caa a22 4500 44583546X CHVBK 20180317145329.0 cr unu---uuuuu 170323e20110801xx s 000 0 eng 10.1007/s00209-010-0690-9 doi (NATIONALLICENCE)springer-10.1007/s00209-010-0690-9 Topological degree for solutions of fourth order mean field equations [Elektronische Daten] [Changshou Lin, Juncheng Wei, Liping Wang] We consider the following fourth order mean field equation with Navier boundary condition $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\,\,{\rm in}\, \Omega,{\quad}u = \Delta u = 0\,\,{\rm on}\,\partial \Omega,\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(*)$$ where h is a C 2,β positive function, Ω is a bounded and smooth domain in $${\mathbb{R}^4}$$ . We prove that for $${\rho \in (32m\sigma_3, 32(m + 1)\sigma_3)}$$ the degree-counting formula for (*) is given by $$d(\rho)=\left\{\begin{array}{ll}\frac{1}{m!} (-\chi (\Omega) +1) \cdot\cdot \cdot (-\chi(\Omega)+m) & {\rm for}\, m >0 ,\\ 1 & {\rm for}\, m=0\end{array}\right.$$ where χ(Ω) is the Euler characteristic of Ω. Similar result is also proved for the corresponding Dirichlet problem $$\Delta^2 u = \rho \frac{h(x) e^{u}}{\int_\Omega h e^{u}}\quad{\rm in}\,\Omega, \quad u = \nabla u = 0 \quad {\rm on}\,\,\partial \Omega.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad(**)$$ Springer-Verlag, 2010 Lin Changshou Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan aut Wei Juncheng Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong aut Wang Liping Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong aut Mathematische Zeitschrift Springer-Verlag 268/3-4(2011-08-01), 675-705 0025-5874 268:3-4<675 2011 268 209 https://doi.org/10.1007/s00209-010-0690-9 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1007/s00209-010-0690-9 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Lin Changshou Department of Mathematics, Taida Institute of Mathematical Sciences, National Taiwan University, Taipei, Taiwan aut NATIONALLICENCE 700 1- Wei Juncheng Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong aut NATIONALLICENCE 700 1- Wang Liping Department of Mathematics, Chinese University of Hong Kong, Shatin, Hong Kong aut NATIONALLICENCE 773 0- Mathematische Zeitschrift Springer-Verlag 268/3-4(2011-08-01), 675-705 0025-5874 268:3-4<675 2011 268 209 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer