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Let B=B1(0) be the unit ball in Rn and r=|x|. We study the poly-harmonic Dirichlet problem
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In this paper, we study two types of weighted Hardy–Littlewood–Sobolev (HLS) inequalities, also known as Stein–Weiss inequalities, on the Heisenberg group. More precisely, we prove the |u| weighted HLS inequality in Theorem 1.1 and the |z| weighted HLS inequality in Theorem 1.5 (where we have denoted u=(z,t) as points on the Heisenberg group). Then we provide regularity estimates of positive solutions to integral systems which are Euler–Lagrange equations of the possible extremals to the Stein–Weiss inequalities. Asymptotic behavior is also established for integral systems associated to the |u| weighted HLS inequalities around the origin. By these a priori estimates, we describe asymptotically the possible optimizers for sharp versions of these inequalities. 相似文献
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Hua Saixiang Zhu Huichao Wang Xiao Wu Mingyuan Wu Qingyun Liu Jiuyi Yang Jianjun Zhang Jianan 《Cellulose (London, England)》2022,29(13):7465-7475
Cellulose - Ultraviolet (UV) protective cotton fabric is a convenient and reliable way to protect human body from sunlight. Herein, a polymerizable UV-absorber,... 相似文献
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In order to investigate bacterium-substratum interactions, understanding of bacterial mass transport is necessary. Comparisons of experimentally observed initial deposition rates with mass transport rates in parallel-plate-flow-chambers (PPFC) predicted by convective-diffusion yielded deposition efficiencies above unity, despite electrostatic repulsion. It is hypothesized that sedimentation is the major mass transport mechanism in a PPFC. The contribution of sedimentation to the mass transport in a PPFC was experimentally investigated by introducing a novel microscopy-based method. First, height-dependent bacterial concentrations were measured at different times and flow rates and used to calculate bacterial sedimentation velocities. For Staphylococcus aureus ATCC 12600, a sedimentation velocity of 240 μm h(-1) was obtained. Therewith, sedimentation appeared as the predominant contribution to mass transport in a PPFC. Also in the current study, deposition efficiencies of S. aureus ATCC 12600 with respect to the Smoluchowski-Levich solution of the convective-diffusion equation were four-to-five fold higher than unity. However, calculation of deposition efficiencies with respect to sedimentation were below unity and decreased from 0.78 to 0.36 when flow rates increased from 0.017 to 0.33 cm(3) s(-1). The proposed analysis of bacterial mass transport processes is simple, does not require additional equipment and yields a more reasonable interpretation of bacterial deposition in a PPFC. 相似文献
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Guozhen Lu Jiuyi Zhu 《Calculus of Variations and Partial Differential Equations》2011,42(3-4):563-577
Let ?? be a real number satisfying 0?<????<?n, ${0\leq t<\alpha, \alpha{^\ast}(t)=\frac{2(n-t)}{n-\alpha}}$ . We consider the integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{u^{{\alpha{^\ast}(t)}-1}(y)}{|y|^t|x-y|^{n-\alpha}}\,dy,\quad\quad\quad\quad\quad\quad\quad(1)$$ which is closely related to the Hardy?CSobolev inequality. In this paper, we prove that every positive solution u(x) is radially symmetric and strictly decreasing about the origin by the method of moving plane in integral forms. Moreover, we obtain the regularity of solutions to the following integral equation $$u(x)=\int\limits_{{\mathbb{R}^n}}\frac{|u(y)|^{p}u(y)}{|y|^t|x-y|^{n-\alpha}}\, dy\quad\quad\quad\quad\quad\quad\quad(2)$$ that corresponds to a large class of PDEs by regularity lifting method. 相似文献
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Guozhen Lu Peiyong Wang Jiuyi Zhu 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2012
Liouville-type theorems are powerful tools in partial differential equations. Boundedness assumptions of solutions are often imposed in deriving such Liouville-type theorems. In this paper, we establish some Liouville-type theorems without the boundedness assumption of nonnegative solutions to certain classes of elliptic equations and systems. Using a rescaling technique and doubling lemma developed recently in Polá?ik et al. (2007) [20], we improve several Liouville-type theorems in higher order elliptic equations, some semilinear equations and elliptic systems. More specifically, we remove the boundedness assumption of the solutions which is required in the proofs of the corresponding Liouville-type theorems in the recent literature. Moreover, we also investigate the singularity and decay estimates of higher order elliptic equations. 相似文献
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Qiuyi Dai Yonggeng Gu Jiuyi Zhu 《Nonlinear Analysis: Theory, Methods & Applications》2011,74(18):7126-7136
This paper concerns a priori estimates and existence of solutions of generalized mean curvature equations with Dirichlet boundary value conditions in smooth domains. Using the blow-up method with the Liouville-type theorem of the p laplacian equation, we obtain a priori bounds and the estimates of interior gradient for all solutions. The existence of positive solutions is derived by the topological method. We also consider the non-existence of solutions by Pohozaev identities. 相似文献
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We investigate the quantitative unique continuation of solutions to higher order elliptic equations with singular coefficients. Quantitative unique continuation described by the vanishing order is a quantitative form of strong unique continuation property. We characterize the vanishing order of solutions for higher order elliptic equations in terms of the norms of coefficient functions in their respective Lebesgue spaces. New versions of quantitative Carleman estimates are established. 相似文献