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Symmetry for solutions of semilinear elliptic equations in \(\mathbb R^ N\) and related conjectures. (English) Zbl 1160.35401
This paper deals with symmetry properties of the solutions of semilinear elliptic equations in \(\mathbb R^N\) and is motivated in monotonicity and symmetry properties of solutions of reaction-convection-diffusion equations naturally arising in many different physical contexts. Here the author proves a stronger version of Gibbon’s conjecture, that is if the level set of \(u\) corresponding to the value of the nonstable equilibrium point is bounded with respect to one direction, then “\(u\) depends only on that direction”.

MSC:
35J60 Nonlinear elliptic equations
35A30 Geometric theory, characteristics, transformations in context of PDEs
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