Symmetry and asymmetry: the method of moving spheres
The first part of the paper deals with symmetry properties of positive solutions of the semilinear elliptic equation \( \Delta u+ (c/|x|^2)u + u ^{(n+2)/(n-2)}=0\) in \(\mathbb R^n\setminus \{0\}\). In partially answering a problem raised by L. Veron, the authors prove, among other results, the following theorem: if \(c \geq (n-2)^2/4\), then the problem has no smooth solutions. If \(c>0\), then any \({\mathcal C}^\infty(\mathbb R^n\setminus \{0\})\) solution must be radially symmetric about the origin and there is a constant \(C>0\) such that \( u(x) \leq C|x|^{-(n-2)/2}\). If \(c \leq (n-2)^2/4\), then the problem has infinitely many smooth radial solutions. If \(c \leq -(n-2)^2/4\), then the problem has non-radial solutions. The second part of the paper is concerned with some nonlinear elliptic equations on the unit sphere.
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