Abstract
We show that positive solutions of a semilinear elliptic problem in the Sobolev critical exponent with Newmann conditions, related to conformal deformation of metrics in , are asymptotically symmetric in a neighborhood of the origin. As a consequence, we prove for a related problem of conformal deformation of metrics in that if a solution satisfies a Kazdan‐Warner‐type identity, then the conformal metric can be realized as a smooth metric on .
Highlights
In recent years there has been a huge interest in studying properties of the positive singular solutions u of the scalar curvature equation
We observe that the interest for studying singular solutions of (1.1) comes from the study of asymptotic behavior of positive solutions of the following problem in Rn, n ≥ 3:
As a consequence of the characterization obtained in the last theorem, we are able to establish a result about the nonexistence of singular solutions for an elliptic problem with Dirichlet and Newmann conditions on the boundary (Theorem 1.2) and a result which is related to the asymptotic symmetry in a neighborhood of the origin (Theorem 1.3)
Summary
Where k(x) is a smooth positive function and B1 is the unit ball of Rn, with n ≥ 3. For any solution of (1.1) in a neighborhood of the origin, where c is a positive constant They established in [4] that the asymptotic symmetry for solutions of (1.1) follows from (1.7). As a consequence of the characterization obtained in the last theorem, we are able to establish a result about the nonexistence of singular solutions for an elliptic problem with Dirichlet and Newmann conditions on the boundary (Theorem 1.2) and a result which is related to the asymptotic symmetry in a neighborhood of the origin (Theorem 1.3).
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