## Radial symmetry of overdetermined boundary-value problems in exterior domains.(English)Zbl 0911.35008

Summary: We extend a classical result by J. Serrin to exterior domains $$\mathbb{R}^n\setminus\overline\Omega$$, where $$\Omega$$ is a bounded domain. We prove, under some hypotheses on $$f$$, that if there exists a solution of $$\Delta u+f(u)=0$$ in $$\mathbb{R}^n\setminus\overline\Omega$$ satisfying the overdetermined boundary conditions that $$\partial u/\partial v$$ and $$u$$ are constant on $$\partial\Omega$$, and such that $$0\leq u\leq u|_{\partial\Omega}$$, then the domain $$\Omega$$ is a ball. Under different assumptions on $$f$$, this result has been obtained by W. Reichel. The main result here covers new cases like $$f(u)=u^p$$ with $$n/(n-2)<p\leq(n+2)/(n-2)$$. When $$\Omega$$ is a ball, almost the same proof allows us to derive the symmetry of positive bounded solutions satisfying only the Dirichlet condition that $$u$$ is constant on $$\partial\Omega$$. Our method relies on Kelvin transforms, various forms of the maximum principle and the device of moving planes up to a critical position.

### MSC:

 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35J65 Nonlinear boundary value problems for linear elliptic equations 35N05 Overdetermined systems of PDEs with constant coefficients
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