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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