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Symmetry properties of solutions of semilinear elliptic equations in the plane

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Abstract.

If g is a nondecreasing nonnegative continuous function we prove that any solution of −Δu+g(u)=0 in a half plane which blows-up locally on the boundary, in a fairly general way, depends only on the normal variable. We extend this result to problems in the complement of a disk. Our main application concerns the exponential nonlinearity g(u)=eau, or power–like growths of g at infinity. Our method is based upon a combination of the Kelvin transform and moving plane method.

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Acknowledgments.

This research has been conducted while the first author was visiting Université de Tours in the framework of the European Project “Nonlinear Parabolic Partial Differential Equations Describing Front Propagation and Other Singular Phenomena”, RTN contract: HPRN-CT-2002-00274.

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Mathematics Subject Classification (1991): 35J60

Revised version: 30 June 2004

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Porretta, A., Véron, L. Symmetry properties of solutions of semilinear elliptic equations in the plane. manuscripta math. 115, 239–258 (2004). https://doi.org/10.1007/s00229-004-0498-1

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  • DOI: https://doi.org/10.1007/s00229-004-0498-1

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