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We show that axially symmetric solutions on\mathbb{S}^4to a constantQ-curvature type equation (it may also be called fourth order mean field equation) must be constant, provided that the parameter\alphain front of the Paneitz operator belongs to the interval[\frac{473 + \sqrt{209329}}{1800}\approx 0.517, 1). This is in contrast to the case\alpha=1, where there exists a family of solutions, known as standard bubbles. The phenomenon resembles the Gaussian curvature equation on\mathbb{S}^2. As a consequence, we prove an improved Beckner's inequality on\mathbb{S}^4for axially symmetric functions with their centers of mass at the origin. Furthermore, we show uniqueness of axially symmetric solutions when\alpha=1/5by exploiting Pohozaev-type identities, and prove the existence of a non-constant axially symmetric solution for\alpha \in (1/5, 1/2)via a bifurcation method.
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Abstract We study the following mean field equation on a flat torus $$T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $$ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $$u$$ provided that $$\rho \leq 8\pi $$. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $$\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.more » « less
