## Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth.(English)Zbl 0702.35085

In connection with the Yamabe problem the importance of the equation (*) $$-\Delta u=u^{(n+2)/(n-2)}$$ for $$u\geq 0$$ has become apparent. In fact, a positive solution u of (*) gives rise to a conformally flat metric g given by $$g_{ij}=u^{4/(n-2)}\delta_{ij}$$ which has constant scalar curvature.
The difficulty of the equation (*) arises from the fact that (*) is the Euler equation of $$\int | \nabla u|^ 2+c(n)\int | u|^{2n/(n-2)}.$$ Here, the growth exponent $$2n/(n-2)$$ is the limit case of the Sobolev embedding $$H^{1,2}\subset L^{2n/(n-2)}$$ which is no longer compact.
In this paper, the authors investigate solutions u of (*) having an isolated singularity at the origin. Then by a “measure theoretic” variation of the Alexandrov reflection technique as developed by B. Gidas, W. M. Ni and L. Nirenberg [Commun. Math. Phys. 68, 209-243 (1979; Zbl 0425.35020)] they show that u is asymptotically radially symmetric at the origin. In fact, the singular radially symmetric function occuring in this way can be exactly identified as the solution of a certain ordinary differential equation. This result is a consequence of a more general theorem where the right hand side $$u^{(n+2)/(n-2)}$$ is replaced by a function g satisfying certain growth conditions.
Reviewer: M.Grüter

### MSC:

 35J60 Nonlinear elliptic equations 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 35C20 Asymptotic expansions of solutions to PDEs

Zbl 0425.35020
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### References:

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