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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

Citations:

Zbl 0425.35020
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References:

[1] Aviles, Comm. Math. Physics 108 pp 177– (1987)
[2] Fowler, Quart, J. Math. 2 pp 259– (1937)
[3] Guzman, Miguel de, Differentiation of Integrals on \(\mathbb{R}\)n, Springer Lecture Notes in Math. 481, 1975. · Zbl 0327.26010
[4] Gidas, Comm. Math. Phys. 68 pp 209– (1979)
[5] , and , Symmetry of positive solutions of nonlinear equations in \(\mathbb{R}\)n, Math. Analysis and Applications, Part A, pp. 369–402, Advances in Math. Supp. Stud. 79, Academic Press, New York–London, 1981.
[6] Gidas, Comm. Pure Appl. Math. 34 pp 525– (1981)
[7] and , Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin, Heidelberg, New York, 1977. · doi:10.1007/978-3-642-96379-7
[8] Schoen, J. Diff. Geometry 20 pp 479– (1984)
[9] Schoen, Comm. Pure Appl. Math. 41 pp 317– (1988)
[10] Schoen, Inventiones Math.
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