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Symmetry properties of solutions of semilinear elliptic equations in the plane. (English) Zbl 1130.35339

Summary: If \(g\) is a nondecreasing nonnegative continuous function we prove that any solution of \(-\Delta u + g(u) = 0\) 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.

MSC:

35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs
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