Symmetry of the solutions of semilinear elliptic equations | Symétrie des solutions d'équations semi-linéaires elliptiques

In this note, we present symmetry and monotonicity results corresponding to two cases where the classical theorem by B. Gidas, W.-M. Ni and L. Nirenberg does not apply. When the equation is non-autonomous, we introduce a comparison with a linear problem. When the nonlinearity is non Lipschitz, we use local moving hyperplanes, local inversion and unique continuation methods. The usual notion of symmetry is then replaced by a notion of local radial symmetry.