Symmetry of the solutions of semilinear elliptic equations

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. Nirenbe rg does not apply. When the equation is non-autonomous, we introduce a comp arison with a linens problem. When the nonlinearity is non-Lipschitz, we us e local moving hyperplanes, local inversion and unique continuation methods . The usual notion of symmetry is then replaced by a notion of local radial symmetry. (C) 1999 Academie des sciences/Editions scientifiques et medical es Elsevier SAS.