A remark on solutions to semilinear equations with Robin boundary conditions

Abstract

Symmetry properties of solutions to elliptic quasilinear equations have been widely studied in the context of Dirichlet boundary conditions. We show that, in the context of Robin boundary conditions, the symmetry property á la Gidas, Ni and Nirenberg does not hold in dimension $n\ge 2$, even for superharmonic functions, and we provide an explicit example.