹Ln=[ψ(Lm),Ln]+β(Lm)nIn+γ(Lm)In=ϕ(m,n)Lm+n, | (23) |
Lm▹Hn=[ψ(Lm),Hn]+α(Lm)Hn=φ(m,n)Hm+n, | (24) |
Lm▹In=[ψ(Lm),In]+α(Lm)2In=χ(m,n)Im+n, | (25) |
Hm▹Ln=[ψ(Hm),Ln]+β(Hm)nIn+γ(Hm)In=ψ(m,n)Hm+n, | (26) |
Hm▹Hn=[ψ(Hm),Hn]+α(Hm)Hn=ξ(m,n)Im+n, | (27) |
Hm▹In=[ψ(Hm),In]+α(Hm)2In=0, | (28) |
Im▹Ln=[ψ(Im),Ln]+β(Im)nIn+γ(Im)In=θ(m,n)Im+n, | (29) |
Im▹Hn=[ψ(Im),Hn]+α(Im)Hn=0, | (30) |
Im▹In=[ψ(Im),In]+α(Im)2In=0. | (31) |
Let
ψ(Lm)=∑i∈Za(m)iLi+∑i∈Zb(m)iHi+∑i∈Zc(m)iIi,ψ(Hm)=∑i∈Zd(m)iLi+∑i∈Ze(m)iHi+∑i∈Zf(m)iIi,ψ(Im)=∑i∈Zg(m)iLi+∑i∈Zh(m)iHi+∑i∈Zx(m)iIi |
where
The "if'' part is a direct checking. The proof is completed.
Lemma 3.2. Let
g(n),h(n)∈{0,−1}for everyn≠0. | (32) |
Proof. By letting
Lemma 3.3. Let
g(Z)=h(Z)=0org(Z)=h(Z)=−1. |
Proof. Since
a(1+g(−1))=0. | (33) |
By letting
(m2−n)(h(m+n)(1+h(n))=0, | (34) |
n(m+n)(g(m+n)(1+g(n))=0, | (35) |
(m−n)(g(m+n)−h(m)h(n)+h(m)g(m+n)+h(n)g(m+n))=0. | (36) |
We now prove the following four claims:
Claim 1. If
By (34) with
Claim 2. If
By (34) with
Claim 3. If
By (35) with
Claim 4. If
By (35) with
Now we consider the values of
Case i. If
Case ii. If
Case iii. If
Case iv. If
Lemma 3.4. Let
(i)
(ii)
(iii)
Proof. By
h(m+n)(h(n)+1)=0 if m⩽1,m2−n≠0, | (37) |
g(m+n)(g(n)+1)=0 if m⩽1,n≠0,m+n≠0, | (38) |
g(m+n)(1+h(m)+h(n))=h(m)h(n) if m≠n. | (39) |
We first prove the following six claims:
Claim 1. If
By (37) with
Claim 2. If
By (37) with
Claim 3. If
By (37) with
Claim 4. If
By (37) with
Next, similar to Claims 1 and 3, we from (38) obtain the following claims.
Claim 5. If
Claim 6. If
Now we discuss the values of
Case i. When
By Claim 1 we have
Case ii. When
By Claim 2 we have
Case iii. When
By Claims 3 and 4 we have
It is easy to check that the values of
Lemma 3.5. Let
(i)
(ii)
(iii)
for some
(iv)
Proof. Take
h(0)(1+f(−n)+h(n))=f(−n)h(n), for all n≠0, | (40) |
a(1+f(−n)+g(n))=0, for all n≠0, | (41) |
a(1+h(−n)+h(n))=0, for all n≠0, | (42) |
g(0)(1+h(−n)+h(n))=h(−n)h(n), for all n≠0. | (43) |
Note that
h(n)(h(m+n)+1)=0 for all m>0,m2−n≠0; | (44) |
h(m+n)(h(n)+1)=0 for all m<0,m2−n≠0; | (45) |
g(n)(g(m+n)+1)=0 for all m>0,n≠0,m+n≠0; | (46) |
g(m+n)(g(n)+1)=0 for all m<0,n≠0,m+n≠0; | (47) |
g(m+n)(1+h(m)+h(n))=h(m)h(n) for all m≠n. | (48) |
For any
Claim 1. If
In fact, by (44) with
Claim 2. If
This proof is similar to Claim 1 by using (44) and (45). Also, similar to Claims 1 and 2, by (46) and (47) we can obtain the following two claims:
Claim 3. If
Claim 4. If
According to (32), by Claims 1 and 2,
(1)
(2)
(3)
(4)
In view of the above result, the next proof will be divided into the following cases.
Case i. When
By taking
Case ii. When
By taking
Case iii. When
By (48) we see that
Case iv. When
Note that
Lemma 3.6. Let
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Proof. The proof of the "if" direction can be directly verified. We now prove the "only if" direction. In view of
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When
When
When
Lemma 3.7. Let (P(ϕi,φi,χi,ψi,ξi,θi),▹i), i=1,2 be two algebras with the same linear space as S and equipped with C-bilinear products x▹iy such that
Lm▹iLn=ϕi(m,n)Lm+n,Lm▹iHn=φi(m,n)Hm+n,Lm▹iIn=χi(m,n)Im+n,Hm▹iLn=ψi(m,n)Hm+n,Hm▹iHn=ξi(m,n)Im+n,Im▹iLn=θi(m,n)Im+n,Hm▹iIn=Im▹iHn=Im▹iIn=0 |
for all m,n∈Z, where ϕi,φi,χi,ψi,ξi,θi, i=1,2 are complex-valued functions on Z×Z. Furthermore, let τ:P(ϕ1,φ1,χ1,ψ1,ξ1,θ1)→P(ϕ2,φ2,χ2,ψ2,ξ2,θ2) be a linear map determined by
τ(Lm)=−L−m,τ(Hm)=−H−m,τ(Im)=−I−m |
for all
{ϕ2(m,n)=−ϕ1(−m,−n);φ2(m,n)=−φ1(−m,−n);χ2(m,n)=−χ1(−m,−n);ψ2(m,n)=−ψ1(−m,−n);ξ2(m,n)=−ξ1(−m,−n);θ2(m,n)=−θ1(−m,−n). | (49) |
Proof. Clearly,
τ(Lm▹iLn)=−ϕi(m,n)L−(m+n),τ(Lm▹iHn)=−φi(m,n)H−(m+n),τ(Lm▹iIn)=−χi(m,n)I−(m+n),τ(Hm▹iLn)=−ψi(m,n)H−(m+n),τ(Hm▹iHn)=−ξi(m,n)I−(m+n),τ(Im▹iLn)=−θi(m,n)I−(m+n) |
for
The remainder is to prove that
τ(Lm▹1Ln)=−ϕ1(m,n)L−(m+n)=ϕ2(−m,−n)L−(m+n)=τ(Lm)▹2τ(Ln),τ(Lm▹1Hn)=−φ1(m,n)H−(m+n)=φ2(−m,−n)H−(m+n)=τ(Lm)▹2τ(Hn),τ(Lm▹1In)=−χ1(m,n)I−(m+n)=χ2(−m,−n)I−(m+n)=τ(Lm)▹2τ(In),τ(Hm▹1Ln)=−ψ1(m,n)H−(m+n)=ψ2(−m,−n)H−(m+n)=τ(Hm)▹2τ(Ln),τ(Hm▹1Hn)=−φ1(m,n)I−(m+n)=φ2(−m,−n)I−(m+n)=τ(Hm)▹2τ(Hn), |
τ(Im▹1Ln)=−θ1(m,n)I−(m+n)=ϕ2(−m,−n)I−(m+n)=τ(Im)▹2τ(Ln) |
and
Theorem 3.8. A graded post-Lie algebra structure on
where
Proof. Suppose that
Conversely, every type of the
Finally, by Lemma 3.7 with maps
The Rota-Baxter algebra was introduced by the mathematician Glen E. Baxter [2] in 1960 in his probability study, and was popularized mainly by the work of Rota [G. Rota1, G. Rota2] and his school. Recently, the Rota-Baxter algebra relation were introduced to solve certain analytic and combinatorial problem and then applied to many fields in mathematics and mathematical physics (see [6,7,19,23] and the references therein). Now let us recall the definition of Rota-Baxter operator.
Definition 4.1. Let
[R(x),R(y)]=R([R(x),y]+[x,R(y)])+λR([x,y]),∀x,y∈L. | (50) |
Note that if
In this section, we mainly consider the homogeneous Rota-Baxter operator
R(Lm)=f(m)Lm, R(Hm)=h(m)Hm, R(Im)=g(m)Im | (51) |
for all
Lemma 4.2. (see [1]) Let
Theorem 4.3. A homogeneous Rote-Baxrer operator
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
(
for all
Proof. In view of Lemma 4.2, if we define a new operation
Lm▹Ln=[R(Lm),Ln]=(m−n)f(m)Lm+n, | (52) |
Lm▹Hn=[R(Lm),Hn]=(m2−n)f(m)Hm+n, | (53) |
Lm▹In=[R(Lm),In]=−nf(m)Im+n, | (54) |
Hm▹Ln=[R(Hm),Ln]=−(n2−m)h(m)Hm+n, | (55) |
Hm▹Hn=[R(Hm),Hn]=(m−n)h(m)Im+n, | (56) |
Im▹Ln=[R(Im),Ln]=mg(m)Im+n | (57) |
and
A similar discussion to Lemma 3.1 gives
(m−n)(f(m+n)−f(n)f(m)+f(m)f(m+n)+f(n)f(m+n))=0,(m2−n)(h(m+n)−f(m)h(n)+f(m)h(m+n)+h(n)h(m+n))=0,n(m+n)(g(m+n)(1+f(m)+g(n))−f(m)g(n))=0,(m−n)(g(m+n)−h(m)h(n)+h(m)g(m+n)+h(n)g(m+n))=0. |
From this we conclude that Equations (10)-(22) hold with
The natural question is: how we can characterize the Rota-Baxter operators of weight zero on the Schrödinger-Virasoro
Definition 4.4. A pre-Lie algebra
(x▹y)▹z−x▹(y▹z)=(y▹x)▹z−y▹(x▹z),∀x,y,z∈A. | (58) |
As a parallel result of Lemma 4.2, one has the following conclusion.
Proposition 1. (see [8]) Let
Using a similar method on classification of Rota-Baxter operators of weight
We would like to express our sincere thanks to the anonymous referees for their careful reading and valuable comments towards the improvement of this article.
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