Symmetry properties of solutions of semilinear elliptic equations in the plane
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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)=e au , or power–like growths of g at infinity. Our method is based upon a combination of the Kelvin transform and moving plane method.
Keywords or PhrasesElliptic equations Kelvin transform Asymptotic expansions
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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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