• Title/Summary/Keyword: variable exponents

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Existence Results for an Nonlinear Variable Exponents Anisotropic Elliptic Problems

  • Mokhtar Naceri
    • Kyungpook Mathematical Journal
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    • v.64 no.2
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    • pp.271-286
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    • 2024
  • In this paper, we prove the existence of distributional solutions in the anisotropic Sobolev space $\mathring{W}^{1,\overrightarrow{p}(\cdot)}(\Omega)$ with variable exponents and zero boundary, for a class of variable exponents anisotropic nonlinear elliptic equations having a compound nonlinearity $G(x, u)=\sum_{i=1}^{N}(\left|f\right|+\left|u\right|)^{p_i(x)-1}$ on the right-hand side, such that f is in the variable exponents anisotropic Lebesgue space $L^{\vec{p}({\cdot})}(\Omega)$, where $\vec{p}({\cdot})=(p_1({\cdot}),{\ldots},p_N({\cdot})){\in}(C(\bar{\Omega},]1,+{\infty}[))^N$.

WEAK HERZ-TYPE HARDY SPACES WITH VARIABLE EXPONENTS AND APPLICATIONS

  • Souad Ben Seghier
    • Journal of the Korean Mathematical Society
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    • v.60 no.1
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    • pp.33-69
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    • 2023
  • Let α ∈ (0, ∞), p ∈ (0, ∞) and q(·) : ℝn → [1, ∞) satisfy the globally log-Hölder continuity condition. We introduce the weak Herz-type Hardy spaces with variable exponents via the radial grand maximal operator and to give its maximal characterizations, we establish a version of the boundedness of the Hardy-Littlewood maximal operator M and the Fefferman-Stein vector-valued inequality on the weak Herz spaces with variable exponents. We also obtain the atomic and the molecular decompositions of the weak Herz-type Hardy spaces with variable exponents. As an application of the atomic decomposition we provide various equivalent characterizations of our spaces by means of the Lusin area function, the Littlewood-Paley g-function and the Littlewood-Paley $g^*_{\lambda}$-function.

FINITE TIME BLOW UP OF SOLUTIONS FOR A STRONGLY DAMPED NONLINEAR KLEIN-GORDON EQUATION WITH VARIABLE EXPONENTS

  • Piskin, Erhan
    • Honam Mathematical Journal
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    • v.40 no.4
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    • pp.771-783
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    • 2018
  • This paper, we investigate a strongly damped nonlinear Klein-Gordon equation with nonlinearities of variable exponent type $$u_{tt}-{\Delta}u-{\Delta}u_t+m^2u+{\mid}u_t{\mid}^{p(x)-2}u_t={\mid}u{\mid}^{q(x)-2}u$$ associated with initial and Dirichlet boundary conditions in a bounded domain. We obtain a nonexistence of solutions if variable exponents p (.), q (.) and initial data satisfy some conditions.

Anisotropic Variable Herz Spaces and Applications

  • Aissa Djeriou;Rabah Heraiz
    • Kyungpook Mathematical Journal
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    • v.64 no.2
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    • pp.245-260
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    • 2024
  • In this study, we establish some new characterizations for a class of anisotropic Herz spaces in which all exponents are considered as variables. We also provide a description of these spaces based on bloc decomposition. As an application, we investigate the boundedness of certain sublinear operators within these function spaces.

BOUNDEDNESS OF THE COMMUTATOR OF THE INTRINSIC SQUARE FUNCTION IN VARIABLE EXPONENT SPACES

  • Wang, Liwei
    • Journal of the Korean Mathematical Society
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    • v.55 no.4
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    • pp.939-962
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    • 2018
  • In this paper, we show that the commutator of the intrinsic square function with BMO symbols is bounded on the variable exponent Lebesgue spaces $L^{p({\cdot})}({\mathbb{R}}^n)$ applying a generalization of the classical Rubio de Francia extrapolation. As a consequence we further establish its boundedness on the variable exponent Morrey spaces $\mathcal{M_{p({\cdot}),u}$, Morrey-Herz spaces $M{\dot{K}}^{{\alpha}({\cdot}),{\lambda}}_{q,p({\cdot})}({\mathbb{R}}^n)$ and Herz type Hardy spaces $H{\dot{K}}^{{\alpha}({\cdot}),q}_{p({\cdot})}({\mathbb{R}}^n)$, where the exponents ${\alpha}({\cdot})$ and $p({\cdot})$ are variable. Observe that, even when ${\alpha}({\cdot}){\equiv}{\alpha}$ is constant, the corresponding main results are completely new.

DUALITIES OF VARIABLE ANISOTROPIC HARDY SPACES AND BOUNDEDNESS OF SINGULAR INTEGRAL OPERATORS

  • Wang, Wenhua
    • Bulletin of the Korean Mathematical Society
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    • v.58 no.2
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    • pp.365-384
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    • 2021
  • Let A be an expansive dilation on ℝn, and p(·) : ℝn → (0, ∞) be a variable exponent function satisfying the globally log-Hölder continuous condition. Let Hp(·)A (ℝn) be the variable anisotropic Hardy space defined via the non-tangential grand maximal function. In this paper, the author obtains the boundedness of anisotropic convolutional ��-type Calderón-Zygmund operators from Hp(·)A (ℝn) to Lp(·) (ℝn) or from Hp(·)A (ℝn) to itself. In addition, the author also obtains the duality between Hp(·)A (ℝn) and the anisotropic Campanato spaces with variable exponents.

THE BOUNDEDNESS OF BILINEAR PSEUDODIFFERENTIAL OPERATORS IN TRIEBEL-LIZORKIN AND BESOV SPACES WITH VARIABLE EXPONENTS

  • Yin Liu;Lushun Wang
    • Bulletin of the Korean Mathematical Society
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    • v.61 no.2
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    • pp.529-540
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    • 2024
  • In this paper, using the Fourier transform, inverse Fourier transform and Littlewood-Paley decomposition technique, we prove the boundedness of bilinear pseudodifferential operators with symbols in the bilinear Hörmander class $BS^{m}_{1,1}$ in variable Triebel-Lizorkin spaces and variable Besov spaces.

Design of Dual-Path Decimal Floating-Point Adder (이중 경로 십진 부동소수점 가산기 설계)

  • Lee, Chang-Ho;Kim, Ji-Won;Hwang, In-Guk;Choi, Sang-Bang
    • Journal of the Institute of Electronics and Information Engineers
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    • v.49 no.9
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    • pp.183-195
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    • 2012
  • We propose a variable-latency Decimal Floating Point(DFP) adder which adopts the dual data path scheme. It is to speed addition and subtraction of operand that has identical exponents. The proposed DFP adder makes use of L. K. Wang's operand alignment algorithm, but operates through high speed data-path in guaranteed accuracy range. Synthesis results show that the area of the proposed DFP adder is increased by 8.26% compared to the L. K. Wang's DFP adder, though critical path delay is reduced by 10.54%. It also operates at 13.65% reduced path than critical path in case of an operation which has two DFP operands with identical exponents. We prove that the proposed DFP adder shows higher efficiency than L. K. Wang's DFP adder when the ratio of identical exponents is larger than 2%.

INFINITELY MANY SMALL SOLUTIONS FOR THE p(x)-LAPLACIAN OPERATOR WITH CRITICAL GROWTH

  • Zhou, Chenxing;Liang, Sihua
    • Journal of applied mathematics & informatics
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    • v.32 no.1_2
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    • pp.137-152
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    • 2014
  • In this paper, we prove, in the spirit of [3, 12, 20, 22, 23], the existence of infinitely many small solutions to the following quasilinear elliptic equation $-{\Delta}_{p(x)}u+{\mid}u{\mid}^{p(x)-2}u={\mid}u{\mid}^{q(x)-2}u+{\lambda}f(x,u)$ in a smooth bounded domain ${\Omega}$ of ${\mathbb{R}}^N$. We also assume that $\{q(x)=p^*(x)\}{\neq}{\emptyset}$, where $p^*(x)$ = Np(x)/(N - p(x)) is the critical Sobolev exponent for variable exponents. The proof is based on a new version of the symmetric mountainpass lemma due to Kajikiya [22], and property of these solutions are also obtained.