On O(k)-invariant solutions of semilinear elliptic equations

Let u be a classical solution of Dirichlet boundary value problem −Δu=f(x,u) in Ω, where Ω is a bounded and O(k)-invariant domain in RN (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.