Asymptotic symmetry and local behavior of semilinear elliptic equations with critical Sobolev growth.

*(English)*Zbl 0702.35085In 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.

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 |

##### Keywords:

critical exponent; asymptotic symmetry; Yamabe problem; conformally flat metric; isolated singularity
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\textit{L. A. Caffarelli} et al., Commun. Pure Appl. Math. 42, No. 3, 271--297 (1989; Zbl 0702.35085)

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