Symmetry for solutions of semilinear elliptic equations in \(\mathbb R^ N\) and related conjectures. (Q1848128)

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''.