Let 0<α<n and b be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator (TΩ,α)mb generated by the function b and the fractional integral operator with the homogeneous kernel TΩ,α on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator (TΩ,α)mb on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator (Tα)mb generated by the function b and the fractional integral operator Tα on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator [b,Tα] generated by the function b and the fractional integral operator Tα on variable Lebesgue spaces and mixed Morrey spaces.
Citation: Heng Yang, Jiang Zhou. Compactness of commutators of fractional integral operators on ball Banach function spaces[J]. AIMS Mathematics, 2024, 9(2): 3126-3149. doi: 10.3934/math.2024152
[1] | Zehra Yucedag . Variational approach for a Steklov problem involving nonstandard growth conditions. AIMS Mathematics, 2023, 8(3): 5352-5368. doi: 10.3934/math.2023269 |
[2] | Aihui Sun, Hui Xu . Decay estimate and blow-up for fractional Kirchhoff wave equations involving a logarithmic source. AIMS Mathematics, 2025, 10(6): 14032-14054. doi: 10.3934/math.2025631 |
[3] | Fugeng Zeng, Peng Shi, Min Jiang . Global existence and finite time blow-up for a class of fractional p-Laplacian Kirchhoff type equations with logarithmic nonlinearity. AIMS Mathematics, 2021, 6(3): 2559-2578. doi: 10.3934/math.2021155 |
[4] | Zhiqiang Li . The finite time blow-up for Caputo-Hadamard fractional diffusion equation involving nonlinear memory. AIMS Mathematics, 2022, 7(7): 12913-12934. doi: 10.3934/math.2022715 |
[5] | Radu Precup, Andrei Stan . Stationary Kirchhoff equations and systems with reaction terms. AIMS Mathematics, 2022, 7(8): 15258-15281. doi: 10.3934/math.2022836 |
[6] | Nanbo Chen, Honghong Liang, Xiaochun Liu . On Kirchhoff type problems with singular nonlinearity in closed manifolds. AIMS Mathematics, 2024, 9(8): 21397-21413. doi: 10.3934/math.20241039 |
[7] | Tao Wu . Some results for a variation-inequality problem with fourth order p(x)-Kirchhoff operator arising from options on fresh agricultural products. AIMS Mathematics, 2023, 8(3): 6749-6762. doi: 10.3934/math.2023343 |
[8] | Jinguo Zhang, Dengyun Yang, Yadong Wu . Existence results for a Kirchhoff-type equation involving fractional p(x)-Laplacian. AIMS Mathematics, 2021, 6(8): 8390-8403. doi: 10.3934/math.2021486 |
[9] | Yun-Ho Kim, Hyeon Yeol Na . Multiplicity of solutions to non-local problems of Kirchhoff type involving Hardy potential. AIMS Mathematics, 2023, 8(11): 26896-26921. doi: 10.3934/math.20231377 |
[10] | Adel M. Al-Mahdi . The coupling system of Kirchhoff and Euler-Bernoulli plates with logarithmic source terms: Strong damping versus weak damping of variable-exponent type. AIMS Mathematics, 2023, 8(11): 27439-27459. doi: 10.3934/math.20231404 |
Let 0<α<n and b be a locally integrable function. In this paper, we obtain the characterization of compactness of the iterated commutator (TΩ,α)mb generated by the function b and the fractional integral operator with the homogeneous kernel TΩ,α on ball Banach function spaces. As applications, we derive the characterization of compactness via the commutator (TΩ,α)mb on weighted Lebesgue spaces, and further obtain a necessary and sufficient condition for the compactness of the iterated commutator (Tα)mb generated by the function b and the fractional integral operator Tα on Morrey spaces. Moreover, we also show the necessary and sufficient condition for the compactness of the commutator [b,Tα] generated by the function b and the fractional integral operator Tα on variable Lebesgue spaces and mixed Morrey spaces.
This paper is dedicated to a Kirchhoff-type equation driven by a nonlocal fractional p-Laplacian as follows:
{M([ψ]ps,p)Lspψ(z)=g(z,ψ)inΩ,ψ>0inΩ,ψ=0onRN∖Ω,[ψ]ps,p∈J, | (P) |
where s∈(0,1), p∈(1,+∞), sp<N, J⊆(0,+∞) is an open interval, Ω⊆RN (N≥2) is an open bounded set with Lipschitz boundary ∂Ω, [ψ]ps,p:=∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy, M is an increasing Kirchhoff-type function on J, and a function g is nonnegative, which will be introduced later. Here, Lsp is a nonlocal operator defined pointwise as follows:
Lspψ(z)=2∫RNK(z,y)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))dyfor all z∈RN, |
where a function K:RN×RN→(0,+∞) fulfills the following assumptions:
(K1) κK∈L1(RN×RN), where κ(z,y)=min{|z−y|p,1};
(K2) There exist positive constants γ0 and γ1 with γ0≥1 such that γ0≤K(z,y)|z−y|N+sp≤γ1 for z≠y and for almost all (z,y)∈RN×RN;
(K3) K(y,z)=K(z,y) for all (y,z)∈RN×RN.
When K(z,y)=|z−y|−(N+sp), Lsp becomes the fractional p-Laplacian operator (−Δ)sp defined as follows:
(−Δ)spψ(z)=2limε↘0∫RN∖Bε(z)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))|z−y|N+spdy,z∈RN, |
where Bε(z):={z∈RN:|z−y|≤ε}.
Over the last few decades, fractional Sobolev spaces and their corresponding nonlocal equations have gained increasing attention because they can be corroborated as models for many physical phenomena arising from studies of Lévy processes, fractional quantum mechanics, optimization, image processing, thin obstacle problems, anomalous diffusion in plasma, American options, game theory, geophysical fluid dynamics, and frame propagation; see [6,14,24,28,34] for comprehensive studies and details on these topics.
The study of Kirchhoff-type problems, which was originally proposed by Kirchhoff [18], has a powerful background in various applications in physics and biology. For this reason, much attention has recently been given to the investigation of elliptic equations related to Kirchhoff coefficients; for example, see [15,16,25,26,29,32] and the references therein. The authors of [11] discussed in detail the physical implications underlying the fractional Kirchhoff model. Particularly, by considering a truncation argument and the mountain pass theorem, the existence of nontrivial solutions to a nonlocal elliptic problem was obtained when an increasing and continuous Kirchhoff term M has the nondegenerate condition infξ∈[0,+∞)M(ξ)≥ξ0>0, where ξ0 is a constant; see also [30] and references therein. However, the existence of at least two different nontrivial solutions to the fractional p-Laplacian equations of the Schrödinger–Kirchhoff type was demonstrated in [32] when the nondegenerate continuous Kirchhoff function M fulfills the hypothesis:
(M1) There is δ∈[1,NN−sp) such that δM(ξ):=δ∫ξ0M(σ)dσ≥M(ξ)ξ for any ξ≥0, where 0<s<1.
The assumption (M1) contains not only the classical example M(ξ)=1+aξδ (a≥0,ξ≥0) but the nonmonotonic cases. In this regard, nonlinear elliptic equations of Kirchhoff type involving (M1) have received widely remarkable attention; see [7,15,16,19,20,35]. Considering these related papers, the functional A:Ws,pK(Ω)→R associated with the principal part in (P) is given by
A(ψ)=1pM(∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy) |
for any ψ∈Ws,pK(Ω), where a solution space Ws,pK(Ω) will be introduced later. Then, in accordance with the fact that M∈C([0,+∞)), it follows that A∈C1(Ws,pK(Ω),R) and its Fréchet derivative is defined as
⟨A′(ψ),ϕ⟩=M([ψ]ps,p)∫RN∫RNK(z,y)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))(ϕ(z)−ϕ(y))dzdy |
for any ψ,ϕ∈Ws,pK(Ω). Specifically, assumptions M∈C([0,∞)) and (M1) play an effective role in deriving some topological properties of functionals A,A′ and the compactness condition of Palais–Smale-type for an energy functional related to (P), which are essential in using variational methods such as Ekeland variational principle, mountain pass theorem, and fountain theorem. But, many examples are eliminated from the continuity of the nondegenerate Kirchhoff function M in [0,∞). For example, let the Kirchhoff functions be defined by
M(ξ)=tanξfor0<ξ<π2 |
and
M(ξ)=(δ−ξ)−ℓforξ∈(−∞,δ),whereδ>0, 0<ℓ<1. |
These functions cannot be covered by any of the results known to date. Recently, to obtain at most one positive solution for the non-local problems with discontinuous Kirchhoff functions, Ricceri [33] discussed a new approach different from those of previous related studies [2,10,11,16,29,32]. The author of [21] recently extended the result of [33] to elliptic equations involving p-Laplacian; see also the paper [22] for problems involving double-phase operators. The primary tools for getting these results in [21,22] are the uniqueness results of the Brézis–Oswald-type problem based on [5] and the abstract global minimum principle in [33]. Especially, the Dìaz–Saa-type inequalities in [8,9] play an essential role in attaining the uniqueness of a positive solution to equations examined in [21,22]. In addition, inspired by previous studies [4,27], the author of [23] determined the existence and uniqueness of a positive solution to nonlinear the Brézis–Oswald type equations involving the fractional Laplacian. For its application, the existence of at most one positive solution to Kirchhoff-type equations driven by the nonlocal fractional Laplacian has been investigated.
The primary aim of this paper is to derive the existence and uniqueness of positive solutions to the fractional p-Laplacian equations involving discontinuous Kirchhoff-type coefficients. In the application of the inequalities of Dìaz–Saa-type in [8,9], the well-known Hopf boundary lemma is required to show that the quotient between solutions is contained in the L∞-space. Though, solutions of fractional-order equations are generally singular at the boundary, making it difficult to work with their quotient between solutions, as Hopf's boundary lemma is not maintained. Hence, in distinction from previous studies [21,22], the major difficulty of this paper is to derive that Brézis–Oswald-type problems involving the fractional p-Laplacian admit at most one positive weak solution. Based on previous studies [4,17,27], we overcome this difficulty by taking into account the discrete Picone inequality in [3,12]. As far as we are aware, the Brézis–Oswald-type result to nonlinear elliptic problems with the Kirchhoff coefficient has not been studied much; we only know of one study [2,23] in this direction. Recently, Biagi and Vecchi [2] obtained uniqueness results for Brézis–Oswald-type Laplacian problems with degenerate Kirchhoff functions M in [0,∞) when M is a continuous, nonnegative and nondecreasing function satisfying M(ξ)>0 for every ξ>0. But, our main result differs from that of [2] because we consider a discontinuous Kirchhoff function M in [0,∞) and solution localization. Although our result is based on previous work [23], problem (P) has more complex nonlinearities than [23] and thus requires a more fastidious analysis to be performed carefully.
The remainder of this paper is organized as follows: In Section 2, we present some essential preliminary knowledge of our considered function spaces to be utilized in this paper. In Section 3, we provide the variational framework associated with problem (P), and then, we will derive the existence and uniqueness results of positive solutions under suitable assumptions.
For the convenience of the reader, in this section we shortly present some practical definitions and fundamental properties of the fractional Sobolev spaces that will be used in the presnt paper. Let s∈(0,1) and p∈(1,∞) be real numbers, and let p∗s be the fractional critical Sobolev exponent, such that is
p∗s:={NpN−spif sp<N,+∞if sp≥N. |
Let Ω⊂RN be an bounded open set with a smooth boundary. Let the fractional Sobolev space Ws,p(Ω) be defined as follows:
Ws,p(Ω):={ψ∈Lp(Ω):∫RN∫RN|ψ(z)−ψ(y)|p|z−y|N+psdzdy<+∞}, |
endowed with the norm
||ψ||Ws,p(Ω):=(||ψ||pLp(Ω)+|ψ|pWs,p(RN))1p, |
where
||ψ||pLp(Ω):=∫Ω|ψ(z)|pdzand|ψ|pWs,p(RN):=∫RN∫RN|ψ(z)−ψ(y)|p|z−y|N+psdzdy. |
Then, Ws,p(Ω) is a reflexive and separable Banach space. In addition, the space C∞0(Ω) is dense in Ws,p(Ω) such that Ws,p0(Ω)=Ws,p(Ω) (see, e.g., [1,28]).
Lemma 2.1. ([28]) Let 0<s<1 and 1<p<+∞. Then, we have the continuous embeddings as follows:
Ws,p(Ω)↪Lr(Ω)for any r∈[1,p∗s], if sp<N;Ws,p(Ω)↪Lr(Ω)for every r∈[1,∞),if sp=N;Ws,p(Ω)↪C0,νb(Ω)for all ν<s−N/p, if sp>N. |
Particularly, the embedding Ws,p(Ω)↪↪Lr(Ω) is compact for any r∈[1,p∗s).
Let us define the fractional Sobolev space Ws,pK(RN) as follows:
Ws,pK(RN):={ψ∈Lp(RN):∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy<+∞}, |
where K:RN×RN∖{(0,0)}→(0,+∞) is a kernel function with the properties (K1)–(K3). By the condition (K1), the function
(z,y)↦K1p(z,y)(ψ(z)−ψ(y))∈Lp(RN) |
for any ψ∈C∞0(RN). We consider the problem (P) in the closed linear subspace defined by
X:={ψ∈Ws,pK(RN):ψ(z)=0 a.e. in RN∖Ω} |
with respect to the norm
||ψ||X:=(||ψ||pLp(Ω)+[ψ]ps,p)1p, |
where
[ψ]ps,p:=∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy. |
In what follows, let 0<s<1 and 1<p<+∞ with ps<N and let the kernel function K:RN×RN∖{(0,0)}→(0,∞) ensure the assumptions (K1)–(K3).
Lemma 2.2. ([35]) If ψ∈X, then ψ∈Ws,p(Ω). Moreover,
||ψ||Ws,p(Ω)≤max{1,γ−1p0}||ψ||X, |
where γ0 is given in (K2).
From Lemmas 2.1 and 2.2, we can obtain the following consequence instantly.
Lemma 2.3. ([35]) For 1≤r≤p∗s and for any ψ∈X, there exists a constant C0=C0(s,N,p)>0 such that
||ψ||pLr(Ω)≤C0∫RN∫RN|ψ(z)−ψ(y)|p|z−y|N+psdzdy≤C0γ0∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy, |
where γ0 is given in (K2). Consequently, the embedding X↪Lr(Ω) is continuous for any r∈[1,p∗s]. In addition, the embedding
X↪↪Lr(Ω) |
is compact for r∈(1,p∗s).
In this section, we introduce the variational setting corresponding to the problem (P). In addition, we present some useful auxiliary consequences and Ricceri's variational principle before delving into our main result.
Definition 3.1. We say that ψ∈X is called a weak solution of (P) if
M([ψ]ps,p)∫RN∫RNK(z,y)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))(φ(z)−φ(y))dzdy=∫Ωg(z,ψ)φ(y)dy |
for any φ∈X.
Let us define the functional A:X→R as
A(ψ):=∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdz. | (3.1) |
Then, it is immediate to obtain that the functional A:X→R belongs to a class of C1(X,R), and its Fréchet derivative is
⟨A′(ψ),φ⟩=p∫RN∫RNK(z,y)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))(φ(z)−φ(y))dzdy |
for any ψ,φ∈X; see [32].
Lemma 3.2. The functional A is convex and weakly lower semicontinuous on X.
Proof. It is trivial that A is convex. Let {wn} be a sequence in X satisfying wn⇀w in X as n→∞. Because A is convex and C1-functional on X, we obtain
A(wn)≥⟨A′(wn),wn−w⟩+A(w). |
Then, it is immediate that
lim infn→∞A(wn)≥A(w)+lim infn→∞⟨A′(wn),wn−w⟩≥A(w). |
Therefore, the conclusion holds.
Meanwhile, g:Ω×R→R is assumed to verify the following conditions:
(G1) g satisfies a Carathéodory condition;
(G2) 0≤g(⋅,ξ)∈L∞(Ω) for every ξ≥0, and there is a constant ρ1>0 such that
g(z,ξ)≤ρ1(1+|ξ|p−1) |
for all ξ≥0 and for almost everywhere z∈Ω;
(G3) The function ξ↦g(z,ξ)ξp−1 is strictly decreasing in (0,+∞) for almost all z∈Ω;
(G4) limξ→+∞g(z,ξ)ξp−1=0 and limξ→0+g(z,ξ)ξp−1=+∞, uniformly in z∈Ω.
Under hypothesis (G1), let us define the functional B0:X→R by
B0(ψ):=∫ΩG(z,ψ(z))dz |
for any ψ∈X, where G(z,ξ)=∫ξ0g(z,t)dt. Thus, it is immediate to prove that B0∈C1(X,R), and its Fréchet derivative is
⟨B′0(ψ),w⟩=∫Ωg(z,ψ)wdz |
for any ψ,w∈X. Next, we define the functional J:X→R by
J(ψ)=1pA(ψ)−λB0(ψ). |
Then, the functional J belongs to C1(X,R), and its Fréchet derivative is
⟨J′(ψ),φ⟩=1p⟨A′(ψ),φ⟩−λ⟨B′0(ψ),φ⟩for any ψ,φ∈X. |
The following is a discrete version of the renowned Picone inequality; see [3, Proposition 4.2] and [12, Lemma 2.6] for a proof.
Lemma 3.3. (Discrete Picone inequality). Let p∈(1,+∞) and let a,b,c,d∈[0,+∞), with a,b>0. Then,
ϕp(a−b)[cpap−1−dpbp−1]≤|c−d|p, | (3.2) |
where ϕp(ξ)=|ξ|p−2ξ for ξ∈R. Moreover, if the equality holds in (3.2), then
ab=cd. |
We prove a practical lemma that will be very usable hereinafter. For any ε>0 and ψj∈X, define the truncation
ψj,ε:=min{ψj,ε−1}. | (3.3) |
Lemma 3.4. Let ψ1,ψ2∈X with ψ1,ψ2≥0 and set
w:=ψp2,ε(ε+ψ1)p−1−ψ1,ε, |
where ψ1,ε,ψ2,ε are as in (3.3). Then, we derive w∈X.
Proof. Let ε>0 be fixed. Because ξ↦min{|ξ|,ε−1} is 1-Lipschitz function, we assert
|ψj,ε(y)−ψj,ε(z)|≤|ψj(y)−ψj(z)|forj=1,2, | (3.4) |
which implies that ψj,ε∈X. On account of the Lagrange theorem, we deduce that
|ar−br|≤r|a−b|max{ar−1,br−1} | (3.5) |
for every r≥0 and for any a,b≥0. Because εp−1≤(ε+ψ1,ε)p−1 and ψ2,ε≤1ε, by considering (3.4) and (3.5), we have
|ψp2,ε(z)(ε+ψ1(z))p−1−ψp2,ε(y)(ε+ψ1(y))p−1|=|ψp2,ε(z)−ψp2,ε(y)(ε+ψ1(z)n)p−1+ψp2,ε(y)(ε+ψ1(y))p−1−(ε+ψ1(z))p−1(ε+ψ1(z))p−1(ε+ψ1(y))p−1|≤pε2p−2|ψ2,ε(z)−ψ2,ε(y)|+1εp|(ε+ψ1(y))p−1−(ε+ψ1(z))p−1(ε+ψ1(z))p−1(ε+ψ1(y))p−1|≤pε2p−2|ψ2,ε(z)−ψ2,ε(y)|+p−1εpmax{(ε+ψ1(z))p−2,(ε+ψ1(y)n)p−2}|ψ1(z)−ψ1(y)|(ε+ψ1(z))p−1(ε+ψ1(y))p−1≤pε2p−2|ψ2(z)−ψ2(y)|+p−1ε2p|ψ1(z)−ψ1(y)| |
for every p>1. Hence, the Gagliardo seminorm of w is finite. In addition, one has
ψp2,ε(ε+ψ1)p−1=ψp−12,ε(ε+ψ1)p−1ψ2,ε≤1ε2p−2ψ2; |
thus,
∫Ω|w|pdz≤2p−1(∫Ω|ψp2,ε(ε+ψ1)p−1|pdz+∫Ω|ψ1,ε|pdz)≤C(ε,p)(||ψ2||Lp(Ω)+||ψ1||Lp(Ω))<+∞, |
where C(ε,p)>0. As a result, we arrive that w∈X.
Definition 3.5. Let X be a topological space. A function h:X→R is inf-compact if the set h−1((−∞,ξ]) is compact for each ξ∈R.
Now, we present the uniqueness result of a nontrivial positive solution for the nonlocal fractional p-Laplacian problem of a Kirchhoff-type. To this end, we employ the abstract global minimum principle introduced by B. Ricceri [33], which plays a crucial role in obtaining our main result.
Theorem 3.6. Let X be a topological space, and let A:X→R, with A−1(0)≠∅ and B:X→R being two functions such that, for each γ>0, the function γA−B is lower semicontinuous, inf-compact, and has a unique global minimum. Moreover, assume that B has no global maxima in X. Further, let J⊆(0,+∞) be an open interval and M:J→R be an increasing function with M(J)=(0,+∞). There exists a unique ˜u∈X such that A(˜u)∈J and
M(A(˜u))A(˜u)−B(˜u)=infu∈X(M(A(˜u))A(u)−B(u)). |
If each assumption of Theorem 3.6 is satisfied, we derive our main result. The fundamental idea of the proof of the uniqueness of positive solutions to problem (P) follows from the paper [4,27]; see also [23].
Theorem 3.7. Assume that an open interval J⊆(0,+∞) exists such that M(J)=(0,+∞) and the restriction of M to J is increasing. Let g:Ω×[0,+∞)→(0,+∞) be a function satisfying conditions (G1)–(G4) and g(z,0)=0 for almost every z∈Ω. Then, problem (P) has a unique positive weak solution ˜w, which is the unique global minimum in X of the functional
ψ↦1pM([˜w]ps,p)∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy−∫Ω(∫ψ+(z)0g(z,t)dt)dz, |
where ψ+:=max{ψ,0}
Proof. First, extend g to R, putting g(z,ξ)=0 for all ξ<0. To utilize Theorem 3.6, consider A given in (3.1) and define B by
B(ψ):=p∫ΩG(z,ψ+(z))dz |
for any ψ∈X. The functional B belongs to a class of C1(X,R) with derivatives given by
⟨B′(ψ),w⟩=p∫Ωg(z,ψ)w(z)dz |
for any ψ,w∈X. Moreover, owing to the fact that g has subcritical growth, the functional B is sequentially weakly continuous on X. Fix η>0. Then, Lemma 3.2 implies the sequentially weakly lower semicontinuity of functional ηA−B on X. Choose
ϵ∈(0,η(C0+γ0)2C0), |
where γ0 and C0 are given in Lemma 2.3. Because limξ→+∞G(z,ξ)ξp=0, there exists a positive real number Cε>0 satisfying
G(z,ξ)≤εp|ξ|p+Cεp | (3.6) |
for almost everywhere z∈Ω and for any ξ∈R. Hence, we obtain
B(ψ)≤ε∫Ω|ψ(z)|pdz+Cεmeas(Ω), |
where meas(Ω) means the Lebesgue measure of Ω on RN. Using this, Lemma 2.3, (3.6) and the definition of the X-norm, we derive that
ηA(ψ)−B(ψ)≥η∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy−ε∫Ω|ψ(z)|pdz−Cεmeas(Ω)≥η(12+γ02C0)||ψ||pX−ε∫Ω|ψ(z)|pdz−Cεmeas(Ω)≥(η(C0+γ0)2C0−ϵ)||ψ||pX−Cεmeas(Ω) |
for any ψ∈X. Thus, owing to the choice of ε, we infer
lim||u||→+∞(ηA(u)−B(u))=+∞. |
This, together with the reflexivity of X and the Eberlein–Smulyan theorem, yields that the sequentially weakly lower semicontinuous functional ηA−B is weakly inf-compact. Now, we claim that it has a unique global minimum in X. As we know, its critical points are exactly the weak solutions to the problem
{Lspψ(z)=1ηg(z,ψ)in Ω,ψ=0on ∂Ω, | (3.7) |
where ψ∈X is said to be a weak solution of problem (3.7) if
∫RN∫RNK(z,y)|ψ(z)−ψ(y)|p−2(ψ(z)−ψ(y))(ϕ(z)−ϕ(y))dzdy=1η∫Ωg(z,ψ)ϕdz | (3.8) |
for any ϕ∈X.
Let us define the energy functional J:X→R as
J(ψ):=1p∫RN∫RNK(z−y)|ψ(z)−ψ(y)|pdzdy−1η∫RNG(z,ψ)dz,ψ∈X, |
and let the modified energy functional ˜J:X→R be defined by
˜J(ψ):=1p∫RN∫RNK(z−y)|ψ(z)−ψ(y)|pdzdy−1η∫RNG+(z,ψ)dz,ψ∈X, |
where
G+(z,τ):=∫τ0g+(z,ξ)dξ and g+(z,τ):={g(z,τ), τ≥0,0, τ<0 |
for any τ∈R and for almost everywhere z∈RN. In compliance with Lemma 3.2 and the argument above, the functional ˜J is also coercive and sequentially weakly lower semicontinuous on X. From this, there is an element ψ0∈X satisfying
˜J(ψ0)=inf{˜J(ψ):ψ∈X}. |
Now, we show that it is possible to assume that ψ0≥0. To this end, we assume that ψ0 is sign-changing. Taking Lemma 3.4 into account, we know ψ+0∈X and thus ˜J(ψ0)≤˜J(ψ+0). Because ˜J(ψ)=J(ψ) when ψ(z)≥0 for almost everywhere z∈Ω, we assert
˜J(ψ+0)=J(ψ+0)=1p∫RN∫RNK(z,y)|ψ+0(z)−ψ+0(y)|pdzdy−1η∫ΩG(z,ψ+0)dz≤1p∫RN∫RNK(z,y)|ψ0(z)−ψ0(y)|pdzdy−1η∫ΩG(z,ψ+0)dz=˜J(ψ0). |
Therefore, ψ+0 is a nonnegative solution to problem (3.7). For simplicity, let us write directly ψ0 instead of ψ+0. Let us claim ψ0>0. As ψ0(z)≥0 for almost everywhere z∈RN, we know that either ψ0(z)>0 or ψ0(z)=0 for almost everywhere z∈RN. Indeed, let us assume that ψ0≢0 in Ω. Then it is enough to prove that ψ0≢0 in all connected components of Ω. Assume to the contrary that there exists a connected component Λ of Ω such that ψ0(z)=0 for almost everywhere z∈Λ. Let us take any nonnegative function ω∈C∞0(Λ) as a test function in (3.8). Then, since g is a nonnegative function and ψ0 is a nonnegative solution of (3.7), we have
0=∫RN∫RNK(z,y)|ψ0(z)−ψ0(y)|p−2(ψ0(z)−ψ0(y))(ω(z)−ω(y))dzdy−1η∫Ωg(z,ψ0)ω(z)dz≤∫RN∫RNK(z,y)|ψ0(z)−ψ0(y)|p−2(ψ0(z)−ψ0(y))(ω(z)−ω(y))dzdy=2∫Λ∫ΛcK(z,y)|ψ0(z)−ψ0(y)|p−2(ψ0(z)−ψ0(y))(ω(z)−ω(y))dzdy=−2∫Λ∫ΛcK(z,y)(ψ0(z))p−1ω(y)dzdy. |
From this, we infer that ψ0(z)=0 for almost everywhere z∈Λc, that is ψ0(z)=0 for almost everywhere z∈RN. This yields a contradiction to the fact that ψ0(z)≠0 for almost everywhere z∈Ω.
Therefore, to show ψ0>0, it suffices to prove that ˜J(ψ0)<0. Now, with consideration for Lemma 2.1 in [13], let us fix any nonnegative function ϱ∈X, with ϱ=0 on ∂Ω, such that
η1∫Ω|ϱ(z)|pdz=∫RN∫RNK(z−y)|ϱ(z)−ϱ(y)|pdzdy, |
where η1 is a positive eigenvalue that can be characterized as
η1=min{ϱ∈X:||ϱ||Lp(Ω)=1}∫RN∫RNK(z−y)|ϱ(z)−ϱ(y)|pdzdy. |
In light of Theorem 3.2 in [13], we assert that ϱ∈L∞(RN). Let α0∈L∞(Ω) with α0>0 and let κ0∈(0,||α0||L∞(Ω)) be fixed. Then, the set
Ωκ0:={z∈Ω:α0(z)≥κ0} |
has a positive measure. Furthermore, fix K>0 so that
K>ηη1∫Ω|ϱ(z)|pdzκ0∫Ωκ0|ϱ(z)|pdz. |
From the first condition in (G4), we can choose a constant ξ0>0 satisfying
G(z,ξ)ξp≥α0(z)Kp |
for any ξ∈(0,ξ0], and for almost everywhere z∈Ω. Then, for small enough ε>0, we get
1η∫ΩG(z,εϱ)εpdz≥Kpη∫Ωα0(z)|ϱ(z)|pdz≥Kκ0pη∫Ωκ0|ϱ(z)|pdz>η1p∫Ω|ϱ(z)|pdz=1p∫RN∫RNK(z−y)|ϱ(z)−ϱ(y)|pdzdy=1p[ϱ]ps,p. | (3.9) |
Hence, using (3.9), we conclude that
[ϱ]ps,p−pη∫RNG(z,εϱ)εpdz<0 |
for any ε>0 sufficiently small, which implies J(εϱ)<0, as required. In consequence, problem (3.7) has a positive solution for any η>0. In particular, this also implies that 0 is not a global minimum of ηA−B.
Next, we prove that problem (3.7) admits at most one positive solution for any η>0. Let ψ1 and ψ2 be two weak positive solutions of (3.7). For any ε>0, we define the truncations ψj,ε as in (3.3) for j=1,2. Let us define the functions
ω1,ε:=ψp2,ε(ε+ψ1)p−1−ψ1,ε |
and
ω2,ε:=ψp1,ε(ε+ψ2)p−1−ψ2,ε. |
In accordance with Lemma 3.4, we assert that ωj,ε∈X for j=1,2. Now, set
ϕp(ξ):=|ξ|p−2ξ. |
Considering the weak formulation (3.8) of ψj, by choosing ϕ=ωj,ε for j=1,2, one has
∫RN∫RNK(z,y)ϕp(ψ1(z)−ψ1(y))(ω1,ε(z)−ω1,ε(y))dzdy=1η∫Ωg(z,ψ1)ω1,ε(z)dz | (3.10) |
and
∫RN∫RNK(z,y)ϕp(ψ2(z)−ψ2(y))(ω2,ε(z)−ω2,ε(y))dzdy=1η∫Ωg(z,ψ2)ω2,ε(z)dz. | (3.11) |
Adding the above two equations (3.10) and (3.11) and utilizing the fact that
ϕp(ψj(z)−ψj(y))=ϕp((ε+ψjn)(z)−(ε+ψj)(y))forj=1,2, |
we obtain
∫RN∫RNK(z,y)ϕp((ε+ψ1)(z)−(ε+ψ1)(y))(ψp2,ε(ε+ψ1)p−1(z)−ψp2,ε(ε+ψ1)p−1(y))dzdy−∫RN∫RNK(z,y)ϕp(ψ1(z)−ψ1(y))(ψ1,ε(z)−ψ1,ε(y))dzdy+∫RN∫RNK(z,y)ϕp((ε+ψ2)(z)−(ε+ψ2)(y))(ψp1,ε(ε+ψ2)p−1(z)−ψp1,ε(ε+ψ2)p−1(y))dzdy−∫RN∫RNK(z,y)ϕp(ψ2(z)−ψ2(y))(ψ2,ε(z)−ψ2,ε(y))dzdy=1η(∫Ω[g(z,ψ1)(ψp2,ε(ε+ψ1)p−1−ψ1,ε)+g(z,ψ2)(ψp1,ε(ε+ψ2)p−1−ψ2,ε)]dz). | (3.12) |
Now, according to the fact that ξ→min{|ξ|,ε−1} is 1-Lipschitz function and the discrete Picone inequality in Lemma 3.3, we derive
ϕp((ε+ψ1)(z)−(ε+ψ1)(y))(ψp2,ε(ε+ψ1)p−1(z)−ψp2,ε(ε+ψ1)p−1(y))≤|ψ2(z)−ψ2(y)|p |
and
ϕp((ε+ψ2)(z)−(ε+ψ2)(y))(ψp1,ε(ε+ψ2)p−1(z)−ψp1,ε(ε+ψ2)p−1(y))≤|ψ1(z)−ψ1(y)|p. |
Because ψj,ε→ψj as ε→0 for j=1,2, by taking to the limit in (3.12) and applying the Fatou Lemma in the first and third terms as well as using the Lebesgue dominated convergence theorem for all the other terms, one has
∫RN∫RNK(z,y)ϕp(ψ1(z)−ψ1(y))(ψp2ψp−11(z)−ψp2ψp−11(y))dzdy−∫RN∫RNK(z,y)|ψ1(z)−ψ1(y)|pdzdy+∫RN∫RNK(z,y)ϕp(ψ2(z)−ψ2(y))(ψp1ψp−12(z)−ψp1ψp−12(y))dzdy−∫RN∫RNK(z,y)|ψ2(z)−ψ2(y)|pdzdy≥1η(∫Ωg(z,ψ1)(ψp2ψp−11−ψ1)+g(z,ψ2)(ψp1ψp−12−ψ2)dz)=−1η∫Ω(g(z,ψ1)ψp−11−g(z,ψ2)ψp−12)(ψp1−ψp2)dz. | (3.13) |
Using Lemma 3.3 on the left-hand side of (3.13), we obtain
∫Ω(g(z,ψ1)ψp−11−g(z,ψ2)ψp−12)(ψp1−ψp2)dz≥0. |
Hence, because the function ξ↦g(z,ξ)ξp−1 is decreasing in (0,+∞), we obtain that ψ1=ψ2. Therefore, we ensure that problem (3.7) possesses at most one positive solution. As a result, we derive that ηA−B admits a unique global minimum in X, since otherwise, in consideration of [31, Corollary 1], it would have at least three critical points. Because 0 is not a global minimum for ηA−B, the global minimum of this functional is consistent with its only nonzero critical point.
Finally, let us show that B has no global maxima. Assume to the contrary that ˆψ∈X is a global maximum of B. Obviously, we know B(ˆψ)>0. Thus, since g is nonnegative, it follows from (G3) that the set
Γ:={z∈Ω:g(z,ˆψ(z))>0} |
has a positive measure. Let us fix a closed set P⊂Γ of positive measures. Let ϱ∈X be such that ϱ≥0 and ϱ(z)=1 for almost everywhere z∈P. Then, we obtain
∫Ωg(z,ˆψ(z))ϱ(z)dz≥∫Pg(z,ˆψ(z))dz>0, |
and so B′(ˆψ)≠0, which is a contradiction.
Hence, each assumption of Theorem 3.6 is satisfied. Therefore, there exists a unique ˜w∈X, with [˜w]ps,p∈J, such that
M([˜w]ps,p)∫RN∫RNK(z,y)|˜w(z)−˜w(y)|pdzdy−p∫ΩG(z,˜w+(z))dz=infψ∈X{M([˜w]ps,p)∫RN∫RNK(z,y)|ψ(z)−ψ(y)|pdzdy−p∫ΩG(z,ψ+(z))dz}. |
Consequently, from what seen above, problem (P) possesses the unique positive weak solution ˜w.
This paper is devoted to deriving the existence and uniqueness of positive solutions to fractional p-Laplacian problems involving discontinuous Kirchhoff-type functions. The main tools for obtaining these results are the uniqueness results of the Brézis–Oswald-type based on [5] and the abstract global minimum principle in [33]. Particularly, based on previous studies [4,27], we obtain the existence of at most one positive weak solution to the fractional p-Laplacian equations of the Brézis–Oswald type by employing the discrete Picone inequality in [3,12]. But, our condition (G4) can be considered a special case of that of [2,27] since the nonlinear term g satisfies the following assumption:
β0(z)=limξ→0+g(z,ξ)ξp−1andβ∞(z)=limξ→+∞g(z,ξ)ξp−1. |
Let us define Λ1(Lsp−β0) and Λ1(Lsp−β∞) as
Λ1(Lsp−β0)=infψ∈X{[ψ]ps,p−∫Ωβ0(z)|ψ(z)|pdz:||ψ||Lp(Ω)=1} |
and
Λ1(Lsp−β0)=infψ∈X{[ψ]ps,p−∫Ωβ∞(z)|ψ(z)|pdz:||ψ||Lp(Ω)=1}. |
If Λ1(Lsp−β0)<0<Λ1(Lsp−β0) in place of (G4) holds, then analogous arguments such as those in [27] implies that problem (3.7) admits at most one positive solution for any η>0. Consequently, explicit modifications of the proof of Theorem 3.7 yield the same consequences concerning problem (P) when Λ1(Lsp−β0)<0<Λ1(Lsp−β0) in place of (G4) is supposed.
Additionally, a new research direction is the investigation of the Brézis–Oswald type fractional p-Laplacian problems involving Hardy potentials:
{M([ψ]ps,p)Lspψ(z)=μ|ψ|p−2ψ|z|p+λg(z,ψ)inΩ,ψ>0inΩ,ψ=0onRN∖Ω, | (4.1) |
where p∈(1,p∗s), μ∈(−∞,μ∗) for a positive constant μ∗. When μ≠0, the classical variational approach is not applicable because of the appearance of the term μ|ψ|p−2ψ|z|−p. The reason is that the Hardy inequality ensures that only the embedding Ws,p0(Ω)↪Lp(Ω,|z|−p) is continuous but not compact. Hence, the situation with μ≠0 would be much more delicate than the situation in the present paper because of the lack of compactness. To the best of our belief, there are no results concerning the localization, existence, and uniqueness of positive solutions to problem (4.1).
All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.
All the authors declare that they have not used artificial intelligence (AI) tools in the creation of this article.
This work was supported by the Incheon National University Research Grant in 2022.
Dr. Yun-Ho Kim is the Guest Editor of special issue “Recent developments in nonlinear equations of Kirchhoff type” for AIMS Mathematics.
Dr. Yun-Ho Kim was not involved in the editorial review and the decision to publish this article.
All the authors declare that there are no conflicts of interest regarding the publication of this paper.
[1] |
F. John, L. Nirenberg, On functions of bounded mean oscillation, Commun. Pure Appl. Math., 14 (1961), 415–426. https://doi.org/10.1002/cpa.3160140317 doi: 10.1002/cpa.3160140317
![]() |
[2] |
R. R. Coifman, R. Rochberg, G. Weiss, Factorization theorems for Hardy spaces in several variables, Ann. of Math., 103 (1976), 611–635. https://doi.org/10.2307/1970954 doi: 10.2307/1970954
![]() |
[3] |
A. Uchiyama, On the compactness of operators of Hankel type, Tôhoku Math. J., 30 (1978), 163–171. https://doi.org/10.2748/tmj/1178230105 doi: 10.2748/tmj/1178230105
![]() |
[4] | A. Karlovich, A. Lerner, Commutators of singular integrals on generalized Lp spaces with variable exponent, Publ. Mat., 49 (2005), 111–125. |
[5] | G. Di Fazio, M. A. Ragusa, Commutators and Morrey spaces, Boll. Unione Mat. Ital. A, 7 (1991), 323–332. |
[6] |
Y. Chen, Y. Ding, X. Wang, Compactness of commutators for singular integrals on Morrey spaces, Can. J. Math., 64 (2012), 257–281. https://doi.org/10.4153/CJM-2011-043-1 doi: 10.4153/CJM-2011-043-1
![]() |
[7] |
J. Tao, D. Yang, W. Yuan, Y. Zhang, Compactness characterizations of commutators on ball Banach function spaces, Potential Anal., 58 (2023), 645–679. https://doi.org/10.1007/s11118-021-09953-w doi: 10.1007/s11118-021-09953-w
![]() |
[8] |
Y. Sawano, K. P. Ho, D. Yang, S. Yang, Hardy spaces for ball quasi-Banach function spaces, Diss. Math., 525 (2017), 1–102. https://doi.org/10.4064/dm750-9-2016 doi: 10.4064/dm750-9-2016
![]() |
[9] | C. Bennett, R. Sharpley, Interpolation of operators, Academic Press, 1988. |
[10] |
H. Yang, J. Zhou, Commutators of parameter Marcinkiwicz integral with functions in Campanato spaces on Orlicz-Morrey spaces, Filomat., 37 (2023), 7255–7273. https://doi.org/10.2298/FIL2321255Y doi: 10.2298/FIL2321255Y
![]() |
[11] |
K. Ho, Fractional integral operators with homogeneous kernels on Morrey spaces with variable exponents, J. Math. Soc. Japan., 69 (2017), 1059–1077. https://doi.org/10.2969/jmsj/06931059 doi: 10.2969/jmsj/06931059
![]() |
[12] |
M. A. Ragusa, Commutators of fractional integral operators on vanishing-Morrey spaces, J. Glob. Optim., 40 (2008), 361–368. https://doi.org/10.1007/s10898-007-9176-7 doi: 10.1007/s10898-007-9176-7
![]() |
[13] |
A. Scapellato, Riesz potential, Marcinkiewicz integral and their commutators on mixed Morrey spaces, Filomat., 34 (2020), 931–944. https://doi.org/10.2298/FIL2003931S doi: 10.2298/FIL2003931S
![]() |
[14] |
H. Yang, J. Zhou, Some characterizations of Lipschitz spaces via commutators of the Hardy-Littlewood maximal operator on slice spaces, Proc. Ro. Acad. Ser. A., 24 (2023), 223–230. https://doi.org/10.59277/PRA-SER.A.24.3.03 doi: 10.59277/PRA-SER.A.24.3.03
![]() |
[15] |
J. Tan, J. Zhao, Rough fractional integrals and its commutators on variable Morrey spaces, C. R. Math., 353 (2015), 1117–1122. https://doi.org/10.1016/j.crma.2015.09.024 doi: 10.1016/j.crma.2015.09.024
![]() |
[16] |
J. Tan, Z. Liu, J. Zhao, On some multilinear commutators in variable Lebesgue spaces, J. Math. Inequal., 11 (2017), 715–734. https://doi.org/10.7153/jmi-2017-11-57 doi: 10.7153/jmi-2017-11-57
![]() |
[17] |
M. A. Ragusa, Local Hölder regularity for solutions of elliptic systems, Duke Math. J., 113 (2002), 385–397. https://doi.org/10.1215/S0012-7094-02-11327-1 doi: 10.1215/S0012-7094-02-11327-1
![]() |
[18] |
Y. Chen, Q. Deng, Y. Ding, Commutators with fractional differentiation for second-order elliptic operators on Rn, Commun. Contemp. Math., 22 (2020), 1950010. https://doi.org/10.1142/S021919971950010X doi: 10.1142/S021919971950010X
![]() |
[19] |
Y. Chen, Y. Ding, G. Hong, Commutators with fractional differentiation and new characterizations of BMO-Sobolev spaces, Anal. PDE, 9 (2016), 1497–1522. https://doi.org/10.2140/apde.2016.9.1497 doi: 10.2140/apde.2016.9.1497
![]() |
[20] |
C. Pérez, G. Pradolini, R. H. Torres, R. Trujillo-González, End-points estimates for iterated commutators of multilinear singular integrals, Bull. London Math. Soc., 46 (2014), 26–42. https://doi.org/10.1112/blms/bdt065 doi: 10.1112/blms/bdt065
![]() |
[21] | A. Bényi, R. H. Torres, Compact bilinear operators and commutators, Proc. Amer. Math. Soc., 141 (2013), 3609–3621. |
[22] | A. Bényi, W. Damián, K. Moen, R. H. Torres, Compact bilinear commutators: the weighted case, Michigan Math. J., 64 (2015), 39–51. |
[23] |
D. Wang, J. Zhou, Z. Teng, Characterization of CMO via compactness of the commutators of bilinear fractional integral operators, Anal. Math. Phys., 9 (2019), 1669–1688. https://doi.org/10.1007/s13324-018-0264-2 doi: 10.1007/s13324-018-0264-2
![]() |
[24] |
T. Hytönen, S. Lappas, Extrapolation of compactness on weighted spaces: Bilinear operators, Indagat. Math., 33 (2022), 397–420. https://doi.org/10.1016/j.indag.2021.09.007 doi: 10.1016/j.indag.2021.09.007
![]() |
[25] |
W. Guo, H. Wu, D. Yang, A revised on the compactness of commutators, Can. J. Math., 73 (2021), 1667–1697. https://doi.org/10.4153/S0008414X20000644 doi: 10.4153/S0008414X20000644
![]() |
[26] | S. Lu, Y. Ding, D. Yan, Singular integrals and related topics, World Scientific, 2007. |
[27] |
A. K. Lerner, S. Ombrosi, I. P. Rivera-Ríos, Commutators of singular integrals revisited, Bull. London Math. Soc., 51 (2019), 107–119. https://doi.org/10.1112/blms.12216 doi: 10.1112/blms.12216
![]() |
[28] |
M. Izuki, T. Noi, Y. Sawano, The John-Nirenberg inequality in ball Banach function spaces and application to characterization of BMO, J. Inequal. Appl., 2019 (2019), 268. https://doi.org/10.1186/s13660-019-2220-6 doi: 10.1186/s13660-019-2220-6
![]() |
[29] |
Y. Zhang, S. Wang, D. Yang, W. Yuan, Weak Hardy-type spaces associated with ball quasi-Banach function spaces Ⅰ: Decompositions with applications to boundedness of Calderón-Zygmund operators, Sci. China Math., 64 (2021), 2007–2064. https://doi.org/10.1007/s11425-019-1645-1 doi: 10.1007/s11425-019-1645-1
![]() |
[30] |
A. Clop, V. Cruz, Weighted estimates for Beltrami equations, Ann. Fenn. Math., 38 (2013), 91–113. https://doi.org/10.5186/aasfm.2013.3818 doi: 10.5186/aasfm.2013.3818
![]() |
[31] |
S. G. Krantz, S. Y. Li, Boundedness and compactness of integral operators on spaces of homogeneous type and applications, Ⅱ, J. Math. Anal. Appl., 258 (2001), 642–657. https://doi.org/10.1006/jmaa.2000.7403 doi: 10.1006/jmaa.2000.7403
![]() |
[32] | L. Grafakos, Classical Fourier analysis, New York: Springer, 2014. |
[33] | J. Garcia-Cuerva, J. L. R. de Francia, Weighted norm inequalities and related topics, North-Holland mathematics studies, 1985. |
[34] |
K. Andersen, R. John, Weighted inequalities for vecter-valued maximal functions and singular integrals, Stud. Math., 69 (1981), 19–31. https://doi.org/10.4064/sm-69-1-19-31 doi: 10.4064/sm-69-1-19-31
![]() |
[35] |
B. Muckenhoupt, R. L. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc., 192 (1974), 261–274. https://doi.org/10.1090/S0002-9947-1974-0340523-6 doi: 10.1090/S0002-9947-1974-0340523-6
![]() |
[36] |
Y. Ding, S. Lu, Higher order commutators for a class of rough operators, Ark. Mat., 37 (1999), 33–44. https://doi.org/10.1007/BF02384827 doi: 10.1007/BF02384827
![]() |
[37] |
D. R. Adams, A note on Riesz potentials, Duke Math. J., 42 (1975), 765–778. https://doi.org/10.1215/S0012-7094-75-04265-9 doi: 10.1215/S0012-7094-75-04265-9
![]() |
[38] |
C. Capone, D. Cruz-Uribe, A. SFO Fiorenza, The fractional maximal operator and fractional integrals on variable Lp spaces, Rev. Mat. Iberoamericana, 23 (2007), 743–770. https://doi.org/10.4171/RMI/511 doi: 10.4171/RMI/511
![]() |
[39] |
C. B. Morrey, On the solutions of quasi-linear elliptic partial differential equations, Trans. Amer. Math. Soc., 43 (1938) 126–166. https://doi.org/10.2307/1989904 doi: 10.2307/1989904
![]() |
[40] |
T. Iida, Weighted estimates of higher order commutators generated by BMO-functions and the fractional integral operator on Morrey spaces, J. Inequal. Appl., 2016 (2016), 4. https://doi.org/10.1186/s13660-015-0953-4 doi: 10.1186/s13660-015-0953-4
![]() |
[41] |
H. Wang, Commutators of singular integral operator on herz-type hardy spaces with variable exponent, J. Korean Math. Soc., 54 (2017), 713–732. https://doi.org/10.4134/JKMS.j150771 doi: 10.4134/JKMS.j150771
![]() |
[42] |
M. Izuki, Commutators of fractional integrals on Lebesgue and Herz spaces with variable exponent, Rend. Circ. Mat. Palermo, 59 (2010), 461–472. https://doi.org/10.1007/s12215-010-0034-y doi: 10.1007/s12215-010-0034-y
![]() |
[43] |
T. Nogayama, Mixed Morrey spaces, Positivity, 23 (2019), 961–1000. https://doi.org/10.1007/s11117-019-00646-8 doi: 10.1007/s11117-019-00646-8
![]() |
[44] |
T. Nogayama, Boundedness of commutators of fractional integral operators on mixed Morrey spaces, Integr. Transf. Spec. F., 30 (2019), 790–816. https://doi.org/10.1080/10652469.2019.1619718 doi: 10.1080/10652469.2019.1619718
![]() |
[45] | H. Zhang, J. Zhou, The Köthe dual of mixed Morrey spaces and applications, 2022. https://doi.org/10.48550/arXiv.2204.00518 |
1. | Yun-Ho Kim, In Hyoun Kim, A Brézis–Oswald-Type Result for the Fractional (r, q)-Laplacian Problems and Its Application, 2025, 9, 2504-3110, 412, 10.3390/fractalfract9070412 |