## Symmetry of positive solutions of $$\Delta u + u^ p = 0$$ in $$\mathbf R^ n$$.(English)Zbl 0844.35028

The Lane-Emden equation $$\Delta u+ u^p= 0$$ is considered in $$\mathbb{R}^n$$, $$n> 2$$. Conditions are established for a nontrivial nonnegative solution of this equation to be radially symmetric. Hölder type and Lipschitz type estimates and the Alexandroff-Serrin moving plane method are applied.

### MSC:

 35J60 Nonlinear elliptic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs

### Keywords:

moving plane method; radially symmetric solution
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