caa a22 4500 445843640 CHVBK 20180317145352.0 cr unu---uuuuu 170323e20111101xx s 000 0 eng 10.1007/s00526-011-0398-7 doi (NATIONALLICENCE)springer-10.1007/s00526-011-0398-7 Symmetry and regularity of extremals of an integral equation related to the Hardy-Sobolev inequality [Elektronische Daten] [Guozhen Lu, Jiuyi Zhu] Let α be a real number satisfying 0<α<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-Sobolev 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. Springer-Verlag, 2011 Lu Guozhen Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut Zhu Jiuyi Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut Calculus of Variations and Partial Differential Equations Springer-Verlag 42/3-4(2011-11-01), 563-577 0944-2669 42:3-4<563 2011 42 526 https://doi.org/10.1007/s00526-011-0398-7 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00526-011-0398-7 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lu Guozhen Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut NATIONALLICENCE 700 1- Zhu Jiuyi Department of Mathematics, Wayne State University, 48202, Detroit, MI, USA aut NATIONALLICENCE 773 0- Calculus of Variations and Partial Differential Equations Springer-Verlag 42/3-4(2011-11-01), 563-577 0944-2669 42:3-4<563 2011 42 526 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer