On -invariant solutions of semilinear elliptic equations

Let u be a classical solution of Dirichlet boundary value problem − Δ u = f ( x , u ) in Ω, where Ω is a bounded and O ( k ) -invariant domain in R N ( 1 ⩽ k ⩽ N ) and the nonlinearity f is O ( k ) -invariant in the first variable and convex in the second variable. In this paper, we introduce sufficient conditions on u to ensure that u is also O ( k ) -invariant.