### Remark on some conformally invariant integral equations: the method of moving spheres

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In this paper we mainly introduce a min-max procedure to prove the existence of positive solutions for certain semilinear elliptic equations in R.

A classical result of A. D. Alexandrov states that a connected compact smooth $n$-dimensional manifold without boundary, embedded in ${\mathbb{R}}^{n+1}$, and such that its mean curvature is constant, is a sphere. Here we study the problem of symmetry of $M$ in a hyperplane ${X}_{n+1}=\text{const}$ in case $M$ satisfies: for any two points $({X}^{\text{'}},{X}_{n+1})$, $({X}^{\text{'}},{\widehat{X}}_{n+1})$ on $M$, with ${X}_{n+1}>{\widehat{X}}_{n+1}$, the mean curvature at the first is not greater than that at the second. Symmetry need not always hold, but in this paper, we establish it under some additional condition for $n=1$. Some variations...

We propose to study a fully nonlinear version of the Yamabe problem on manifolds with boundary. The boundary condition for the conformal metric is the mean curvature. We establish some Liouville type theorems and Harnack type inequalities.

We prove some results on the existence and compactness of solutions of a fractional Nirenberg problem. The crucial ingredients of our proofs are the understanding of the blow up profiles and a Liouville theorem.

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